![Preface
The motion of fluids has intrigued scientists since antiquity and we may say that the
field of mathematical fluid dynamics originated more than two centuries ago. In 1755
Euler [2] gave a mathematical formulation of the principle of conservation of mass in
terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear
and angular momentum by PDEs. Material symmetry and frame invariance were used by
Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of
internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint-
Venant [11] and Stokes [12].
In the 19th century no sharp distinction was drawn between mathematicians and
physicists as we sometime see in more recent times. The formulation of the equations
of fluid motion could be considered as either mathematics or physics. The first work
in fluid dynamics that has a "modern" mathematical flavor may have been done by
Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous
problem. Riemann recognized the mathematical nature of the entropy. This notion led
him to his duality method for solving the non-characteristic Cauchy problem for linear
hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his
contemporaries. What we now call the Cauchy problem for a PDE and the search for its
solution did not have the significance that it is accorded nowadays. Even Poincar6 did not
raise that kind of question in his Th~orie des tourbillons.
For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly
accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the
Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28.
Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of
the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much
as Riemann, Leray developed new mathematical tools which proved to have independent
interest: e.g., weak solutions (that we now call Leray's solutions in this context) and
topological degree (a joint work with Schauder [4]).
Since the 1930s, the interest that mathematicians devote to fluid dynamics has
unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von
Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a
Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on
the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic
displays such a variety of models and questions that thousands of scientists, among them
many mathematicians, focus their research on fluid dynamics.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-1-320.jpg)
![Preface vii
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
A.-L. Cauchy, Bull. Soc. Philomathique (1823), 9-13; Exercices de Math6matiques 2 (1827), 42-56, 108-
111; 4 (1829), 293-319.
L. Euler, M6m. Acad. Sci. Berlin 11 (1755), 274-315; 15 (1759), 210-240.
J. Leray, J. Math. Pures Appl. 12 (1933), 1-82; 13 (1934), 331-418; Acta Math. 63 (1934), 193-248.
J. Leray and J. Schauder, Ann. Sci. Ecole Norm. Sup. (3) 51 (1934), 45-78.
L. Lichtenstein, Math. Z. 23 (1925), 89-154; 26 (1926), 196-323; 28 (1928), 387-415; 32 (1930), 608-725.
P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vols. 1, 2, Oxford Univ. Press (1998).
J. Nash, Bull. Soc. Math. France 90 (1962), 487-497.
C.L.M.H. Navier, M6m. Acad. Sci. Inst. France 6 (1822), 375-394.
S.D. Poisson, J. Ecole Polytechnique 13 (1831), 1-174.
B. Riemann, G6tt. Abh. Math. C1.8 (1860), 43-65.
B. de Saint-Venant, C. R. Acad. Sci. Paris 17 (1843).
G.G. Stokes, Trans. Cambridge Philos. Soc. 8 (1849), 207-319.
W. Wolibner, Math. Z. 37 (1933), 698-726.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-3-320.jpg)
![Preface
The motion of fluids has intrigued scientists since antiquity and we may say that the
field of mathematical fluid dynamics originated more than two centuries ago. In 1755
Euler [2] gave a mathematical formulation of the principle of conservation of mass in
terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear
and angular momentum by PDEs. Material symmetry and frame invariance were used by
Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of
internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint-
Venant [11] and Stokes [12].
In the 19th century no sharp distinction was drawn between mathematicians and
physicists as we sometime see in more recent times. The formulation of the equations
of fluid motion could be considered as either mathematics or physics. The first work
in fluid dynamics that has a "modern" mathematical flavor may have been done by
Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous
problem. Riemann recognized the mathematical nature of the entropy. This notion led
him to his duality method for solving the non-characteristic Cauchy problem for linear
hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his
contemporaries. What we now call the Cauchy problem for a PDE and the search for its
solution did not have the significance that it is accorded nowadays. Even Poincar6 did not
raise that kind of question in his Th~orie des tourbillons.
For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly
accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the
Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28.
Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of
the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much
as Riemann, Leray developed new mathematical tools which proved to have independent
interest: e.g., weak solutions (that we now call Leray's solutions in this context) and
topological degree (a joint work with Schauder [4]).
Since the 1930s, the interest that mathematicians devote to fluid dynamics has
unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von
Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a
Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on
the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic
displays such a variety of models and questions that thousands of scientists, among them
many mathematicians, focus their research on fluid dynamics.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-6-320.jpg)
![The Boltzmann equation andfluid dynamics 3
1. Introduction
We say that a gas flow is rarefied when the so-called mean free-path of the gas molecules,
i.e., the average distance covered by a molecule between to subsequent collisions, is not
completely negligible with respect to a typical geometric length (the radius of curvature
of the nose of a flying vehicle, the radius of a pipe, etc.). The most remarkable feature of
rarefied flows is that the Navier-Stokes equations do not apply. One must then resort to the
concepts of kinetic theory of gases and the Navier-Stokes equations must be replaced by
the Boltzmann equation [43].
Thus the Boltzmann equation became a practical tool for the aerospace engineers, when
they started to remark that flight in the upper atmosphere must face the problem of a
decrease in the ambient density with increasing height. This density reduction would
alleviate the aerodynamic forces and heat fluxes that a flying vehicle would have to
withstand. However, for virtually all missions, the increase of altitude is accompanied
by an increase in speed; thus it is not uncommon for spacecraft to experience its peak
heating at considerable altitudes, such as, e.g., 70 km. When the density of a gas decreases,
there is, of course, a reduction of the number of molecules in a given volume and, what is
more important, an increase in the distance between two subsequent collisions of a given
molecule, till one may well question the validity of the Euler and Navier-Stokes equations,
which are usually introduced on the basis of a continuum model which does not take into
account the molecular nature of a gas. It is to be remarked that, as we shall see, the use of
those equations can also be based on the kinetic theory of gases, which justifies them as
asymptotically useful models when the mean free path is negligible.
In the area of environmental problems, the Boltzmann equation is also required.
Understanding and controlling the formation, motion, reactions and evolution of particles
of varying composition and shapes, ranging from a diameter of the order of 0.001 gm to
50 gm, as well as their space-time distribution under gradients of concentration, pressure,
temperature and the action of radiation, has grown in importance, because of the increasing
awareness of the local and global problems related to the emission of particles from electric
power plants, chemical plants, vehicles as well as of the role played by small particles in the
formation of fog and clouds, in the release of radioactivity from nuclear reactor accidents,
and in the problems arising from the exhaust streams of aerosol reactors, such as those
used to produce optical fibers, catalysts, ceramics, silicon chips and carbon whiskers.
One cubic centimeter of atmospheric air at ground level contains approximately 2.5 x
1019 molecules. About a thousand of them may be charged (ions). A typical molecular
diameter is 3 x 10-10 m (3 x 10-4 gm) and the average distance between the molecules
is about ten times as much. The mean free path is of the order of 10-8 m, or 10-2 l,tm.
In addition to molecules and ions one cubic centimeter of air also contains a significant
number of particles varying in size, as indicated above. In relatively clean air, the number
of these particles can be 105 or more, including pollen, bacteria, dust, and industrial
emissions. They can be both beneficial and detrimental, and arise from a number of natural
sources as well as from the activities of all living organisms, especially humans. The
particles can have complex chemical compositions and shapes, and may even be toxic
or radioactive.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-7-320.jpg)
![4 C. Cercignani
A suspension of particles in a gas is known as an aerosol. Atmospheric aerosols are of
global interest and have important impact on our lives. Aerosols are also of great interest
in numerous scientific and engineering applications [175].
A third area of application of rarefied gas dynamics has emerged in the last quarter of
the twentieth century. Small size machines, called micromachines, are being designed and
built. Their typical sizes range from a few microns to a few millimiters. Rarefied flow
phenomena that are more or less laboratory curiosities in machines of more usual size can
form the basis of important systems in the micromechanical domain.
A further area of interest occurs in the vacuum industry. Although this area existed
for a long time, the expense of the early computations with kinetic theory precluded
applications of numerical methods. The latter could develop only in the context of the
aerospace industry, because the big budgets required till recently were available only there.
The basic parameter measuring the degree of rarefaction of a gas is the Knudsen number
(Kn), the ratio between the mean free path )~ and another typical length. Of course,
one can consider several Knudsen numbers, based on different characteristic lengths,
exactly as one does for the Reynolds number. Thus, in the flow past a body, there are
two important macroscopic lengths: the local radius of curvature and the thickness of the
viscous boundary layer 8, and one can consider Knudsen numbers based on either length.
Usually the second one (Kn~ = )~/8), gives the most severe restriction to the use of Navier-
Stokes equations in aerospace applications.
When Kn is larger than (say) 0.01, the presence of a thin layer near the wall, of thickness
of the order )~ (Knudsen layer), influences the viscous profile in a significant way.
This and other effects are of interest in both high altitude flight and aerosol science; in
particular they are all met by a shuttle when returning to Earth. Another phenomenon of
importance is the formation of shock waves, which are not discontinuity surfaces, but thin
layers (the thickness is zero only if the Euler model is adopted).
When the mean free path increases, one witnesses a thickening of the shock waves,
whose thickness is of the order of 6)~. The bow shock in front of a body merges with
the viscous boundary layer; that is why this regime is sometimes called the merged layer
regime by aerodynamicists. We shall use the other frequently used name of transition
regime.
When Kn is large (few collisions), phenomena related to gas-surface interaction play an
important role. They enter the theory in the form of boundary conditions for the Boltzmann
equation. One distinguishes between free-molecule and nearly free-molecule regimes. In
the first case the molecular collisions are completely negligible, while in the second they
can be treated as a perturbation.
2. The basic molecular model
According to kinetic theory, a gas in normal conditions (no chemical reactions, no
ionization phenomena, etc.) is formed of elastic molecules rushing hither and thither at
high speed, colliding and rebounding according to the laws of elementary mechanics.
Monatomic molecules of a gas are frequently assumed to be hard, elastic, and perfectly
smooth spheres. One can also consider these molecules to be centers of forces that move](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-8-320.jpg)
![The Boltzmann equation andfluid dynamics 5
according to the laws of classical mechanics. More complex models are needed to describe
polyatomic molecules.
The rules generating the dynamics of many spheres are easy to describe: thus, e.g., if
no body forces, such as gravity, are assumed to act on the molecules, each of them will
move in a straight line unless it happens to strike another molecule or a solid wall. The
phenomena associated with this dynamics are not so simple, especially when the number
of spheres is large. It turns out that this complication is always present when dealing with
a gas, because the number of molecules usually considered is extremely large: there are
about 2.7.1019 in a cubic centimeter of a gas at atmospheric pressure and a temperature
of 0~
Given the vast number of particles to be considered, it would of course be a hopeless task
to attempt to describe the state of the gas by specifying the so-called microscopic state, i.e.,
the position and velocity of every individual sphere; we must have recourse to statistics. A
description of this kind is made possible because in practice all that our typical observations
can detect are changes in the macroscopic state of the gas, described by quantities such as
density, bulk velocity, temperature, stresses, heat-flow, which are related to some suitable
averages of quantities depending on the microscopic state.
3. The Boltzmann equation
The exact dynamics of N particles is a useful conceptual tool, but cannot in any way
be used in practical calculations because it requires a huge number of real variables (of
the order of 102~ The basic tool is the one-particle probability density, or distribution
function P(1)(x, ~, t). The latter is a function of seven variables, i.e., the components of
the two vectors x and ~ and time t.
Let us consider the meaning of p(1) (x, ~, t); it gives the probability density of finding
one fixed particle (say, the one labelled by 1) at a certain point (x, ~) of the six-dimensional
reduced phase space associated with the position and velocity of that molecule. In order
to simplify the treatment, we shall for the moment assume that the molecules are hard
spheres, whose center has position x. When the molecules collide, momentum and kinetic
energy must be conserved; thus the velocities after the impact, ~'l and ~'2, are related to
those before the impact, ~l and ~2, by
~' In (~1 '
1 =~1--n 9 --~2)]
'= In (~1 '
~2 ~2 "+- n 9 -- ~2)]
(3.1)
where n is the unit vector along ~1 -- ~t Note that the relative velocity
1"
V --" ~1 -- ~2 (3.2)
satisfies
V'= V - 2n(n. V), (3.3)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-9-320.jpg)
'nld~zdn"
(3.6)
The calculation of the gain term G is exactly the same as the one for L, except for the fact
that we have to integrate over the hemisphere B +, defined by V. n > 0 (the particles are
moving away one from the other after the collision). Thus we have:
G--(N-1)O-2 fR3 f13+P(2)(Xl'Xl+o-n'~l'~2't)l(~2-~l)'nld~2dn"
(3.7)
We can now insert in Equation (3.4) the information that the probability density p(2) is
continuous at a collision; in other words, although the velocities of the particles undergo
the discontinuous change described by Equations (3.1), we can write"
p(2) (x1, ~ 1, x2, ~2, t) -- p(2)(x1, ~1 -- n(n. V), x2, ~2 -+-n(n. V), t)
if Ix1 - x21 -- o-. (3.8)
For brevity, we write (in agreement with Equations (3.1))"
~fl -- ~1 -- n(n. V), ~2 -- ~2 -+- n(n. V). (3.9)
Inserting Equation (3.8) in Equation (3.5) we thus obtain:
G--(N- 1)o 2 JR3 ft~+ p(2)(Xl, x, + o-n, ~'1,~2, t)[(~2 - ~l)" n[ d~2 dn
(3.10)
which is a frequently used form. Sometimes n is changed into -n in order to have the same
integration range as in L; the only change (in addition to the change in the range) is in the
second argument of p(2), which becomes Xl - o-n.
At this point we are ready to understand Boltzmann's argument. N is a very large number
and o- (expressed in common units, such as, e.g., centimeters) is very small; to fix the ideas,
let us consider a box whose volume is 1 cm 3 at room temperature and atmospheric pressure.
Then N ~ 1020 and o- ~ 10-8 cm. Then (N - 1)o-2 ~ No 2 ~ 104 cm2 -- 1 m2 is a sizable
quantity, while we can neglect the difference between Xl and Xl -4-o-n. This means that the
equation to be written can be rigorously valid only in the so called Boltzmann-Grad limit,
when N --+ cxz,o- --+ 0 with No 2 finite.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-11-320.jpg)
.nld~2dn ,
G = Na 2 p(1) (Xl, ~:1,t) p(1) (Xl, ~2' 2 1
3 -
(3.13)
L--Nff2 fR3 fl~- P(1)(Xl'l~l't)p(1)(Xl'~2't)[(~2-~l)'ll]d~2dll" (3.14)
By inserting these expressions in Equation (3.6) we can write the Boltzmann equation in
the following form:
Op(1) Op(1)
+ ~.
at OXl
s 't)
= Na 2 [P(1)(Xl,l~'l,t)p(1)(Xl,l~2 '
3 -
-- P(1)(Xl,l;1,t)p(1)(Xl,l~2, t)]](l;2 - ~1)" n] d~2 dn. (3.15)
We remark that the expressions for ~'1 and ~2t given in Equations (3.1) are by no means the
only possible ones. In fact we might use a different unit vector to, directed as V', instead
of n. Then Equations (3.1) is replaced by:
, - 1
, 1
~J2- ~- ~l~Jl- ~J21to,
(3.16)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-12-320.jpg)
![The Boltzmann equation andfluid dynamics 9
where ~ = 89
(~j1+ ~2) is the velocity of the center of mass. The relative velocity V satisfies
v' : ~lVI. (3.17)
The recipes (3.13) and (3.14) can be justified at various levels of rigor [36,113,39,47].
We finally mention that we have for simplicity neglected any body force acting on the
molecules, such as gravity. It is not hard to take them into account; if the force per unit
mass acting on the molecules is denoted by X, then a term X. 0p(1)/0~1 must be added to
the left-hand side of Equation (3.8).
4. Molecules different from hard spheres
In the previous section we have discussed the Boltzmann equation when the molecules
are assumed to be identical hard spheres. There are several possible generalizations of this
molecular model, the most obvious being the case of molecules which are identical point
masses interacting with a central force, a good general model for monatomic gases. If
the range of the force extends to infinity, there is a complication due to the fact that two
molecules are always interacting and the analysis in terms of "collisions" is no longer
possible. If, however, the gas is sufficiently dilute, we can take into account that the
molecular interaction is negligible for distances larger than a certain a (the "molecular
diameter") and assume that when two molecules are at a distance smaller than a, then no
other molecule is interacting with them and the binary collision analysis considered in the
previous section can be applied. The only difference arises in the factor o-21(~2 - ~1)" nl
which turns out to be replaced by a function of V = I~2 -- ~ I I and the angle 0 between n
and V ([39,35,42]). Thus the Boltzmann equation for monatomic molecules takes on the
following form:
Op(l) Op(1)
+ ~.
Ot Oxl
=NfR3 it3_[V(1)(x1 , ~tl,/) e(1)(x1 , ~j~, t)
- e(1)(Xl,~,t)p(1)(Xl,~2, t)]n(o, 1~2 -- ~1) d~2d0 d~, (4.1)
where e is the other angle which, together with 0, identifies the unit vector n. The function
B(O,V) depends, of course, on the specific law of interaction between the molecules. In
the case of hard spheres, of course
B(O,1~2 - ~ll) = cos0 sin01~2 - ~ll- (4.2)
In spite of the fact that the force is cut at a finite range cr when writing the Boltzmann
equation, infinite range forces are frequently used. This has the disadvantage of making
the integral in Equation (4.1) rather hard to handle; in fact, one cannot split it into the
difference of two terms (the loss and the gain), because each of them would be a divergent](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-13-320.jpg)
![10 C. Cercignani
integral. This disadvantage is compensated in the case of power law forces, because one
can separate the dependence on 0 from the dependence upon V. In fact, one can show [39,
35] that, if the intermolecular force varies as the n-th inverse power of the distance, then
B(O, 1/~2-/~11) = ffi(O)1/~2- I~11
(n-5)/(n-1), (4.3)
where fl(O) is a non-elementary function of 0 (in the simplest cases it can be expressed
by means of elliptic functions). In particular, for n = 5 one has the so-called Maxwell
molecules, for which the dependence on V disappears.
Sometimes the artifice of cutting the grazing collisions corresponding to small values
of l0 - zr/2l is used (angle cutoff). In this case one has both the advantage of being able
to split the collision term and of preserving a relation of the form (4.3) for power-law
potentials.
Since solving of the Boltzmann equation with actual cross sections is complicated, in
many numerical simulations use is made of the so-called variable hard sphere model in
which the diameter of the spheres is an inverse power law function of the relative speed V
(see [43]).
Another important case is when we deal with a mixture rather than with a single gas.
In this case we have n unknowns, if n is the number of the species, and n Boltzmann
equations; in each of them there are n collision terms to describe the collision of a molecule
with other molecules of all the possible species [43,39].
If the gas is polyatomic, then the gas molecules have other degrees of freedom in addition
to the translation ones. This in principle requires using quantum mechanics, but one can
devise useful and accurate models in the classical scheme as well. Frequently the internal
energy Ei is the only additional variable that is needed; in which case one can think of
the gas as of a mixture of species [43,39], each differing from the other because of the
value of Ei. If the latter variable is discrete we obtain a strict analogy with a mixture;
otherwise we have a continuum of species. We remark that in both cases, kinetic energy is
not preserved by collisions, because internal energy also enters into the balance; this means
that a molecule changes its "species" when colliding. This is the simplest example of a
"reacting collision", which may be generalized to actual chemical species when chemical
reactions occur. The subject of mixture and polyatomic gases will be taken up again in
Section 16.
5. Collision invariants
Before embarking in a discussion of the properties of the solutions of the Boltzmann
equation we remark that the unknown of the latter is not always chosen to be a probability
density as we have done so far; it may be multiplied by a suitable factor and transformed
into an (expected) number density or an (expected) mass density (in phase space, of
course). The only thing that changes is the factor in front of Equations (3.1) which is no
longer N. In order to avoid any commitment to a special choice of that factor we replace
NB(O, V) by B(O, V) and the unknown P by another letter, f (which is also the most
commonly used letter to denote the one-particle distribution function, no matter what its](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-14-320.jpg)
![The Boltzmann equation and fluid dynamics 11
normalization is). In addition, we replace the current velocity variable ~1 simply by ~ and
2 by ~,. Thus we rewrite Equation (4.1) in the following form:
Of Of ~ fB (f'ft*-ff*)B(O V)d~ dOde,
o-; +~~x= ~ _ ' *
(5.1)
where V - [~ - ~, 1.The velocity arguments ~i and ~, in f' and f,~ are of course given by
Equations (3.1) (or (3.15)) with the suitable modification.
The right-hand side of Equation (5.1) contains a quadratic expression Q(f, f), given
by:
Q(f' f)= fR3fs2 (f' f',- ff,)B(O, V)dl~,dOde. (5.2)
This expression is called the collision integral or, simply, the collision term and the
quadratic operator Q goes under the name of collision operator. In this section we study
some elementary properties of Q. Actually it turns out that it is more convenient to study
the slightly more general bilinear expression associated with Q(f, f), i.e.:
1~ fs (f'g1*+g' f*- fg*-gf*)13(O' V)dl;*dOde"
Q(f' g) = -2 3 2
(5.3)
It is clear that when g = f, Equation (5.3) reduces to Equation (5.2) and
Q(f, g) -- Q(g, f). (5.4)
Our first aim is to indicate a basic property of the eightfold integral:
fR Q(f, g)r
3
where f, g and ~b are functions such that the indicated integrals exist and the order of
integration does not matter. Simple manipulations (see [43,39,35]) give the following
result:
n O(f, g)~b(~) d/~
,, , ,
= _ (f g, + g f, - fg, - gf,)
8 3 3 -
x (4)+ 4), - dp'- r V) dl~,dl~dOde. (5.6)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-15-320.jpg)
![12 C. Cercignani
This relation expresses a basic property of the collision term, which is frequently used. In
particular, when g = f, Equation (5.6) reads
R3 Q(f' f)r
= (f f, - ff,)(dp + ~b, - ~b' - ~bl,)B(0, V) d~j d~jd0 de. (5.7)
3 3 - *
We now observe that the integral in Equation (5.6) is zero independent of the particular
functions f and g, if
r + r = r + r (5.8)
is valid almost everywhere in velocity space. Since the integral appearing in the left-hand
side of Equation (5.7) is the rate of change of the average value of the function 4~ due
to collisions, the functions satisfying Equation (5.8) are called "collision invariants". It
can be shown (see, e.g., [39]) that a continuous function 4~has the property expressed by
Equation (5.8) if and only if
~b(~) =a +b.~ +cl~l 2, (5.9)
where a and c are constant scalars and b a constant vector. The assumption of continuity
can be considerably relaxed [5,40,6]. The functions 7t0 = 1, (Tel, 7t2, ~P3)= ~, 7t4 = I~12
are usually called the elementary collision invariants; they span the five-dimensional
subspace of the collision invariants.
6. The Boltzmann inequality and the Maxwell distributions
In this section we investigate the existence of positive functions f which give a vanishing
collision integral:
= (f f, - f f,) 13(0, V) d~,, dOde - O.
3 -
(6.1)
In order to solve this equation, we prove a preliminary result which plays an important
role in the theory of the Boltzmann equation: if f is a nonnegative function such that
log f Q(f, f) is integrable and the manipulations of the previous section hold when
q~= log f, then the Boltzmann inequality:
fR log f Q(f, f) d~j ~<0 (6.2)
holds; further, the equality sign applies if, and only if, log f is a collision invariant, or,
equivalently:
f = exp(a + b. ~j+ cl~j12). (6.3)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-16-320.jpg)
![The Boltzmann equation and fluid dynamics 15
Equation (7.6) shows that the momentum flow is described by the components of a
symmetric tensor of second order. The defining integral can be re-expressed in terms of
c and v. We have [43,39,35]:
mij =- pvi l)j -+-Pij, (7.9)
where:
pij -- fR3 CiCj f d~j (i, j -- 1, 2, 3) (7.10)
plays the role of the stress tensor (because the microscopic momentum flow associated
with it is equivalent to forces distributed on the boundary of any region of gas, according
to the macroscopic description).
Similarly one has [43,39,35]:
1
w -- plvl 2 + pe, (7.11)
where e is the internal energy per unit mass (associated with random motions) defined by:
1 s le12fd~j. (7.12)
pe-- -~ 3
and:
(') k
,,
ri -- pvi _v_2 -+-e + vj Pij -+-qi
j=l
(i = 1, 2, 3), (7.13)
where qi are the components of the heat-flow vector:
if
qi -- -~ ci ICl2f d~j. (7.14)
The decomposition in Equation (7.13) shows that the microscopic energy flow is a sum of
a macroscopic flow of energy (both kinetic and internal), of the work (per unit area und
unit time) done by stresses, and of the heat-flow.
In order to complete the connection, as a simple mathematical consequence of the
Boltzmann equation, one can derive five differential relations satisfied by the macroscopic
quantities introduced above; these relations describe the balance of mass, momentum and
energy and have the same form as in continuum mechanics. To this end let us consider the
Boltzmann equation
Of Of
+/~. - Q (f, f) (7 15)
at "](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-19-320.jpg)
![18 C. Cercignani
homogeneous solutions. In this case the various quantities do not depend on x and
Equation (8.2) reduces to
07-t
= S ~<0. (8.7)
Ot
This implies the so-called H-theorem (for the space homogeneous case): 7-/is a decreasing
quantity, unless f is a Maxwellian (in which case the time derivative of 7-/ is zero).
Remember now that in this case the densities p, pv and pe are constant in time; we can
thus build a Maxwellian M which has, at any time, the same p, v and e as any solution
f corresponding to given initial data. Since 7-/decreases unless f is a Maxwellian (i.e.,
f = M), it is tempting to conclude that f tends to M when t ~ cx~. This conclusion
is, however, unwarranted from a purely mathematical viewpoint, without a more detailed
consideration of the source term S in Equation (8.7), for which [47] should be consulted.
If the state of the gas is not space-homogeneous, the situation becomes more complicated.
In this case it is convenient to introduce the quantity
H- fs2 7-/dx, (8.8)
where s is the space domain occupied by the gas (assumed here to be time-independent).
Then Equation (8.2) implies
dt ~< J. n &r, (8.9)
~2
where n is the inward normal and dcr the surface element on OS2. Clearly, several situations
may arise (see [43] and [47] for a detailed discussion).
It should be clear that H has the properties of entropy (except for the sign); this
identification is strengthened when we evaluate H in an equilibrium state (see [43,39,
35]) because it turns out to coincide with the expression of a perfect gas according to
equilibrium thermodynamics, apart from a factor -R. A further check of this identification
is given by an inequality satisfied by the right-hand side of Equation (8.9) when the gas is
able to exchange energy with a solid wall bounding S2 (see Section 11 and [43,39,47]).
9. Model equations
When trying to solve the Boltzmann equation for practical problems, one of the major
shortcomings is the complicated structure of the collision term, Equation (4.2). When one
is not interested in fine details, it is possible to obtain reasonable results by replacing the
collision integral by a so-called collision model, a simpler expression J (f) which retains
only the qualitative and average properties of the collision term Q(f, f). The equation for
the distribution function is then called a kinetic model or a model equation.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-22-320.jpg)
![The Boltzmann equation andfluid dynamics 19
The most widely known collision model is usually called the Bhatnagar, Gross and
Krook (BGK) model, although Welander proposed it independently at about the same time
as the above mentioned authors [14,173]. It reads as follows:
,l(f) -- v[~(l~) - f (l~)], (9.1)
where the collision frequency v is independent of ~ (but depends on the density p and the
temperature T) and q~ denotes the local Maxwellian, i.e., the (unique) Maxwellian having
the same density, bulk velocity and temperature as f:
-- p(2yr RT) -3/2exp[- I/~- vI2/(2RT)]. (9.2)
Here p, v, T are chosen is such a way that for any collision invariant ~pwe have
~3 r d~ -- JR3 r d~. (9.3)
It is easily checked that, thanks to Equation (9.3):
(a) f and q~ have the same density, bulk velocity and temperature;
(b) J (f) satisfies conservation of mass, momentum and energy; i.e., for any collision
invariant:
s ~(/~)J(f) d~ -- O; (9.4)
(c) J (f) satisfies the Boltzmann inequality
fRlog f J(f) d/~ ~<0
3
(9.5)
the equality sign holding if and only if, f is a Maxwellian.
It should be remarked that the nonlinearity of the BGK collision model, Equation (9.1),
is much worse than the nonlinearity in Q(f, f); in fact the latter is simply quadratic in
f, while the former contains f in both the numerator and denominator of an exponential,
because v and T are functionals of f, defined by Equations (6.2) and (6.27).
The main advantage in the use of the BGK model is that for any given problem one
can deduce integral equations for p, v, T, which can be solved with moderate effort on a
computer. Another advantage of the BGK model is offered by its linearized form (see [43,
113,35]).
The BGK model has the same basic properties as the Boltzmann collision integral, but
has some shortcomings. Some of them can be avoided by suitable modifications, at the
expense, however, of the simplicity of the model. A first modification can be introduced in
order to allow the collision frequency v to depend on the molecular velocity, more precisely
on the magnitude of the random velocity c (defined by Equation (6.5)), while requiting that
Equation (9.4) still holds. All the basic properties, including Equation (9.5), are retained,
but the density, velocity and temperature appearing in q~ are not the local ones of the gas,](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-23-320.jpg)
![20 C. Cercignani
but some fictitious local parameters related to five functionals of f different from p, v, T;
this follows from the fact that Equation (9.3) must now be replaced by
fR3 v(Icl)~r(~J)cP(~J)d~J- fR3 v(Icl)!#(~j)f(~j) d~j. (9.6)
A different kind of correction to the BGK model is obtained when a complete agreement
with the compressible Navier-Stokes equations is required for large values of the collision
frequency. In fact the BGK model has only one parameter (at a fixed space point and time
instant), i.e., the collision frequency v; the latter can be adjusted to give a correct value for
either the viscosity/z or the heat conductivity x, but not for both. This is shown by the fact
that the Prandtl number Pr = lZ/CpX (where Cp is the specific heat at constant pressure)
turns out [39,35] to be unity for the BGK model, while it is about to 2/3 for a monatomic
gas (according to both experimental data and the Boltzmann equation). In order to have a
correct value for the Prandtl number, one is led [87,62] to replacing the local Maxwellian
in Equation (9.1) by
rP(I~) = p(rc)-3/2(detA)I/e exp(-(~j - v). [A(/~- v)]), (9.7)
where A is the inverse of the matrix
A-1 = (2RT/Pr)I- 2(1 - Pr)p/(p Pr), (9.8)
where Iis the identity and p the stress matrix. If we let Pr -- 1, we recover the BGK model.
Only recently [3] this model (called ellipsoidal statistical (ES) model) has been shown
to possess the property expressed by Equation (9.5). Hence the H-theorem holds for the
ES model.
Other models with different choices of q5 have been proposed [151,35] but they are not
so interesting, except for linearized problems (see [43,39,35]).
Another model is the integro-differential model proposed by Lebowitz, Frisch and
Helfand [114], which is similar to the Fokker-Planck equation used in the theory of
Brownian motion. This model reads as follows:
3 [02f
J (f)= D ~ -UfT~2-~-~ - o,)y]],
RT O~k
(9.9)
where D is a function of the local density p and the local temperature T. If we take D
proportional to the pressure p = pRT, Equation (9.9) has the same kind of nonlinearity
(i.e., quadratic) as the true Boltzmann equation.
The idea of kinetic models can be naturally extended to mixtures and polyatomic gases
[151,127,81,43].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-24-320.jpg)
![22 C. Cercignani
Equations (10.5) and (10.6) indicate that the operator L is symmetric and non-positive
in the aforementioned Hilbert space.
11. Boundary conditions
The Boltzmann equation must be accompanied by boundary conditions, which describe
the interaction of the gas molecules with the solid walls. It is to this interaction that one
can trace the origin of the drag and lift exerted by the gas on the body and the heat transfer
between the gas and the solid boundary.
The study of gas-surface interaction may be regarded as a bridge between the kinetic
theory of gases and solid state physics and is an area of research by itself. The difficulties
of a theoretical investigation are due, mainly, to our lack of knowledge of the structure
of surface layers of solid bodies and hence of the effective interaction potential of the
gas molecules with the wall. When a molecule impinges upon a surface, it is adsorbed
and may form chemical bonds, dissociate, become ionized or displace surface molecules.
Its interaction with the solid surface depends on the surface finish, the cleanliness of the
surface, its temperature, etc. It may also vary with time because of outgassing from the
surface. Preliminary heating of a surface also promotes purification of the surface through
emission of adsorbed molecules. In general, adsorbed layers may be present; in this case,
the interaction of a given molecule with the surface may also depend on the distribution
of molecules impinging on a surface element. For a more detailed discussion the reader
should consult [39,110] and [41].
In general, a molecule striking a surface with a velocity ~ reemerges from it with a
velocity ~ which is strictly determined only if the path of the molecule within the wall
can be computed exactly. This computation is very hard, because it depends upon a great
number of details, such as the locations and velocities of all the molecules of the wall and
an accurate knowledge of the interaction potential. Hence it is more convenient to think in
terms of a probability density R(~ ~--+ ~; x, t; r) that a molecule striking the surface with
velocity between ~ and ~j~+ d~~at the point x and time t will re-emerge at practically
the same point with velocity between ~ and ~ + d~ after a time interval r (adsorption or
sitting time). If R is known, then we can easily write down the boundary condition for the
distribution function f (x, ~, t). To simplify the discussion, the surface will be assumed to
be at rest. A simple argument ([43,39,35]) then gives:
f(x,/~, t)I/~ .n[
= dr R(~j' --+ ~; x, t; r)f(x, ~', t - r)l~'' n[ d~'
'.n<O
(x ~ 0S'2,~. n > 0). (11.1)
The kernel R can be assumed to be independent of f under suitable conditions which
we shall not detail here [39,110,41]. If, in addition, the effective adsorption time is small
compared to any characteristic time of interest in the evolution of f, we can let r = 0 in](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-26-320.jpg)
![The Boltzmann equation andfluid dynamics 23
the argument of f appearing in the right-hand side of Equation (3.4); in this case the latter
becomes:
f (x, ~j, t)I~j" nl
= f~'.n<0e(~t ~/~; x, t)f(x,/~', t)l~'" nl d~j' (x 6 Y2, ~ 9n > 0), (11.2)
where
fO G
R(~j' --+ ~j; x,t)= drR(~j' --+ ~j; x,t; r). (11.3)
Equation (11.2) is, in particular, valid for steady problems.
Although the idea of a scattering kernel had appeared before, it is only at the end of
1960's that a systematic study of the properties of this kernel appears in the scientific
literature [35,110,41]. In particular, the following properties were pointed out [36,35,110,
41,34,108,48,109,37]:
(1) Non-negativeness, i.e., R cannot take negative values:
R(~j' --+ ~j; x, t; r) ~>0 (11.4)
and, as a consequence:
R(se' -+/~; x, t) ~>0. (11.5)
(2) Normalization, if permanent adsorption is excluded; i.e., R, as a probability density
for the totality of events, must integrate to unity:
R(/~' --+ ~; x,t" r)dse -- 1
dr '.n~>0 (11.6)
and, as a consequence:
f~ R (~' ~ se; x, t) d~ -- 1. (11.7)
'-n>/O
(3) Reciprocity; this is a subtler property that follows from the circumstance that the
microscopic dynamics is time reversible and the wall is assumed to be in a local equilibrium
state, not significantly disturbed by the impinging molecule. It reads as follows:
les'.nIMw(~')R(es' ~ se;x, t; r)= I~ .nlMw(~)R(-~ --~ -se'; x, t; r) (11.8)
and, as a consequence:
les'. nlMw(fj')R(es' ~/~; x,t) = I/~"nlMw(es)R(-es -+ -/~'; x,t). (11.9)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-27-320.jpg)
![24 C.Cercignani
Here Mw is a (non-drifting) Maxwellian distribution having the temperature of the wall,
which is uniquely identified apart from a factor.
We remark that the reciprocity and the normalization relations imply another property:
(3') Preservation of equilibrium, i.e., the Maxwellian Mw must satisfy the boundary
condition (11.1):
Mw(~)l~. nl = dr
'.n<O
R(~j' --+/~;x, t; r)Mw(/~')l~j'' nl d~j' (11.10)
equivalent to:
Mw(~)l~" nl = f~'.n<O R(8' ~ ~; x, t)Mw (/~')l/~'. nl dS'. (11.11)
In order to obtain Equation (11.10) it is sufficient to integrate Equation (11.8) with respect
to ~' and r, taking into account Equation (11.6) (with -~ and -~' in place of ~' and
~, respectively). We remark that one frequently assumes Equation (11.10) (or (11.11)),
without mentioning Equation (11.8) (or (11.9)); although this is enough for many purposes,
reciprocity is very important when constructing mathematical models, because it places a
strong restriction on the possible choices. A detailed discussion of the physical conditions
under which reciprocity holds has been given by B~irwinkel and Schippers [10].
The basic information on gas-surface interaction, which should be in principle obtained
from a detailed calculation based on a physical model, is summarized in a scattering kernel.
The further reduction to a small set of accommodation coefficients can be advocated for
practical purposes, provided this concept is firmly related to the scattering kernel (see [43,
39,47] for further details).
In view of the difficulty of computing the kernel R(~j' --~ ~) from a physical model
of the wall, frequently one constructs a mathematical model in the form of a kernel
R (~' ~ ~j) which satisfies the basic physical requirements expressed by Equations (11.5),
(11.7), (11.9) and is not otherwise restricted except by the condition of not being too
complicated.
One of the simplest kernels is
R(~j' ~ ~j) = ctMw(~)1~. nl + (1 - ct)3(~ - ~' + 2n(~' 9~)). (11.12)
This is the kernel corresponding to Maxwell's model [121], according to which a fraction
(1 -ct) of molecules undergoes a specular reflection, while the remaining fraction c~ is
diffused with the Maxwellian distribution of the wall Mw. This is the only model for the
scattering kernel that appeared in the literature before the late 1960's. We refer to original
papers and standard treatises for details on more recent models [110,41,34,108,48,109,37,
10,121,135,91,100,112,99,68,174,38,54,53,22,43,39], among which the most popular in
recent years has been the so-called Cercignani-Lampis (CL) model.
It is remarkable that, for any scattering kernel satisfying the three properties of
normalization, positivity and preservation of equilibrium, a simple inequality holds. The](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-28-320.jpg)
![The Boltzmann equation andfluid dynamics 25
latter was stated by Darrozbs and Guiraud [70] who also sketched a proof. More details
were given later [37,39]. It reads as follows
fin--- fR~'3 n flog f d~<~-(2RTw)-lfR ~
.3 nl~ 12fd~j (x ~ 0S'2). (11.13)
Equality holds if and only if f coincides with Mw (the wall Maxwellian) on 0S-2 (unless
the kernel in Equation (11.2) is a delta function). We remark that if the gas does not slip
upon the wall, the right-hand side of Equation (11.13) equals -qn/(RTw) where qn is the
heat-flow along the normal, according to its definition given in Section 5. If the gas slips on
the wall, then one must add the power of the stresses Pn "v to qn. In any case, however, the
right-hand side equals qn
(w), where q(nw)is the heat-flow in the solid at the interface, because
the normal energy flow must be continuous through the wall and stresses have vanishing
power in the solid, because the latter is at rest. If we identify the function H introduced in
Section 8 with -rl/R (where r/is the entropy of the gas), the inequality in Equation (11.13)
is exactly what one would expect from the Second Principle of thermodynamics.
12. The continuum limit
In this section we investigate the connection between the Navier-Stokes equations and
the Boltzmann equation by using a method, which originated with Hilbert [85] and
Enskog [74].
The discussion is made complicated by the various possible scalings. For example, if we
denote by (~,,7) the microscopic space and time variables (those entering in the Boltzmann
equation) and by (x, t) the macroscopic variables (those entering in the fluid dynamical
description), we can study scalings of the following kind
= s-ix, (12.1)
7 - s -~ t, (12.2)
where ot is an exponent between 1 and 2. For ot -- 1, this is called the compressible scaling.
If c~ > 1, we are looking at larger "microscopic" times. We now investigate the limiting
behavior of solutions of the Boltzmann equation in this limit.
Notice first that the compressible Euler equations,
8tp + div(pv) = O,
Ot(pvi) + div pvvi + -~peei - 0,
1 1 5
Ot[p(e + -~lvl2)] + div[pv(-~lvl2 + -~e)] - 0,
(12.3)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-29-320.jpg)
![The Boltzmann equation andfluid dynamics 27
From (12.8) and the perfect gas law p0 - p0T0, which we assume to hold at zeroth order,
it follows that poTo is constant as a function of the space variables. The first and third
equations in (12.9) now imply
2
Ot(poTo)+ div(p0T0v0) -- - ~-p0T0divv0. (12.10)
.5
As poTois only a function of t, say A(t), Equation (12.10) implies that
3AI
divv0--5 (t)/A(t). (12.11)
Under suitable assumptions, Equation (12.11) implies that A1(t) = 0 and then divv0 = 0.
This is the case, for example, if we are in a box with nonporous walls, because then the
normal component of v0 is 0 and we can use the divergence theorem. A similar argument
applies to the case of a box with periodicity boundary conditions; or if we are in the entire
space 9~3 and the difference between v0 and a constant vector decays fast enough at infinity.
Assuming that we have conditions which imply that div v0 = 0, we easily get from the
continuity equation that P0 will be independent of x if the initial value is, and the same
for To. But with this knowledge Equations (12.9) then actually entail that P0 and To are
constant.
Therefore, if the initial conditions are "well prepared" in the sense that v~(x, 0) is a
divergence-free vector field and p~ (x, 0), Te(x, 0) are constant, we expect as a first-order
approximation for p~, v~, T ~ the solution of the equations
div v0 - 0,
Otvo + (vo. Ox)VO- - ~
OxPl (12.12)
p0
which are the incompressible Euler equations. This limit, known as the "low velocity
limit", is well known at the macroscopic level. We refer to Majda's book [120] for
references and a detailed discussion. The variable r/-1 enters into the theory as the square
of the speed of sound. If this parameter is large compared to typical speeds of the fluid, then
the incompressible model is well suited to describe the time evolution, provided that the
initial velocity field is divergence-free and the initial density and temperature are constant.
The incompressible fluid limit was met in the last section in connection with the Stokes
paradox. In fact, it seems that this limit and the derivation of the steady incompressible
Navier-Stokes equations from the Boltzmann equation were first considered in connection
with the flow past a body at small values of the Mach number [32]. We remark that in the
steady case div v0 - 0 and P0 and To turn out to be constant without using the boundary
conditions.
Let us examine the kinetic picture as described by the Boltzmann equation. The above
discussion suggests that if ot e (1,2), then in the scaling (12.1)-(12.2) the solutions
of the Boltzmann equation will converge to a local Maxwellian distribution whose
parameters satisfy the incompressible Euler equations. This assertion can actually be
proved rigorously [71,9].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-31-320.jpg)
![28 C. Cercignani
For ct = 2 something special happens. Of course, the incompressible Euler equations are
invariant under the scaling (12.1)-(12.3); however, for c~= 2 the incompressible Navier-
Stokes equations,
Otv + (v. 0x)V= -Oxr + v A v,
0x. v = 0 (12.13)
(where r = p/p and v = Iz/p is the kinematic viscosity) are also invariant under the same
scaling. It is therefore of great interest to understand whether the Boltzmann dynamics
"chooses" in this limit the Euler or the Navier-Stokes evolution.
We can expect that the answer is Navier-Stokes. In other words, considering larger times
than those typical for Euler dynamics (e-zt instead of e-c~t, c~ < 2), dissipation becomes
nonnegligible. We can see an illustration of this behavior in the following example:
consider two parallel layers of fluid moving with velocities v and v + 6v. Suppose that
we want to decide whether there is any momentum transfer between these layers (which is
expected for the Navier-Stokes equations, but not for the Euler equations). The momentum
transfer can in principle be affected by the trend to thermalization typical of the Boltzmann
collision term, but a scaling argument shows that it is proportional to e~-Z6v, with the
consequence that it remains only relevant for c~= 2.
These considerations can be put on a rigorous basis, and, of course, the viscosity
coefficient can be computed in terms of kinetic expressions (see [43,39,35]).
The incompressible Navier-Stokes equations can be derived from the Boltzmann
equation if the time interval is such that smooth solutions of the continuum equations
exist. The tool which yields this result is a truncated Hilbert expansion and one gets local
convergence for the general situation [71] or global and uniform convergence if the data
are small in a suitable sense [9].
Let us consider now the cases in which compressibility is not negligible. If we scale
space and time in the same way, the scaled Boltzmann equation becomes:
Otf e + I~. Oxfe = _1Q(fe, fe). (12.14)
E
We will use the abbreviation Dt f := Otf + ~ 9Oxf. Of course, we expect that
eDtf e -+ 0 as e --+ O, (12.15)
and if
fe __+ fo, (12.16)
the limit f0 must satisfy
Q(fO, f0) = o. (12.17)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-32-320.jpg)
![TheBoltzmannequationandfluiddynamics 29
This implies, as we know, that f0 is a local Maxwellian distribution:
f~
p(x,t) ( I~-v(x,t)l 2)
(2zrRT(x, t))3/2 exp - 2RT(x, t) "
(12.18)
The fields (p, x, T), which characterize the behavior of the local Maxwellian distribution
M in space and time, are expected to evolve according to continuum equations which we
are going to derive. First, let us emphasize again that these fields are varying slowly on
the space-time scales which are typical for the gas described in terms of the Boltzmann
equation.
From the conservation laws (I. 12.36)
f g/uQ(f, f)d~j-O (c~=O ..... 4), (12.19)
we readily obtain, as we know,
f ~Po~(Ot
f + I~. Oxf) d/~ --0. (12.20)
This is a system of equations for the moments of f which is in general not closed. However,
if we assume f = M and use the identities (for M they are identities; for a general f they
are definitions given in (I.12.3-7), where e = 3RT)
-- f M ds
e, (12.21)1
, v--fMseds
e, (12.21)2
3 1 if 2M ,
w = -~pRT + ~plv] 2- ~ s
e ds
e (12.21)3
we readily obtain from (12.20) that
Otp + div(pv) = O, (12.22)
Ot(pvi) + div(fM~i) --O, (12.23)
(3 1)l(f )
Ot -~peZ+ ~p[vl 2 + ~div M~jI~I
2 --0. (12.24)
These equations are nothing but Equations (12.19)-(12.21), specialized to the case
of a Maxwellian distribution. This is of crucial importance if we want to write](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-33-320.jpg)
![The Boltzmann equation and fluid dynamics 31
have to take Nff 2 -- 1/e ~ oo. This, at first glance, seems contradictory, but there is really
no problem. The Boltzmann equation holds for a perfect gas, i.e., for a gas such that the
density parameter 6 - N~r3/V, where V is the volume containing N molecules, tends to
zero. The parameter
1 Nff 2
-- -- N1/3S 2/3 (12.31)
Kn V 2/3
may tend to zero, to cxz or remain finite in this limit. These are the three cases which occur
if we scale N as 3-m (m ~>0), for m < 2, m > 2 and m = 2 respectively. In the first case
the gas is in free-molecular flow and we can simply neglect the collision term (Knudsen
gas), in the second we are in the continuum regime which we are treating here, and we
cannot simply "omit" the "small" term, i.e., the left-hand side of the Boltzmann equation,
because the limit is singular. In the third case the two sides of the Boltzmann equation are
equally important (Boltzmann gas) and this is the case dealt with before for solutions close
to an absolute Maxwellian distribution.
In spite of the fact that we face a singular perturbation problem, Hilbert [85] proposed
an expansion in powers of e. In this way, however, we obtain a Maxwellian distribution
at the lowest order, with parameters satisfying the Euler equations and corrections to this
solution which are obtained by solving inhomogeneous linearized Euler equations [85,43,
39,35]. In order to avoid this and to investigate the relationship between the Boltzmann
equation and the compressible Navier-Stokes equations, Enskog introduced an expansion,
usually called the Chapman-Enskog expansion [74,39,35]. The idea behind this expansion
is that the functional dependence of f upon the local density, bulk velocity and internal
energy can be expanded into a power series. Although there are many formal similarities
with the Hilbert expansion, the procedure is rather different.
As remarked by the author [35,39], the Chapman-Enskog expansion seems to introduce
spurious solutions, especially if one looks for steady states. This is essentially due to the
fact that one really considers infinitely many time scales (of orders e, e2..... e n .... ).
The author [35,39] introduced only two time scales (of orders e and e2) and was able
to recover the compressible Navier-Stokes equations. In order to explain the idea, we
remark that the Navier-Stokes equations describe two kinds of processes, convection and
diffusion, which act on two different time scales. If we consider only the first scale we
obtain the compressible Euler equations; if we insist on the second one we can obtain
the Navier-Stokes equations only at the price of losing compressibility. If we want both
compressibility and diffusion, we have to keep both scales at the same time and think of f
as
f(x, g;, t) -- f (ex, ~,, et, e2t). (12.32)
This enables us to introduce two different time variables tl - et, t2 - e2t and a new space
variable Xl - ex such that f - f(xl, ~, tl, t2). The fluid dynamical variables are functions](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-35-320.jpg)
![32 c. Cercignani
of xl, tl, t2, and for both f and the fluid dynamical variables the time derivative is given
by
O Of 2 Of
m = e~ + e ~. (12.33)
Ot Otl Ot2
In particular, the Boltzmann equation can be rewritten as
Of 62 Of
+ + e!~"Oxf = Q(f, f)
Otl -~2
If we expand f formally in a power series in e, we find that at the lowest order f is
a Maxwellian distribution. The compatibility conditions at the first order give that the
time derivatives of the fluid dynamic variables with respect to tl is determined by the
Euler equations, but the derivatives with respect to t2 are determined only at the next
level and are given by the terms of the compressible Navier-Stokes equations describing
the effects of viscosity and heat conductivity. The two contributions are, of course, to be
added as specified by (12.33) in order to obtain the full time derivative and thus write the
compressible Navier-Stokes equations.
It is not among the aims of this article to describe the techniques applied to and the
results obtained from the computations of the transport coefficients, such as the viscosity
and heat conduction coefficients, for given molecular interaction. For this we refer to
standard treatises [64,86,75].
The results discussed in this section show that there is a qualitative agreement between
the Boltzmann equation and the Navier-Stokes equations for sufficiently low values of the
Knudsen number.
There are however flows where this agreement does not occur. They have been especially
studied by Sone [154]. New effects arise because the no-slip and no temperature jump
boundary condition do not hold. In addition to the thermal creep induced along a boundary
with a nonuniform temperature, discovered by Maxwell, two new kinds of flow are induced
over boundaries kept at uniform temperatures. They are related to the presences of thermal
stresses in the gas.
The first effect [154,153,138] is present even for small Mach numbers and small
temperature differences and follows from the fact that there are stresses related to the
second derivatives of the temperature (see Section 15). Although these stresses do not
change the Navier-Stokes equations, they change the boundary conditions; the gas slips on
the wall, and thus a movement occurs even if the wall is at rest. This effect is particularly
important in small systems, such as micromachines, since the temperature differences are
small but may have relatively large second derivatives; it is usually called the thermal stress
slip flow [154,153,138].
The second effect is nonlinear [101,155] and occurs when two isothermal surfaces do
not have constant distance (thus in any situation with large temperature gradients, in the
absence of particular symmetries). In fact, if we assume that in the Hilbert expansion
the velocity vanishes at the lowest order, i.e., the speed is of the order of the Knudsen
number, the terms of second order in the temperature show up in the momentum equation.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-36-320.jpg)
![The Boltzmann equation andfluid dynamics 33
These terms are associated with thermal stresses and are of the same importance as those
containing the pressure and the viscous stresses. A solution in which the gas does not move
can be obtained if and only if:
grad T A grad(I grad TIe) = 0. (12.34)
Since Igrad T I measures the distance between two nearby isothermal lines, if this quantity
has a gradient in the direction orthogonal to grad T, the distance between two neighboring
isothermal lines varies and we must expect that the gas moves.
These effects may occur even for sufficiently large values of the Knudsen number; they
cannot be described, however, in terms of the local temperature field. They rather depend
by the configuration of the system. They should not be confused with flows due to the
presence of a temperature gradient along the wall, such as the transpiration flow [139] and
the thermophoresis of aerosol particles [162].
Numerical examples of simulations of this kind of flow are discussed in [43].
13. Free-molecule and nearly free-molecule flows
After discussing the behavior of a gas in the continuum limit, in this section we consider
the opposite case in which the small parameter is the Knudsen number (or the inverse of
the mean free path).
By analogy with what we did in the previous section, we might be tempted to use a series
expansion of the form (12.2), albeit with a different meaning of the expansion parameter.
This, however, does not work in general, for a reason to be presently explained. The factor
multiplying the gradient of f in Equation (5.1) takes all possible values and hence also
values of order e; thus we should expect troubles from the molecules travelling with low
speeds, because then certain terms in the left-hand side can become smaller than the fight-
hand side, in spite of the small factor e. This is confirmed by actual calculations, especially
for steady problems.
Let us now consider the limiting case when the collisions can be completely neglected.
This, by itself, does not pose many problems.
The Boltzmann equation (in the absence of a body force) reduces to the simple form
Dtf = atf + I~ . Oxf =0. (13.1)
Since the molecular collisions are negligible, the gas-surface interaction discussed in
Section 11 plays a major role. This situation is typical for artificial satellites, since the
mean free path is 50 meters at 200 kilometers of altitude.
The general solution of Equation (13.1) is in terms of an arbitrary function of two vectors
g(., .):
f (x, l~, t) = g(x- I~t, ~,). (13.2)
In the steady case, Equation (13.1) reduces to
I~. Oxf = 0, (13.3)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-37-320.jpg)
![34 C. Cercignani
and the general solution becomes:
f (x, ~, t) = g(x A l~, l~). (13.4)
Frequently it is easier to work with the property that f is constant along the molecular
trajectories than with the explicit solutions given by Equations (13.3)-(13.4).
The easiest problem to deal with is the flow past a convex body. In this case, in fact,
the molecules arriving at the surface of the wall have an assigned distribution function f~,
usually a Maxwellian distribution with the density p~, bulk velocity v~, and temperature
Too, prevailing far away from the body, and the distribution function of the molecules
leaving the surface is given by the boundary conditions. The distribution function at any
other point P, if needed, is simply obtained by the following rule: if the straight line
through P having the direction of ~ intersects the body at a point Q and ~j points from
Q towards P, then the distribution function at P is the same as that at Q; otherwise it
equals f~.
Interest is usually confined to the total momentum and energy exchanged between the
molecules and the body, which, in turn, easily yield the drag and lift exerted by the gas on
the body and the heat transfer between the body and the gas.
In practice, the temperature of a body is determined by a balance of all forms of heat
transfer at the body surface. For an artificial satellite, a considerable part of heat is lost by
radiation and this process must be duly taken into account in the balance.
The results take a particularly simple form in the case of a large Mach number since
we can let the latter go to infinity in the various formulas. One must, however, be careful,
because the speed is multiplied by sin 0 in many terms and thus the aforementioned limit
is not uniform in 0. Thus the limiting formulas can be used, if and only if, the area where
S sin 0 ~< 1 is small.
The standard treatment is based on the definition of accommodation coefficients, but
calculations based on other models are available [39,50,49].
The case of nonconvex boundaries is, of course, more complicated and one must solve an
integral equation to obtain the distribution function at the boundary. If one assumes diffuse
reflection according to a Maxwellian, the integral equation simplifies in a considerable
way, because just the mass flow at the boundary must be computed [39].
In particular the latter equation can be used to study free-molecular flows in pipes
of arbitrary cross section with a typical diameter much smaller than the mean free path
(capillaries). If the cross section is circular the equation becomes particularly simple and
is known as Clausing's equation [39].
The perturbation of free-molecular flows is not trivial for steady problems because of
the abovementioned non-uniformity in the inverse Knudsen number. If one tries a na'fve
iteration, the singularity arising in the first iterate may cancel when integrating to obtain
moments (cancellation is easier, the higher is the dimensionality of the problem, because a
first-order pole is milder, if the dimension is higher). The singularity is always present and,
although it may be mild, it can build up a worse singularity when computing subsequent
steps. The difficulties are enhanced in unbounded domains where the subsequent terms
diverge at space infinity. The reason for the latter fact is that the ratio between the mean
free path )~ and the distance d of any given point from the body is a local Knudsen](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-38-320.jpg)
![The Boltzmann equation andfluid dynamics 35
number which tends to zero when d tends to infinity; hence collisions certainly arise in
an unbounded domain and tend to dominate at large distances. On this basis we are led to
expect that a continuum behavior takes place at infinity, even when the typical lengths
characterizing the size of the body are much smaller than the mean free path; this is
confirmed by the discussion of the Stokes paradox for the steady linearized Boltzmann
equation (see [43,39,35]).
Both difficulties are removed by the so-called collision iteration: the loss term is partly
considered to be unknown in the iteration, thus building an exponential term which controls
the singularity. The presence of the latter is still felt through the presence of logarithmic
terms in the (inverse) Knudsen number. In higher dimensions this is multiplied by a power
of (Kn)-1 which typically equals the number of space dimensions relevant for the problem
under consideration in a bounded domain. In particular the dependence upon coordinates
will show the same singularity (we can think of local Knudsen numbers based on the
distance from the nearest wall); as a consequence first derivatives will diverge at the
boundary in one dimension and the same will occur for second, or third derivatives, in two,
or three, space dimensions, respectively. In an external domain we have, in addition to the
low speed effects, the effect of particles coming from infinity, which actually dominates. In
particular in one dimension (half-space problems) the terms coming from iterations are of
the same order as the lowest order terms; actually for a half-space problem there is hardly
a Knudsen number (the local one is an exception). In two dimensions the corrections in the
moments are of order Kn-1 log Kn. In three dimensions a correction of order Kn-2 log Kn
is preceded by a correction of order Kn -1 .
Care must be exercised when applying the aforementioned results to a concrete
numerical evaluation, as mentioned above. In fact, for large but not extremely large
Knudsen numbers (say 10 ~< Kn ~< 100)logKn is a relatively small number, although
log Kn ---> oQ for Kn ---> oo. Hence terms of order log Kn/Kn, though mathematically
dominating over terms of order 1/Kn are of the same order as the latter for practical
purposes. As consequence, the two kinds of terms must be computed together if numerical
accuracy is desired for the aforementioned range of Knudsen numbers.
Related to this remark is the fact that any factor appearing in front of Kn in the argument
of the logarithm is meaningless unless the term of order Kn -I is also computed. This
is particularly important when the factor under consideration depends upon a parameter
which can take very large (or very small) values (typically a speed ratio). Thus Hamel
and Cooper [70,85] have shown that the first iterate of the integral iteration is incapable
of describing the correct dependence upon the speed ratio and have applied the method
of matched asymptotic expansions [81] to regions near a body and far from a body. In
particular, for the hypersonic flow of a gas of hard spheres past a two-dimensional strip,
they find for the drag coefficient
elog e]
CD = CDf.m. 1 + 2zr ' (13.5)
where the inverse Knudsen number e is based on the mean free path )~ = 7r3/2cr2nooS w
(or is the molecular diameter and S~, = S~(Tw/Too), whereas n~ and S~ are the number
density at infinity).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-39-320.jpg)
![The Boltzmann equation andfluid dynamics 37
little loss of generality, that the bulk velocity in the Maxwellian is zero and we shall denote
the unperturbed density and temperature by P0 and To.
Although the equation is now linear, and hence all the weapons of linear analysis
are available, it is far from easy to solve for a given boundary value problem, such as
Couette flow. Yet it is possible to gain an insight on the behavior of the general solution
of Equation (14.3) (see [43,39,35]). This insight gives the following picture for a slab
problem, provided the plates are sufficiently far apart (several mean free paths). There are
two Knudsen layers near the boundaries, where the behavior of the solution is strongly
dependent on the boundary conditions, and a central core (a few mean free paths away
form the plates), where the solution of the Navier-Stokes equations holds (with a slight
reminiscence of the boundary conditions). If the plates are close in terms of the mean free
path, then this picture does not apply because the core and the kinetic layers merge.
One can give evidence for the above statements just in the case of the linearized
Boltzmann equation, but there is a strong evidence that this qualitative picture applies to
nonlinear flows as well, with a major exception. In general, compressible flows develop
shock waves at large speeds and these do not appear in the linearized description. As
already remarked, these shocks are not surfaces of discontinuity as for an ideal fluid,
governed by the Euler equations, but layers of rapid change of the solution (on the scale
of the mean free path). One can obtain solutions for flows containing shocks from the
Navier-Stokes equations, but, since they change significantly on the scale of the mean free
path, they are inaccurate. Other regions where this picture is inaccurate are the zones of
high rarefaction, where nearly free-molecular conditions may prevail, even if the rest of
the flow is reasonably described in terms of Navier-Stokes equations, Knudsen layers and
shock layers.
The theory of Knudsen layers can be essentially described by the linearized Boltzmann
equation. The main result concerns the boundary conditions for the Navier-Stokes
equations. They turn out to be different from those of no-slip and no temperature jump.
In fact, the velocity slip turns out to be proportional to the normal gradient of tangential
velocity and the temperature jump to the normal gradient of temperature. When one can
use the Navier-Stokes equations but must use the slip and temperature-jump boundary
conditions, one talks of the slip regime; this typically occurs for Knudsen numbers
between 10-1 and 10-2.
Subtler phenomena may occur if the solutions depend on more than one space
coordinate. The most important change with respect to traditional continuum mechanics
is the presence of the term with the second derivatives of temperature in the expression
of the stress deviator and of the term with the second derivatives of bulk velocity in the
expression of the heat flow. These terms were already known to Maxwell [121]. In recent
times, their importance has been stressed by Kogan et al. [101] and by Sone et al. [155] (as
already mentioned in Section 12).
Even in fully three-dimensional problems the solution of the linearized Boltzmann
equation reduces to the sum of two terms, one of which, h8, is important just in the
Knudsen layers and the other, hA, is important far from the boundaries. The latter has a
stress deviator and a heat flow with constitutive equations different from those of Navier-
Stokes and Fourier. In spite of this, the bulk velocity, pressure, and temperature satisfy
the Navier-Stokes equations when steady problems are considered. In fact, when we take](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-41-320.jpg)
![38 C. Cercignani
the divergence of the heat flow vector a term proportional to the Laplacian of v vanishes,
thanks to the continuity equation, and thus just a term proportional to the temperature
gradient survives; then, taking the divergence of the stress, a term grad(AT) vanishes,
because of the energy equation. Yet, the new terms in the constitutive relations may produce
physical effects in the presence of boundary conditions different from those of no-slip and
no temperature jump. In fact, we must expect the velocity slip to be proportional to the
shear stress and the temperature jump to the heat flow.
15. Approximate methods for linearized problems
Linearization combined with the use of models lends itself to the use of analytical methods,
which turn out to be particularly useful for a preliminary analysis of certain problems.
Closed form solutions are not so frequent and are practically restricted to the case of half-
space problems [43,39,35]. The latter, in turn, are useful to investigate Knudsen layers and
compute the slip and temperature jump coefficients.
The use of BGK or similar models permits reducing the solution of Boltzmann's integro-
differential equation in phase-space to solving integral equation in ordinary space. This is
obtained because in the BGK model the distribution function f occurs only in two ways:
explicitly in a linear, simple way and implicitly through a few moments (appearing in
the local Maxwellian and the collision frequency). Then one can express f in terms of
these moments by integrating a linear, simple partial differential equation; then, using the
definitions of these moments and the expression of f one can obtain integral equations for
the same moments [43,39,35]. These equations can be solved numerically in a much easier
way than the Boltzmann equation. This is particularly true in the linearized case.
The integral equation approach lends itself to a variational solution. The main idea of
this method (for linearized problems) is the following. Suppose that we must solve the
equation:
12h=S, (15.1)
where h is the unknown,/2 a linear operator and S a source term. Assume that we can form
a bilinear expression ((g, h)) such that ((s h)) = ((g, Eh)),for any pair {g, h} in the set
where we look for a solution. Then the expression (functional):
J(/~) -- ((h, C/~)) - 2((S,/Tt)) (15.2)
has the property that if set/t - h + r/, then the terms of first degree in ~ disappear and
J(h) reduces to J(h) -+-((r/,/2r/)) if and only if h is a solution of Equation (15.1). In other
words if r/is regarded as small (an error), the functional in Equation (15.2) becomes small
of second order in the neighborhood of h, if and only if h is a solution of Equation (15.1).
Then we say that the solutions of the latter equation satisfy a variational principle, or
make the functional in Equation (15.2) stationary. Thus a way to look for solutions of
Equation (15.1) is to look for solutions which make the functional in Equation (15.2)
stationary (variational method).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-42-320.jpg)
![The Boltzmann equation andfluid dynamics 39
The method is particularly useful if we know that ((0,/20)) is non-negative (or non-
positive) because we can then characterize the solutions of Equation (15.1) as maxima or
minima of the functional (15.2). But, even if this is not the case, the property is useful. First
of all, it gives a non-arbitrary recipe to select among approximations to the solution in a
given class. Second, if we find that the functional J is related to some physical quantity,
we can compute this quantity with high accuracy, even if we have a poor approximation to
h. If the error 0 is of order 10%, then J will be in fact computed with an error of the order
of 1%, because the deviation of J (/t) from J (h) is of order 02, as we have seen.
The integral formulation of the BGK model lends itself to the application of the
variational method [58]. Thus in the case of Couette the functional is related to the stress
component p12 which is constant and gives the drag exerted by the gas on each plate. Thus
this quantity can be computed with high accuracy [58,43].
This method can be generalized to other problems and to the more complicated mod-
els [39]. It can also be used to obtain accurate finite ordinate schemes, by approximating
the unknowns by trial functions which are piecewise constant [44].
In the case of the steady linearized Boltzmann equation, Equation (14.3), a similar
method can be used. Let us indicate by Dh the differential part appearing in the left-hand
side (Dh = ~ 90xh for steady problems) and assume that there is a source term as well
(an example of a source occurs in linearized Poiseuille flow, see [43,39,35]) and write our
equation in the form:
Dh-Lh--S. (15.3)
If we try the simplest possible bilinear expression
f0L g(x, s
e)h(x,/j) dx d,~ (15.4)
((g,h)) -- 3
and we use it with Eh = Dh - Lh we cannot reproduce the symmetry property ((Eg, h)) =
((g, Eh)). It works for Lh but not for Dh. There is however a trick [33] which leads to the
desired result.
Let us introduce the parity operator in velocity space, P, such that P[h(~j)] -- h(-~).
Then we can think of replacing Equation (15.3) by
PDh- PLh = PS (15.5)
because this is completely equivalent to the original equation. In addition, because of the
central symmetry of the molecular interaction PLh = LPh and the fact that we had no
problems with L is not destroyed by the fact that we use P. On the other hand we have by
a partial integration:
((g, PDh)) --((PDg, h)) + ((g+, Ph-)) B -((Pg-,h+)) B. (15.6)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-43-320.jpg)
![42 C. Cercignani
and, finally, neglecting again terms of order higher than first:
h0 = ap+ - K ~p_. (15.19)
Then Equation (15.13) with S = 0 gives
J(h) = -((Ph-, ~P+))B+ ((Ph-, K~_)) B
= -((Ph-, g/+))B + ((Kh-, Pqt-))B
= -((Ph-, g/+))B + ((h+ -- ~+ + K~-, P~-))B
= -((Ph-, lp+))B + ((h+-, P~_))B
+((KTt---O+,P~-)) B. (15.20)
The last term is a known quantity, whereas the first and the second can be combined to give
unknown quantities of physical importance. In fact, if we take into account the expression
of gr (Equation (15.18)), we obtain:
-((Ph-, ~+))B + ((h+-' P~-))B
Vii)
To ~j" nh(x, ~j)M d~jda
f(
4- 2RTo 2 To
~j. nh(x, ~j)M d~jda
l(f f
RTo Pn "Vwdo" + q(n)
Tw - To da~. (15.21)
}
To
Here Pn is the normal stress vector and qn the normal component of the heat flow. The fact
that the mass flow vanishes at the wall has been taken into account.
Because of the linearity of the problem, it is possible and convenient, without loss of
generality, to consider separately the two cases vw = 0 and Tw = To. Then the two terms
in the expression above occur in two different problems. For some typical problems Vw
vanishes on just one part of the boundary, whereas it is a constant on the remaining part
of the latter; then the factor multiplying this constant is the drag on the corresponding part
of the boundary. Similarly one can consider the case in which the factor in front of qn
vanishes on just one part of the boundary, whereas it is a constant on the remaining part of
the latter, and relate the value of the functional to the heat transfer.
The two variational principles which have been discussed are related to each other [39].
The integral equation approach and the variational method have been applied with great
success to many linearized problems [43,39,35]. Among the most interesting results we
mention the calculation of the minimum in the flow rate for Poiseuille flow [30,61,31,
44], first experimentally discovered by Knudsen [98], and the calculation of the drag on
a sphere (at low Mach numbers) where the results agree very well [59] with the semi-
empirical formula derived by Millikan from his experimental data [126].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-46-320.jpg)
![The Boltzmann equation andfluid dynamics 43
16. Mixtures
As is well known, air at room pressure and temperature is a mixture, its main components
being two diatomic gases, nitrogen and oxygen. This immediately calls for an immediate
change in our basic equation, Equation (5.1), which is only suitable for a single monatomic
gas.
A remarkable feature of aerodynamics at the molecular level is that the evolution
equations change in a significant way, when dealing with polyatomic rather than
monatomic gases. This is not the case when the gas is treated as a continuum, where,
at least at room conditions, only a few changes in the equations occur, the most remarkable
being the change of the ratio of specific heats y.
If we consider a mixture of monatomic gases, the differences between the various species
occur in the values of the masses and in the law of interaction between molecules of
different species; in the simplest case, when the molecules are pictured as hard spheres,
the second difference is represented by unequal values of the molecular diameters. In the
mathematical treatment, a first difference will be in the fact that we shall need n distribution
functions f/ (i = 1, 2..... n) if there are n species. The notation becomes complicated,
but there is no new idea, except, of course, for the fact that we must derive a system of n
coupled Boltzmann equations for the n distribution functions. The arguments are exactly
the same, with obvious changes, and the result is
a f/ a f/ a f/
-~- ~" if- Xi"
at ~x a~
-- ~ fR3 ft~ (fi'f[* - f~f~*)Bik(n" V' lVl)dl;*dn'
k=l +
(16.1)
where ]~ik is computed from the interaction law between the i-th and k-th species, while
in the k-th term in the left-hand side, V = ~ - ~j. is the relative velocity of the molecule of
the i-th species (whose evolution we are following) with respect to a molecule of the k-th
species (against which the former is colliding). The arguments ~f and ~jf.are computed, as
before, from the laws of conservation of mass and energy in a collision with the following
result:
~j, ~j _ 2/zik n[(~j - ~j.). n],
mi
2lZik
~', -- ~j, + n[(~ - ~,). n],
mk
(16.2)
where lZik : mimk/(mi -+-m~) is the reduced mass [39].
To prepare some material for the description of polyatomic gases and chemical reactions,
we remark that Equation (16.1) can be rewritten as follows [39]:
oj5 ~ ~. oj5 oj5
~)---7- - -~X -p- xi " O~](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-47-320.jpg)
![44 C. CercQnani
n
-zf. 3 R 3
k=l x xR 3
(fi'f/~, - fi fk,)Wik(l~,/j, 1/~
t,/j~,) d/~, d/~'d/~,. (16.3)
Now ~j~,~,, ~, ~, are independent variables (i.e. they are not related by the conservation
laws) and
Wik(~, ~,[~'~,) ~- Sik(n" V, [Y[)8(mi~, + mk~, -- mi~' -- mk~t,)
x 8(mi[~,[ 2 + mk[~,[ 2 -- mi [~[ 2 -- mk[~t,[2), (16.4)
where n = (/~ -/~t)/[/~ - ~J'l and
Si~(n. V, IVl)= 13ik(n. V, IVl) (mi q- mk)3mimk. (16.5)
2n. V
Conservation of momentum and energy is now taken care of by the delta functions
appearing in Equation (16.4) [39].
With a slight modification, Equation (16.3) can be extended to the case of a mixture
in which a collision can transform the two colliding molecules of species j, 1 into two
molecules of different species k, i (a very particular kind of chemical reaction). In this
case the relations between the velocities before and after the encounter are different from
the ones used so far, but we may still write a set of equations for the n species:
aj~ aj~ afi
O---t--[- ~" -~X -'[-X i" 0
3 R 3
k,l,j--1 • •
(f/fj, - fi fk,)Wi~ (~, l~,l~z, l~Z,)d~, d~Zdl~,, (16.6)
where Wi~ gives the probability density that a transition from velocities ~', ~jt,to velocities
~, ~, takes place in a collision which transforms two molecules of species l, j, respectively,
into two molecules of species i, k, respectively. It is clear how the previous model is
included into the new one when the species change does not occur.
The idea of kinetic models analogous to the BGK model can be naturally extended to
mixtures and polyatomic gases [151,127,81 ]. A typical collision term of the BGK type will
read
n
Ji(fr) -" Jij(fr) - Z 1)ij[(I)ij(~) - fi(~)],
j--1 j=l
(16.7)
where Vij are the collision frequencies and ~ij is a Maxwellian distribution to be
determined by suitable conditions that generalize Equation (9.4).
There are some important changes concerning the collision invariants and the definition
of macroscopic functions in the case of mixtures. First of all, the collision invariants in](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-48-320.jpg)
![46 C. Cercignani
The stress tensor for the i-th species is given by:
p(i) fR
jk -- mi CjCk fi dl~
3
(i = 1..... n; j,k= 1,2,3); (16.15)
and the stress tensor for the mixture is the sum of the various stresses:
~_(i)
Pjk -- Pjk
i=1
(j, k = 1, 2, 3). (16.16)
It is to be remarked that, though these definitions are the most common and natural, they
are not used by all the authors. One might, e.g., define a peculiar velocity for each species
and use it to define the partial stresses. Then it is no longer true that the stress tensor for
the mixture is the sum of the partial stresses.
Similarly the thermal energy per unit mass (associated with random motions) is defined
for each species by:
n(i)e(i)- 2 3 Icl2fi d~ (i = 1..... n) (16.17)
and for the mixture by:
pe = ~p(i)e(i)
i=1
(16.18)
A similar procedure can be applied to the heat flow.
The pressure is, as usual, 1/3 of the trace of the stress matrix and is related to the
temperature by p -- nkB T. Please remark that there is not a constant R such that p = p R T.
17. Polyatomic gases
A possible picture of a molecule of a polyatomic gas, suggested by quantum mechanics, is
as follows [172]. The molecule is a mechanical system, which differs from a point mass by
having a sequence of internal states, which can be identified by a label, assuming integral
values. In the simplest cases these states differ from each other because the molecule
has, besides kinetic energy, an internal energy taking different values Ei in each of the
different states. A collision between two molecules, besides changing the velocities, can
also change the internal states of the molecules and, as a consequence, the internal energy
enters in the energy balance. From the viewpoint of writing evolution equations for the
statistical behavior of the system, it is convenient to think of a single polyatomic gas as
a mixture of different monatomic gases. Each of these gases is formed by the molecules
corresponding to a given internal energy, and a collision changing the internal state of
at least one molecule is considered as a reactive collision of the kind considered above,](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-50-320.jpg)
![The Boltzmann equation andfluid dynamics 47
Wi/~(~J,/~,I~J',~J',) giving the probability density of a collision transforming two molecules
with internal states l, j respectively, and velocities ~t ~jt respectively, into molecules with
internal states i, k, respectively, and velocities ~j, ~,, respectively.
This model is amply sufficient to discuss aerodynamic applications. We want to mention,
however, that it requires nondegenerate levels of internal energy, if there are, e.g., strong
magnetic fields which can act on the internal variables such as (typically) the spin of the
molecules. In that case, if the molecule has spin s, the distribution function f becomes
a square matrix of order 2s + 1 and the kinetic equation reflects the fact that matrices in
general do not commute and, as remarked by Waldmann [170,171] and Snider [152], the
collision term contains not simply the cross-section but the scattering amplitude, which
may not commute with f.
It is appropriate now to enquire why we started talking about quantum rather than
classical mechanics. The main reason is not related to practice, but rather to history.
Classical models of polyatomic molecules are regarded with suspicion since 1887 when
Lorentz found a mistake in the proof of the H theorem of Boltzmann [23] for general
polyatomic molecules. The question arises from the fact that when one proves the H
theorem for a monatomic gas one usually does not explicitly underline (because it is
irrelevant in that case) that the velocities ~jt and ~t, are not the velocities into which a
collision transforms the velocities ~j and ~,, but the velocities which are transformed by a
collision into the latter ones; this is conceptually very important, but the lack of a detailed
discussion does not lead to any inconvenience because the expressions for ~t, and ~t are
invariant with respect to a change of sign of the unit vector n, which permits an equivalence
between velocity pairs that are carried into the pair ~,, ~ and those which originate from
the latter pair, as a consequence of a collision. The remarkable circumstance which we
have just recalled is related to the particular symmetry of a collision described by a central
force, which allows to associate to a collision [~j,, ~] -+ [~jt,,~t] another collision, the so-
called "inverse collision" [~t,, ~jt] __+ [~,, ~], which differs from the former just because
of the transformation of the unit vector n into -n. When polyatomic molecules are dealt
with, the states before and after a collision require more than just the velocities of the
mass centers to be described (the angular velocity, e.g., if the molecule is pictured as
a solid body). Let us symbolically denote by [A, B] the state of the pair of molecules.
Then there is no guarantee that one can correlate an "inverse collision" [At, B t] ~ [A, B],
differing from the previous one just because of the change of n into -n with the collision
[A, B] ~ [At, Bt]. Now in the original proof of the H theorem for polyatomic molecules
proposed by Boltzmann [23], the assumption was implicitly made that there is always
such collision. Lorentz remarked [119,24,26] that this is not true in general. Boltzmann
recognized his blunder and proposed another proof based on the so-called "closed cycles
of collisions" [119,24,26,166]; the initial state [A,B] is reached not through a single
collision but through a sequence of collisions. This proof, although called unobjectionable
by Lorentz and Boltzmann [26], never satisfied anybody [167,171 ].
For a while the matter was forgotten till a quantum mechanical proof showed that the
required property followed from the unitarity of the S matrix [171]. A satisfactory proof
of the inequality required to prove the H theorem for a purely classical, but completely
general, model was given only in 1981 [51].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-51-320.jpg)
![48 c. Cercignani
For aerodynamic applications all these aspects are not so relevant and, in fact, the main
problem is to find a sufficiently handy model for practical calculations.
Lordi and Mates [118] studied the "two centers of repulsion" model and found that a
rather complicated numerical solution was required for a given set of impact parameters.
The lack of closed form expressions makes the model impractical for applications, where
the numerical solution describing the collision should be repeated many millions of times.
Curtiss and Muckenfuss [132] developed the collision mechanics of the so-called sphero-
cylinder model, consisting of a smooth elastic cylinder with two hemispherical ends.
Whether or not two molecules collide depends on more than one parameter; in addition,
there are several "chattering" collisions in a single collision event. The loaded-sphere
model had been already developed by Jeans [94] in 1904 and was subsequently developed
by Dahler and Sather [69] and Sandier and Dahler [146]. Although it is spherical in
geometry, the molecules rotate about the center of mass, which does not coincide with the
center of the sphere, with the consequence that it has essentially the same disadvantages as
the sphero-cylinder model.
The only exact model which is amenable to explicit calculations is the perfectly rough
sphere model, first suggested by Bryan [28] in 1894. The name is due to the fact that the
relative velocity at the point of contact of the two molecules is reversed by the collision.
This model has some obvious disadvantages. First, a glancing collision may result in a
large deflection; second, all collisions can produce a large interchange of rotational and
translational energy, with the consequence that the relaxation time for rotational energy is
unrealistically short; third, the number of internal degrees of freedom is three, rather than
two, which makes the model inappropriate for a description of the main components of
air, which are diatomic gases. One can disregard the first disadvantage and put a remedy to
the second by assuming that a fraction of collisions follow the smooth-sphere rather that
rough-sphere dynamics; but there is obviously no escape from the third difficulty.
In practical calculations, one has learned, since long time, that one must compromise
between the faithful adherence to a microscopic model and the computational time required
to solve a concrete problem. This was true in the early days of rarefied gas dynamics
(and may still be true nowadays when one tries to find approximate closed form solutions
or spare computer time) even for monatomic gases, as we discussed in Section 9 (in
connection with the BGK model) and shall discuss later (in connection with Direct
Simulation Monte Carlo). As a matter of fact, when trying to solve the Boltzmann equation,
one of the major shortcomings is the complicated structure of the collision term; if to this
problem, present even in the simplest case, one adds the complication of the presence
of internal degrees of freedom, any practical problem becomes intractable, unless one is
ready to accept the aforementioned compromise. Fortunately, when one is not interested
in fine details, it is possible to obtain reasonable results by replacing the collision integral
by a phenomenological collision model, e.g., a simpler expression which retains only the
qualitative and average properties of the true collision term. As computers become more
and more powerful, the amount of phenomenological simplification diminishes and the
calculations may more closely mimic the microscopic models.
For polyatomic gases, the basic new fact with respect to the monatomic ones is that
the total energy is redistributed between translational and internal degrees of freedom at
each collision. Those collisions for which this redistribution is negligible are called elastic,](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-52-320.jpg)
![TheBoltzmann equationandfluid dynamics 49
while the others are called inelastic. The simplest approach would be to calculate the effect
of collisions as a linear combination of totally elastic and completely inelastic collisions,
the second contribution being described by a model analogous to the BGK model which
was described in Section 9.
There are, of course, problems related to the molecule spins and their alignment; they
are particularly important when we put the molecules in a magnetic field and peculiar
phenomena, which go under the general name of Senftleben-Beenakker effects, arise.
There is an entire book devoted to this topic [122] and we shall not deal with these
problems.
Let us now consider in more detail the case of a continuous internal-energy variable. In
this case, it is convenient to take the unit vector n in the center-of-mass system and use
the internal energy Ei and Ei, of the colliding molecules. As usual, the values before a
collision will be denoted by a prime. Equation (1.2) is replaced by
/ fo
Q(f, f)= d~, Ei 2 dEi, (E~)~ dE~
3 2
f E-E~ t t t
x (E;,) n~__.._~2
dEi,(f f~_ ff,)
Jo
• n 9n~; E i E(~,~ Ei, Ei,). (17.1)
Here E -- m[tj[2/4 + Ei -k- Ei, is the total energy in the center-of-mass system which is
conserved in a collision. The kernel B satisfies the reciprocity relation
IVIB(E'n n' ' E(
, 9 ;E i, ~,~ Ei, Ei,)
9 rE! )
--[Vt[/3(E; n. n~ Ei, Ei, ~ E i t, (17.2)
Here we follow a paper by Kug~er [111] and look for a one-parameter family of models,
assuming that the scattering is isotropic in the center-of-mass system. The second crucial
assumption will be that the redistribution of energy among the various degrees of freedom
only depends upon the ratios of the various energies to the total energy E, Ei -- Ei/E,
etc. This assumption is valid for collisions of rigid elastic bodies and can be considered as
a good approximation for steep repulsive potentials9 It is then possible to write 13 in the
following form:
IVIB(E n n' ' '
; 9 ;E i,Ei,~ Ei, Ei,)
iv, i2
IVl
O'tot(E) 0(el ,
- - 4roE n ' ei* ~ Ei , ei,; "g). (17.3)
The denominator on the fight takes care of normalization. Then the function 0 satisfies the
following relations:
1 n-2 f01-6 n-2 ~ !
S--~--de E, 2 ds,0(s , ei, ~ ei, ei,; "g)-~- 1, (17.4)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-53-320.jpg)
![50 C. Cercignani
' ')0(e~, ' .r)
(1 -- 6 i -- 6i, 6i, ~ 8i, 6i,,
! ! .15).
= (1 - si - si,)O(si, si, ~ s i, si,, (17.5)
The dependence of O'tot on E makes it possible to adjust the model to the correct
dependence of the viscosity on temperature. The parameter 15is chosen in such a way as to
represent the degree of inelasticity of the collisions, r - 0 corresponds to elastic collisions"
! o - ~
0(8~, Ei, ~ Si, Ei,, 0) = 6
n-2 n-2
l
2 6, 2 = 6(8i -- e~)~(E~ -- 8i, ). (17.6)
15-- cx~ corresponds to maximally inelastic collisions:
n22 --~-~-~F(n + 2)
' 9cx~)= 6 6, (1 -- 8i -- 8,i)
0(8~, 8i, ~ 8i, 8i,, (F(n)) 2 9 (17.7)
A mixture of the two extreme cases gives the model first proposed by Borgnakke and
Larsen [27] in 1975"
O(E , Ei, ~ 8i, Ei,, 15)
! 90) + 1 - e-r O(s 8i, ~ 6i, 6i,,
= e-r 0(6 , 6i, --+ 6i, 6i,, , 9 (17.8)
Ku~6er [111 ] notices an analogy between this model and Maxwell's model for gas-surface
interaction, as discussed Section 11, and introduces another model, called the theta model,
which would correspond to the Cercignani-Lampis model in this analogy.
The Larsen-Borgnakke model has become a customary tool in numerical simulations of
polyatomic gases. It can be also applied to the vibrational modes through either a classical
procedure that assigns a continuously distributed vibrational energy to each molecule, or
through a quantum approach that assigns discrete vibrational levels to each molecule. It
would be out of place here to discuss this point in more detail, for which we refer to the
book of Bird [20]. We also refrain from discussing the interesting recent developments [12,
13] of an old idea of Boltzmann [25,42] to interpret, in the frame of classical statistical
mechanics, the circumstance that at low temperatures the internal degrees of freedom
appear to be frozen, as due to the extremely long relaxation times for the energy transfer
process.
The Larsen-Borgnakke model suffers from the limitation that it considers all the
collisions as a mixture of the elastic or completely inelastic collisions, disregarding the
possibility of a partially inelastic one. In order to construct a more general model we use a
procedure which was first used to deal with models for boundary conditions [43,39,35]. We
l
start from a sensible approximate kernel 00(s~, 6il ~ 6i, C/l), which is chosen on the basis
of intuition, but does not satisfy the basic properties (17.4) and (17.5). At this point we add
to it some other terms, which ensure that the three fundamental properties are satisfied, as
follows:
!
0(61 , 6il ~ 6i, 6il)
!
=- 00(6 I, Eil ~ 8i, Eil)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-54-320.jpg)
![The Boltzmann equation and fluid dynamics 51
+ F(n -+-2)(1 -- ei- eil)(1 -- U(ei eil))(1 -- H(e~ e~l))/I,
(/-,(n))2 ' ,
fo,fo,i
=
n-2 n-2
! 2 dei de/l,
O0(e~, eil --'+ ei, eil) ei 2 eil (17.9)
I--1-
F' (n + 2)
(/-'(n)) 2
L1L 1-6i n-2 n-2
2 dei deil (17.10)
• (1 -- ei -- eil) H(ei, eil) e i 2 ei 1 .
As in the aforementioned case, Equation (17.9) may be interpreted as the linear
!
combination of two normalized kernels 01(...) and 02(...), while H(e~,eil ) is a sort
of accommodation coefficient depending on the energies of the impinging molecules
I
(H(e I, ell) must lay in the interval [0, 1]); we write
! !
0(...) -- H(e~, 8il)01 (...) --}- (1 - H(E~, 8il))02(...),
l
0o(...) -- 01 (...)H(e;, C/l);
F(n + 2) (1 - ei - eil)(1 -- H(ei eil))/I.
02(. 9.) -- (/_,(n))2
(17.11)
The expression of 02(...) is suggested by the requirement that the product (1-
t1))02(.. ") must satisfy reciprocity, the definition of I is chosen in order to get
the normalization of 02(...).
l
Cercignani and Lampis [52] proposed the following kernel 00(eI, ei] ~ ei, eil;a, b),
containing two parameters a and b.
!
00(e~, eil ~ ei, eil" a, b) --
ba ,,-2 n-2 ,,-2 ,,-2 (1 - ei - eil) 1/2
4 e~-- T !--T
ei 4 eil eil ! ! 1/2
7r (l--c/--eil )
• exp[-a (ei - e!) 2 - a (eil -- e;1)2]. (17.12)
In order to avoid a singularity in the expression of the kernel (17.12) and therefore also in
t t I _ mCfr 2/4 > r, where r is a little positive
that of H(e~, eil), it is assumed that E- E i - Eil
!
parameter: then 00(...) and H(e~, eil) are limited. This approximate kernel 00(...) for
!
a --+ cx~tends to bOel(e~, eil, ei, eil), so that the full kernel given by Equation (10) tends to
the Larsen-Borgnakke model. For finite values of a, 00(...) describes a collision in which
l
! (and similarly for ell, eil ), but may be different, according to
ei is not exactly equal to e i
a Gaussian distribution.
By fitting the theoretical expression for viscosity with some experimental data (from
standard handbooks), the aforementioned authors [52] have obtained the values of the
parameters in the case of N2 and 02.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-55-320.jpg)
![52 C Cercignani
The aforementioned authors, in a joint paper with J. Struckmeyer [56,57,55], found that
a good fitting of some experimental data given in [99] can be obtained. However, they
were unable to obtain a unique determination of the parameters a and b. Some examples
of possible choices of a and b have been given in [52], in the case of N2 and 02. In those
examples, low values of a, for instance, a = 1, a - 0.1, and for each of them a value of
b close to its maximum value, were chosen, but it is also possible to choose much higher
values of a.
In order to obtain more information about the values of the parameters a and b, an
obvious way would be to try to fit a second transport coefficient. Unfortunately, the
experimental data for fly versus temperature are very scanty [141,73]. The bulk viscosity
of gases can only be measured by the attenuation and dispersion of an ultrasonic, acoustic
signal. Moreover, it is a difficult method subject to experimental error [73]. Because of
this, it is not possible to draw conclusions on the range of applicability of the model. In the
application of the kernel to the calculation of transport coefficients, everything does work
also without introducing the cut-off. The situation may be different in other problems,
for instance in the application of DSMC. Therefore a similar model that does not present
singularities and does not require a cut-off was introduced [56], based on the following
kernel:
!
00(8~, 8il ---> 8i, 8il; a, b)
2ba n-2 n-2 n-2 n-2 (1 -- 8i -- 8il)
4 t---T-- !-----T
8i 4 8il Ei 8il t t
re (1 -- 8i -- eil) -1- (1 - 8 i -- 8il )
X exp[--a(8i 8') 2 t 2]
-- --a(8il--8il) . (17.13)
Using this kemel, the authors repeated the calculation of heat conductivity, which follows
the same procedure as before. The results given in [52] about heat conductivity versus
temperature are identical to those calculated with the new kemel.
We end this review of models for polyatomic gases by remarking that a model
generalizing the ES model (see Section 9) to polyatomic gases has been discussed by
Andries et al. [3], who have also shown that the H-theorem holds for this model for
polyatomic gases as well.
Conceming the boundary conditions for the distribution function for polyatomic gases,
we remark that there is not much material published on this subject, perhaps because the
theory is not so different from that holding in the monatomic case. Concerning specific
models, one should mention an extension of the CL model to polyatomic molecules
proposed by Lord [117] as a generalization of the Cercignani-Lampis model [48].
18. Chemistry and radiation
Chemical reactions are important in high altitude flight because of the high temperatures
which develop near a vehicle flying at hypersonic speed (i.e., at Mach numbers larger
than 5). Up to 2000 K, the composition of air can be considered to be the same as at
standard conditions. Beyond this temperature, N2 and 02 begin to react and form NO. At](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-56-320.jpg)
![The Boltzmann equation andfluid dynamics 53
2500 K diatomic oxygen begins to dissociate and form atomic oxygen O, till 02 completely
disappears at about 4000 K. Nitrogen begins to dissociate at a slightly higher temperature
(about 4250 K). NO disappears at about 5000 K. Ionization phenomena start at about
8500 K.
As we implicitly remarked when we wrote Equation (16.6), the kinetic theory of gases is
an ideal tool to deal with chemical reactions of a particular kind, i.e. bimolecular reactions,
which can be written schematically as
A + B +-~ C + D, (18.1)
where A, B, C and D represent different molecular species. We already used the term
"molecule", as usual in kinetic theory, to mean also atom (a monatomic molecule); in
this section we shall further enlarge the meaning of this term to include ions, electrons
and photons as well, when we have to deal with ionization reactions and interaction with
radiation.
As long as the reaction takes place in a single step with the presence of no other species
than the reactants, it is a well-known circumstance that the change of concentration of a
given species (A, say) in a space-homogeneous mixture can be written as follows:
dnA
dt
-- kb(T)ncnD - k f(T)nAnB. (18.2)
We remark that, in chemistry, one uses the molar density in place of the number density
used here, the two being obviously related through Avogadro's number.
The rate coefficients kf and kb for the forward and backward (or reverse) reactions,
respectively, are functions of temperature and are usually written by a semiempirical
argument, which generalizes the Arrhenius formula, in the form:
( Ea)
k(T) = AT ~ exp -~-~- , (18.3)
where A and r/ (= 0 in the Arrhenius equation) are constants, and Ea is the so-called
activation energy of the reaction. It is clear that these equations, though having a flavor
of kinetic theory, are essentially macroscopic and can be assumed to hold when the
distribution function is essentially Maxwellian.
In fact, the above reaction theory can be obtained by assuming that the distribution
functions are Maxwellians, whereas the role of internal degrees of freedom may be ignored
and the reaction cross-section vanishes if the translational energy Et in the center-of-mass
system is less than Ea and equals a constant aR if the energy is larger than Ea. A more
accurate theory is obtained [18-20] by assuming that the ratio of the reaction cross-section
to the total cross-section is zero when the total collision energy Ec (equal to the sum of
Et and the total internal energy of the two colliding molecules Ei) is less than Ea and
proportional to the product of a power of Ec - Ea and a power of Ec. The exponents and the
proportionality factor are essentially dictated by the number of internal degrees of freedom,
the exponent of the temperature in the diffusion coefficient of species A in species B and](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-57-320.jpg)
![The Boltzmann equation and fluid dynamics 55
This simple theory is based on a molecular interaction that is attractive at large distance
and repulsive at short distances. One can assume a highly simplified scheme by taking a
hard sphere core with diameter cr and a scattering of the square-well type at some larger
distance cr,. The potential energy is, say, -Q (Q > 0) between cr and cr,. Then if m is
the mass of a molecule A, for a distance r > or,, the trajectory of one molecule with
respect to the other before the interaction begins, will be a straight line r cos 0 = b (in
polar coordinates), where b is the impact parameter. The condition b < or, must be satisfied
if the molecules actually interact. Then, assuming for simplicity that A = B (as in the case
of recombination of oxigen) the conservation of energy and angular momentum give
(d!) 2 1 4Q 1
-~ r 2 m g 2 : b
--5
(or < r < a,), (18.6)
where V is, as usual, the relative speed. We easily verify that the trajectory has a comer
point at a distance r = or, and the molecule we are following is deflected toward the other
by an angle
Oo cos-l(b) ,[b ( 4Q )-,/2]
- -- -cos- -- 1+ V2 . (18.7)
a, a, m
The orbiting pair P has in this case a very simple motion: in fact the molecules A approach
each other and then have a hard sphere collision. After that they tend to separate again;
and the "molecule" P will disappear and two molecules A emerge again, unless a third
molecule X collides with P, which is stabilized into a A2. The unstable pair P is endowed
with an internal energy
mv2
EB---~ + Q. (18.8)
This energy is stored to be, possibly, converted into kinetic energy (and hence, from a
macroscopic viewpoint, heat of reaction) through the process (18.4).
Even for this simple scheme we must write three Boltzmann equations: one for the
species A, one for the species A2 and one for the unstable species P; even if A is
monatomic A2 and P are diatomic and hence have an internal energy. The species X
used in the above argument can be any of the three aforementioned species. The species A
loses particles when colliding with A (formation of P) and P gains in the same process,
but loses in most collisions undergone by a P molecule.
This model can be slightly complicated by assuming that there is a potential barrier
between or, and cr** (or** > or,). If the potential energy is Ea > 0 for these distances
between the A molecules, then the formation can occur if, and only if, the relative speed V
is larger than 4Ea/m. Then Ea plays the role of the activation energy.
The resulting system of Boltzmann equations reads as follows:
-- + ~. -- -2
at Ox 3 +
f f, BrAAd/~ dn + (f F, - f g,)BAA2 ,
3 +](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-59-320.jpg)
![56 C.Cercignani
+ fR fB (ftf~--ff*)BeAAd~j*dn
3 +
+ fB (ft f1* - f f *)13AA2d~* dn'
3 +
(18.9)~
OF
Ot
+ ~.
OF
Ox y.f f.f .. d~
2 ' ' d~j dn + f g,13Bades,
= F g, BBA2 ,
3 + 3 +
-fl~3ft3+Fg, Bt~A2dl~,dn
+ (F f~ - Ff,)]~AA2 d~j, dn
3 +
+ fB (FIFI* - FF*)I3A2A2 d~* dn'
3 +
(18.9)2
Og
Ot ag fR f13 ft t r fR f13
9 = f*]~aad~, dn - gf,13Ba d~, dn
~XX 3 + 3 +
-fR3ft3+gF*131~a2df~,an, (18.9)3
where f, F, and g denote the distribution functions for species A, A2, and B, whereas the
superscripts r and e are used to discriminate between reactive and elastic collisions when
necessary. For simplicity, we have omitted indicating the internal energies of particles A2
and B. The factors 2 take into account the fact that 2 particles of a species disappear or
appear at the same time. Models to deal with a chemically reactive gas, akin to the BGK
one have been discussed by Yoshizawa [178].
Ionization reactions involve the electronic states and it is unlikely that a purely classical
theory will be successful in describing them, because of the selection rules. Yet, one can
use the phenomenological approach to provide at least an upper bound for the reaction
rates9
As mentioned above, one can, in principle, think of interaction with radiation as if it
were a reaction involving photons as "molecules". Here spontaneous emission should also
be taken into account. It becomes harder to develop phenomenological models, because
one should consider as many species as there are excited levels for each molecule9
Photons can be described by means of the so-called radiative transfer (or radiation
transport) equation, which looks like a Boltzmann equation. The analogy is, however, in a
sense, artificial, because the number of photons is not conserved.
Of 3f
~ + c~o.
at ax](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-60-320.jpg)
![The Boltzmann equation andfluid dynamics 57
-- fR3 K(x, k' ~ k)f(x, k') dk' - v(x, k)f(x, k) + s(x, k). (18.10)
Here K is the scattering probability from a wavevector k' to a wavevector k. The direction
of k is given by o~and its magnitude is the radiation frequency multiplied by the speed of
light c, not to be confused with v(x, k), the total frequency of scattering and absorption
events. The term s(x, k) describes the volume radiation source, due to photon emission.
If inelastic scattering effects, like fluorescence and stimulated emission, are neglected,
then there is no interaction between photons of different frequency and we can replace
the arguments by k' and k by the corresponding unit vectors oJ and oY. The emission
term can be expressed as a product of the Planck distribution by the volume emission
coefficient elkl,T.
Boundary conditions for radiation can be described in a way similar to that used for
gas-surface interaction. Of course, emission and absorption of radiation occur, along with
reflection [149,96].
19. The DSMC method
Kinetic models and perturbation methods are very useful in obtaining approximate
solutions and forming qualitative ideas on the solutions of practical problems, but in
general they are not sufficient to provide detailed and precise answers for practical
problems. Various numerical procedures exist which either attempt to solve for f by
conventional techniques of numerical analysis or efficiently by-pass the formalism of
the integrodifferential equation and simulate the physical situation that the equation
describes (Monte Carlo simulation). Only recently proofs have been given that these partly
deterministic, partly stochastic games provide solutions that converge (in a suitable sense)
to solutions of the Boltzmann equation. Numerical solutions of the Boltzmann equation
based on finite difference methods meet with severe computational requirements due to
the large number of independent variables.
The only method that has been used for space-inhomogeneous problems in more than
one space dimension is the technique of Hicks, Yen and Nordsiek [136,177], which is based
on a Monte Carlo quadrature method to evaluate the collision integral. This method was
further developed by Aristov and Tcheremissine [4,164] and has been applied with some
success to a few two-dimensional flows [165,45].
An additional difficulty for traditional numerical methods is the fact that chemically
reacting and thermally radiating flows (and even simpler flows of polyatomic gases)
are hard to describe with theoretical models having the same degree of accurateness
as the Boltzmann equation for monatomic nonreacting and nonradiating gases. These
considerations paved the way to the development of simulation schemes, which started with
the work of Bird on the so-called Direct Simulation Monte Carlo (DSMC) method [17]
and have become a powerful tool for practical calculations. There appear to be very few
limitations to the complexity of the flow fields that this approach can deal with. Chemically
reacting and ionized flows can be and have been analysed by these methods.
A problem which arises in the applications of the DSMC method is the choice of a model
for the molecular collisions. The issue is to have simple computing rules by discarding what](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-61-320.jpg)
![58 C. Cercignani
is physically insignificant. In the case of monatomic gases, the relation of the deflection
angle to the impact parameter and the relative speed appears to be the most important
piece of physics. It turns out, however, that the scattering law has little effect and that
the observable effects are strongly correlated with the cross-section change with relative
speed. This realization led to the idea of the variable hard-sphere (VHS) model [19] which
combines the scattering simplicity of the hard-sphere model with a variable cross-section
based on a molecular diameter proportional to some power co - 1/2 of the relative speed
(co being the power of absolute temperature ruling the change of the viscosity coefficient).
This does not produce problems in a single gas, because the heat conductivity varies with
approximately the same law as the viscosity coefficient, but problems arise for mixtures.
If one wants the correct diffusion coefficient, another modification is needed. Koura and
Matsumoto [107] developed the variable soft sphere (VSS) model, which introduces an
additional power law parameter and gives the necessary flexibility for mixtures. More
complicated models can be devised when the attractive part of the intermolecular force
is taken into account [84]. A simpler method has also been proposed [97].
Before discussing the DSMC method and some of its applications in some detail, we
remark that, although the DSMC has no rivals for practical computations, some other
methods may turn out to be of interest in the future if much more powerful computers will
be available. Thus, e.g., discrete velocity models have been an intensely studied subject
for many years, before becoming a systematic method of approximating the Boltzmann
equation.
The DSMC method, the molecular collisions are considered on a probabilistic rather
than a deterministic basis. The main aim is to calculate practical flows through the use
of the collision mechanics of model molecules. In fact, the real gas is modeled by some
hundred thousands or millions of simulated molecules on a computer. For each of them
the space coordinates and velocity components (as well as the variables describing the
internal state, if we deal with polyatomic molecules) are stored in the memory and are
modified with time as the molecules are simultaneously followed through representative
collisions and boundary interactions in the simulated region of space. In most applications,
the number of simulated molecules is extremely small in comparison with the number of
molecules that would be present in a real gas flow. Thus, in the simulation, each model
molecule is representing the appropriate number of real molecules, The calculation is
unsteady and the steady solutions are obtained as asymptotic limits of unsteady solutions.
The flow field is subdivided into cells, which are taken to be small enough for the solution
to be approximately constant through the cell. The time variable is advanced in discrete
steps of size At, small with respect to the mean free time, i.e., the time between two
subsequent collisions of a molecule. This permits a separation of the inertial motion of the
molecules from the collision process: one first moves the molecules according to collision-
free dynamics and subsequently the velocities are modified according to the collisions
occurring in each cell. The rate of occurrence of collisions is given by (hard spheres):
(19.1)
r- ~ rij,
i,j<i](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-62-320.jpg)
![The Boltzmann equation andfluid dynamics 59
P ft3 n-(~i-~j)dn,
rij m Nm + (19.2)
where N is the number of molecules in the sample and ~i is the velocity of the i-th
molecule (i = 1..... N). Each time a collision occurs, the velocities of a collision pair are
modified (and, as a consequence, r also varies). Let Tk be the length of the time interval
between the (k - 1)-th and the k-th collision in the time interval [0, At] (t = 0 is the time
of the 0-th collision by definition). The time intervals Tk are chosen in this way: after the
(k - 1)-th collision, one samples a pair (i, j) on the basis of the probability distribution
pij --rij/r (with a fixed velocity for each pair); then one takes Tk = 2N/[(N- 1)rij].
Then one samples a direction 6oof the post-collisional velocity ~i -- ~ j with the probability
+
distribution p(6o) - rij/r (fixed i, j and I~i - ~Jj l) and replaces ~Ji and ~j by ~j+ and ~j,
given by
1
~? -- ~(~i -Jr-~j -~- 6o[~i -- ~j[), (19.3)
1 (19.4)
The operation is repeated until Y~'~kTk exceeds AT.
Some variations of Bird's method have appeared, due to Koura [104], Belotserkovskii
and Yanitskii [11], Deshpande [72]. They differ from Bird's method because of the
procedure used to sample the time interval between two subsequent collisions. In particular
Koura [104] introduced the so-called "null collision technique", which uses, for models
different from hard spheres, the maximum of the cross-section, when estimating the
possible collisions. Then for values of the cross-section less than the maximum, some
collisions produce a null effect.
A different method was proposed by Nanbu [133]. One does not subdivide At and works
with the probability Pi that the i-th molecule collides in [0, At]. One has
N
Pi -- At Z rij.
j-1
(19.5)
One starts with rij evaluated at t -- 0 and samples a random number coin [0, 1]. According
to whether co is smaller or larger than Pi, the i-th molecule collides or does not collide in
[0, At]. If a collision occurs, one samples a collision partner of the i-th molecule from the
probability distribution
p)i) ._ rij (19.6)
Y~L l rik "
Then one samples a direction 6o of the post-collisional velocity ~+- ~+ with the
+
probability distribution p(6o) - rij/r (for fixed values of i, j and I/~+ - ~j I) and replaces
/~i by/~+ given by Equation (19.3). Then the procedure is repeated for all the values of i.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-63-320.jpg)
![60 C. Cercignani
The method of Nanbu was criticized by Koura [103], who asserted that momentum and
energy are not conserved by collisions, because one does not change the velocity of the j-th
molecule, when one changes the velocity of the i-th molecule. The criticism is not well-
founded, however, because, as Nanbu pointed out [134], the Boltzmann equation requires
the overall conservation of momentum and energy of the system at each space point, not the
conservation of the same quantities at each single simulated event. What is new in Nanbu's
method is precisely the circumstance that it does not try to simulate the N-body dynamics,
but rather the description of the system described by the Boltzmann equation. Nanbu's
method is now well understood, from both a physical and mathematical standpoint and has
been rigorously proven to yield approximations to solutions of the Boltzmann equation,
provided the number of test molecules is sufficiently large. The relevant theorem reads as
follows [7,8]:
THEOREM. If the Boltzmann equation with initial data fo has a smooth, nonnegative
solution f (x,/~) ~ L l, then the solution f of Nanbu's method converges weakly in L 1
to f, in the sense that, for any test function q~(x,~j) 6 L~176
f~ f dx d~ ~ f~f dx d~ (19.7)
as N --+ ~, Ax, At --+ O.
Subsequently, a similar result was proved for Bird's method [169,142].
The fluctuations inevitably occurring in a DSMC calculation can cause problems if the
number of molecules is not large enough to have a representative sample in each area of
the flow domain. They can, however, have a physical significance and contain information
on those occurring in a real gas. The comparisons between the fluctuations in DSMC and
the predictions of fluctuating hydrodynamic theory have been reviewed by Garcia [76] and
appear to be consistent with the fluctuations in a real gas.
The computing task of a simulation method varies with the molecular model. For models
other than Maxwell's it is proportional to N for Bird's method, while it is proportional to
N 2 for Nanbu's method. Babovsky [7] found a procedure to reduce the computing task of
Nanbu's method and make it proportional to N. His modification is based on the idea of
subdividing the interval [0, 1] into N equal subintervals. If the random number colies in the
j-th segment, one calculates only eij -~- rij At and there is no collision if co < (j/n) - Pij,
there is a collision with the j-th molecule if co >~(j/n) - Pij. There is also a condition that
eij must satisfy, i.e., eij < 1/N, but this is usually automatically verified, given the size
of At. Application of Nanbu's method in the form modified by Babovsky shows that the
the computing task is not only theoretically but also practically comparable to that of Bird's
method [80]. This modification eventually evolved into what is called the "Finite-Pointset"
method.
We remark that one may take advantage of flow symmetries in physical space, but all
collisions are calculated as three-dimensional events.
As for the boundary conditions, Maxwell's model of diffuse reflection (see Section 11)
is adequate for many problems. There are many cases. The CL model [48] has been](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-64-320.jpg)
![The Boltzmann equation andfluid dynamics 61
adapted and extended by Lord [117] for application in DSMC studies. The resulting CLL
model [117,150] has been shown to provide a realistic boundary condition with incomplete
accommodation [176]. More complicated models would be required to describe chemical
reactions which can occur at the surface for high impact energies.
20. Some applications of the DSMC method
The first significant application of DSMC method dealt with the structure of a normal shock
wave [121], but only a few years later Bird was able to calculate shock profiles [15] that
allowed meaningful comparisons with the experimental results then available [16] and with
subsequent experiments [147,2]. This long time span is understandable: the method is very
demanding of computer resources. In 1964, even with the fastest computers, the restriction
on the number of molecules which could be used was such that large random fluctuations
had to be expected in the results, and it was difficult to arrive at definite conclusions. Thus
the number of simulated molecules and the sample sizes in the computations that could
be performed in those years were extremely small in comparison with those that have
been routinely employed by an increasing number of workers. The problem of the shock
wave structure has continued to be an important test case. Later studies have included
comparisons of measured and computed velocity distribution functions within strong shock
waves in helium [140].
Early DSMC studies were also devoted to the problem of hypersonic leading edge. This
arises in connection with the flow of a gas past a very sharp plate, parallel to the oncoming
stream. When the Reynolds number Re = p~ V~L/#~, based on the plate length is very
large, the picture, familiar from continuum mechanics, of a potential flow plus a viscous
boundary layer is valid everywhere except near the leading and the trailing edge. Estimates
obtained already in the late 1960s by Stewartson [161] and Messiter [124] showed that the
Knudsen number at the trailing edge is of order Ma~ Re-3/4, where Ma~ is the upstream
Mach number. As a consequence, kinetic theory is not needed (for large values of Re) at
the trailing edge. For the leading edge, the Knudsen number is of order Ma~; hence in
supersonic, or, even more, hypersonic flow (Ma~ ~>5), the flow in the region about the
leading edge must be considered as a typical problem in kinetic theory.
In particular, the viscous boundary layer and the outer flow are no longer distinct from
each other, although [123,82,95] a shock-like structure may still be identified. It is in
this connection that the name of merged-layer regime, mentioned in Section 1, arose.
There are several methods based on simplified continuum models, represented by the
papers of Oguchi [137], Shorenstein and Probstein [148], Chow [66,67], Rudman and
Rubin [145], Cheng et al. [65], and Kot and Turcotte [102], which usefully predict surface
and other gross properties in this regime. The good agreement between these approaches
and experiment gave new evidence for the the importance of the Navier-Stokes equations.
Nevertheless, if we go sufficiently close to the leading edge, the Navier-Stokes equations
must be given up in favor of the Boltzmann equation. Huang and coworkers [90,88,89]
carried out extensive computations based on discrete ordinate methods for the BGK model
and were able to show the process of building the flow picture assumed in the simplified
continuum models mentioned above.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-65-320.jpg)
![62 c. Cercignani
The first DSMC is due to Vogenitz et al. [168] and exhibits a flow structure qualitatively
different from the predictions of earlier studies. Their results are supported by the
experiments of Metcalf et al. [125]. Validation studies of the DSMC method were also
conducted at the Imperial College [83].
Hypersonic flows past blunt bodies were also the object of many simulations, most
of the calculations being those made for the Shuttle Orbiter re-entry, for which useful
comparisons with measured data were possible [128]. This comparison was concerned
with the windward centerline heating and employed an axially symmetric equivalent body.
Later comparisons [143] with Shuttle data were for the aerodynamic characteristics of the
full three-dimensional shape.
Another interesting problem which has been simulated by Ivanov and his coworkers is
the reflection on a plane wall of an oblique shock wave generated by a wedge [92,93].
Three-dimensional DSMC calculations have also been made for the flow past a delta
wing [29]. The results compare well with wind-tunnel measurements [116] of the flow
field under the same conditions.
Other important problems are related to separated flows, especially wake flows and
flows involving viscous boundary layer separation and reattachment. The first calculations
referred to the two-dimensional flow over a sharp flat plate followed by an angled
ramp [129]. The results were in a reasonably good agreement with wind tunnel studies,
which is not truly two-dimensional because of inevitable sidewall effects. Similar
experiments were therefore performed [63] for the corresponding axially symmetric flow,
less subject to the aforementioned non-uniformity. The DSMC calculations for these
cases [130] show excellent agreement with experimental results. In particular, separation
and reattachment of a viscous boundary layer in the laminar regime are correctly predicted.
The most remarkable wake flow simulation was for a 70~ spherically blunted cone model
that had been tested in several wind tunnels [1,115]. The results of the calculations [131] of
the lee side flow that contains the vortex are in good agreement with the experiments and
with Computational Fluid Dynamics (CFD) studies of the flow based on the Navier-Stokes
equations.
In the case of polyatomic gases one has several cross-sections, such as elastic, rotational,
vibrational, and also reactive, if chemical reactions occur. Koura [105] has extended his
null collision technique [104] to these cases and improved it later [106]. He applied this
method to simulate the hypersonic rarefied nitrogen flow past a circular cylinder [106],
with particular attention to the simulation of the vibrational relaxation of the gas; he also
investigated the effect of changing the number of molecules in each (adaptive) cell and the
truncation in the molecular levels.
The Direct Simulation Monte Carlo method is not only a practical tool for engineers,
but also a good method for probing into uncovered areas of the theory of the Boltzmann
equation, such as stability of the solutions of this equation and the possible transition to
turbulence [156,60,77,78,157,159,158,21,144,160,20].
We finally remark that the Direct Simulation Monte Carlo method has been used even
to uncover the analytical nature of a singularity in a limiting solution of the Boltzmann
equation, the structure of an infinitely strong shock wave. The latter arises when the
temperature upstream of the shock is taken to be zero; then the solution of the Boltzmann
equation is the sum of a delta function term and a more regular distribution. The latter was](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-66-320.jpg)
![The Boltzmann equation andfluid dynamics 63
approximated by a Maxwellian by H. Grad [79] but turns out to go to infinity [46] when
equals the velocity upstream. The DSMC solution gives strong evidence on the nature of
the singularity, which is confirmed by a deterministic method [163].
21. Concluding remarks
The use of the Boltzmann equation to study rarefied flows has reached a mature stage.
The qualitative features are well understood, new phenomena have been uncovered,
powerful numerical methods have been developed. Further progress, such as the possibility
to indicate that turbulence for gases has features different from turbulence in liquids,
depends on the computing power available. The same can be said for the development
of deterministic numerical methods as opposed to Monte Carlo. We have not treated all
the possible subjects: among the most important omissions, we mention wave propagation,
expansion into a vacuum and the application of the Boltzmann equation to the important
problems of evaporation and condensation. For these flows, as well as for details on other
topics we refer to relevant monographs [43,39] and the literature quoted therein.
References
[1] J. Allegre and D. Bisch, Experimental study of a blunted cone at rarefied hypersonic conditions, CNRS
Report RC 94-7 (1994).
[2] H. Alsmeyer, Density profiles in argon and nitrogen shock waves measured by the absorption of an electron
beam, J. Fluid Mech. 74 (1976), 497-513.
[3] E Andries, E Le Tallec, J.E Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation
with small Prandtl number, submitted to Eur. J. Mech. B (2000).
[4] V. Aristov and EG. Tcheremissine, The conservative splitting methodfor solving the Boltzmann equation,
U.S.S.R. Comput. Math. Math. Phys. 20 (1980), 208-225.
[5] L. Arkeryd, On the Boltzmann equation. Part II."Thefull initial value problem, Arch. Rational Mech. Anal.
45 (1972), 17-34.
[6] L. Arkeryd and C. Cercignani, On afunctional equation arising in the kinetic theory of gases, Rend. Mat.
Acc. Lincei 9 11 (1990), 139-149.
[7] H. Babovsky, A convergence prooffor Nanbu's Boltzmann simulation scheme, Eur. J. Mech. B 8 (1) (1989),
41-55.
[8] H. Babovsky and R. Illner, A convergence prooffor Nanbu's simulation method for the full Boltzmann
equation, SIAM J. Numer. Anal. 26 (1989), 45-65.
[9] C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math.
Models Methods Appl. Sci. 1 (1991), 235-257.
[10] K. B~winkel and S. Schippers, Nonreciprocity in noble-gas metal-surface scattering, Rarefied Gas
Dynamics: Space-Related Studies, E.E Muntz, D.E Weaver and D.H. Campbell, eds, AIAA, Washington
(1989), 487-501.
[11] O.M. Belotserkovskii and V. Yanitskii, Statistical particle-in-cell methodfor solving rarefied gas dynamics
problems, Zh. Vychisl. Mat. i Mat. Fiz. 15 (1975), 1195-1203 (in Russian).
[12] G. Benettin, L. Galgani and A. Giorgilli, Realization of holonomic constraints and freezing of high
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[13] G. Benettin, L. Galgani and A. Giorgilli, Realization of holonomic constraints and freezing of high
frequency degrees of freedom in the light of classical perturbation theory. Part II, Comm. Math. Phys.
121 (1989), 557-601.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-67-320.jpg)
![64 C. Cercignani
[14] EL. Bhatnagar, E.E Gross and M. Krook, A model for collision processes in gases. Small amplitude
processes in charged and neutral one-component systems, Phys. Rev. 94 (1954), 511-525.
[15] G.A. Bird, Shock wave structure in a rigid sphere gas, Rarefied Gas Dynamics, Vol. 1, J.H. deLeeuw, ed.,
Academic Press, New York (1965), 216-222.
[16] G.A. Bird, Aspects of the structure of strong shock waves, Phys. Fluids 13 (1970), 1172-1177.
[17] G.A. Bird, Direct Simulation of the Boltzmann equation, Phys. Fluids 13 (1970), 2676-2681.
[18] G.A. Bird, Simulation of multi-dimensional and chemically reacting flows, Rarefied Gas Dynamics,
R. Campargue, ed., CEA, Paris (1979), 365-388.
[19] G.A. Bird, Monte-Carlo simulation in an engineering context, Rarefied Gas Dynamics, Part I, AIAA,
Washington (1981), 239-255.
[20] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford
(1994).
[21] G.A. Bird, The initiation of centrifugal instabilities in an axially symmetric flow, Rarefied Gas Dynamics
Symposium 20, Ching Shen, ed., Peking Univ. Press, Beijing (1997), 624-629.
[22] R.C. Blanchard, Rarefied flow lift to drag measurement of the Shuttle Orbiter, Paper No. ICAS 86-
118.10.1, 15th ICAS Congress, London (September 1986).
[23] L. Boltzmann, Weitere Studien iiber das Wiirmegleichgewicht unter Gasmolekiilen, Sitzungsber. Akad.
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[174] M.M.R. Williams, Aphenomenological study ofgas-surface interactions, J. Phys. D. Appl. Phys. 4 (1971),
1315-1319.
[175] M.M.R. Williams and S.K. Loyalka, Aerosol Science. Theory & Practice, Pergamon Press, Oxford (1991).
[176] M.S. Woronowicz and D.F.G. Rault, Cercignani-Lampis-Lord gas-surface interaction model: comparison
between theory and simulation, J. Spacecraft and Rockets 31 (1994), 532-534.
[177] S.M. Yen, B. Hicks and R.M. Osteen, Further development of a Monte Carlo method for the evaluation
of Boltzmann collision integral, Rarefied Gas Dynamics, Vol. I, A. 12, M. Becker and M. Fiebig, eds,
DFVLR-Press, Porz-Wahn (1974), 1-10.
[178] Y. Yoshizawa, Wave structures of a chemically reacting gas by the kinetic theory of gases, Rarefied Gas
Dynamics, Part I, J.L. Potter, ed., AIAA, New York (1977), 501-517.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-73-320.jpg)
![A review of mathematical topics in collisional kinetic theory 77
The goal of this chapter is to introduce, and make a preliminary discussion of, the
mathematical models and problems which will be studied in more detail thereafter. The first
section addresses only physical issues, starting from scratch. We begin with an introduction
to kinetic theory, then to basic models for collisions.
Then in Section 2, we start describing the mathematical problems which arise in
collisional kinetic theory, restricting the discussion to the ones that seem to us most
fundamental. Particular emphasis is laid on the Boltzmann equation. Each paragraph
contains at least one major problem which has not been solved satisfactorily.
Next, a specific section is devoted to the classification of collision kernels in the
Boltzmann collision operator. The variety of collision kernels reflects the variety of
possible interactions. Collision kernels have a lot of influence on qualitative properties
of the Boltzmann equation, as we explain.
In the last two sections, we first present some basic general tools and considerations
about the Boltzmann operator, then give an overview of existing mathematical theories for
collisional kinetic theory.
1. Models for collisions in kinetic theory
1.1. Distributionfunction
The object of kinetic theory is the modelling of a gas (or plasma, or any system made up
of a large number of particles) by a distribution function in the particle phase space. This
phase space includes macroscopic variables, i.e., the position in physical space, but also
microscopic variables, which describe the "state" of the particles. In the present survey, we
shall restrict ourselves, most of the time, to systems made of a single species of particles
(no mixtures), and which obey the laws of classical mechanics (non-relativistic, non-
quantum). Thus the microscopic variables will be nothing but the velocity components.
Extra microscopic variables should be added if one would want to take into account non-
translational degrees of freedom of the particles: internal energy, spin variables, etc.
Assume that the gas is contained in a (bounded or unbounded) domain X C I~u (N = 3
in applications) and observed on a time interval [0, T], or [0, +ec). Then, under the
above simplifying assumptions, the corresponding kinetic model is a nonnegative function
f(t, x, v), defined on [0, T] x X x I~N. Here the space Ii~u -- ~N is the space of possible
velocities, and should be thought of as the tangent space to X. For any fixed time t, the
quantity f(t, x, v)dx dv stands for the density of particles in the volume element dx dv
centered at (x, v). Therefore, the minimal assumption that one can make on f is that for
all t/> 0,
f(t, ", ") E L~oc(X; L 1(It{N));
or at least that f (t,., .) is a bounded measure on K x R N, for any compact set K C X. This
assumption means that a bounded domain in physical space contains only a finite amount
of matter.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-80-320.jpg)
![78 C. Villani
Underlying kinetic theory is the modelling assumption that the gas is made of so many
particles that it can be treated as a continuum. In fact there are two slightly different ways
to consider f: it can be seen as an approximation of the true density of the gas in phase
space (on a scale which is much larger than the typical distance between particles), or it
can reflect our lack of knowledge of the true positions of particles. Which interpretation is
made has no consequence in practice. 1
The kinetic approach goes back as far as Bernoulli and Clausius; in fact it was introduced
long before experimental proof of the existence of atoms. The first true bases for kinetic
theory were laid down by Maxwell [335,337,336]. One of the main ideas in the model is
that all measurable macroscopic quantities ("observables") can be expressed in terms of
microscopic averages, in our case integrals of the form f f(t, x, v)qg(v)dr. In particular
(in adimensional form), at a given point x and a given time t, one can define the local
density p, the local macroscopic velocity u, and the local temperature T, by
P = fRN f (t, x, v) dv, pu -- fRu f (t, x, v)v dv,
Plu] 2 + NpT -- foNf (t, x, l))1l)12dr.
(1)
For much more on the subject, we refer to the standard treatises of Chapman and
Cowling [154], Landau and Lipschitz [304], Grad [250], Kogan [289], Uhlenbeck and
Ford [433], Truesdell and Muncaster [430], Cercignani and co-authors [141,148,149].
1.2. Transport operator
Let us continue to stick to a classical description, and neglect for the moment the interaction
between particles. Then, according to Newton's principle, each particle travels at constant
velocity, along a straight line, and the density is constant along characteristic lines dx/dt -
v, dv/dt = 0. Thus it is easy to compute f at time t in terms of f at time 0:
f (t, x, v) = f (O, x- vt, v).
1For instance, assume that the microscopic description of the gas is givenby a cloud of n points x1..... Xn
in I~N, with velocities Vl..... Vn in I~N. A microscopic configuration is an element (Xl,Vl..... Xn,Vn) of
(R N • I~N)n. The "density" of the gas in this configuration is the empirical measure (l/n)y~nt=l ~(xi,vi)
it is a probability measure on RN x RN. In the first interpretation, f (x, v)dx dv is an approximation of the
empirical measure.In the secondone,there is a symmetric probability density fn on the space (RN x RN)n of
all microscopic configurations,and f is an approximation of the one-particle marginal
plfn(xl, Vl) -- f fn(xl, Vl ..... Xn, vn)dx2dv2"'" dxndvn.
Thus the first interpretation is purely deterministic, while the second one is probabilistic. It is the second
interpretation which was implicitlyused by Boltzmann, and which is needed by Landford's validation theorem,
see Section2.1.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-81-320.jpg)
![A review of mathematical topics in collisional kinetic theory 81
Under these five assumptions, in 1872 Boltzmann (cf. [93]) was able to derive a
quadratic collision operator which accurately models the effect of interactions on the
distribution function f"
of
Ot coll
(t, x, v) = Q(f, f) (t, x, v) (6)
= f~N dr, fSN-1 do" B(v- v,, a)(f~f~, - ff,). (7)
Here we have used standard abbreviations: f' = f(t, x, v'), f, = f(t, x, v,), f,~ =
f (t, x, v~,). Moreover, the nonnegative function B(z, cr), called the Boltzmann collision
kernel, depends only on Izl and on the scalar product (z/lz[, or). Heuristically, it can be
seen as a probability measure on all the possible choices of the parameter cr 6 SN-l, as
a function of the relative velocity z = v - v,. But truly speaking, this interpretation is in
general false, because B is not integrable .... The Boltzmann collision kernel is related
to the cross-section F, by the identity B(z, a) -- Izlr (z, a). By abuse of language, B is
often called the cross-section.
Let us explain (7) a little bit. This operator can formally be split, in a self-evident way,
into a gain and a loss term,
Q(f, f)= Q+(f, f)- Q-(f, f).
The loss term "counts" all collisions in which a given particle of velocity v will encounter
another particle, of velocity v,. As a result of such a collision, this particle will in general
change its velocity, and this will make less particles with velocity v. On the other hand,
t, then the vt particle may
each time particles collide with respective velocities v~ and v,
acquire v as new velocity after the collision, and this will make more particles with
velocity v: this is the meaning of the gain term.
It is easy to trace back our modelling assumptions: (1) the quadratic nature of this
operator is due to the fact that only binary collisions are taken into account; (2) the fact
that the variables t, x appear only as parameters reflects the assumption that collisions are
localized in space and time; (3) the assumption of elastic collisions results in the formulas
giving v~and v~,; (4) the microreversibility implies the particular structure of the collision
kernel B; (5) finally, the appearance of the tensor products 3 f~f/, and ff, is a consequence
of the chaos assumption.
On the whole, the Boltzmann equation reads
of
Ot
+ v. Vxf = Q(f, f), t ~ 0, X E ]1~N, U E ~N., (8)
or, when a macroscopic force F (x) is also present,
of
Ot
-- + v . Vx f + F(x) . Vvf = Q(f, f), t >~0, x E]I~N vE]~ N ,
3In the sense that ff, (f | f)(v, v,), , t I
= f f, = (f | f)(v', v,).
(9)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-84-320.jpg)
![82 C.Villani
The deepest physical and mathematical properties of the Boltzmann equation are linked
to the subtle interaction between the linear transport operator and the nonlinear collision
operator.
We note that this equation was written in weak formulation by Maxwell as early as 1866:
as recalled in Section 2.3 below, Maxwell [335,337] wrote down the equation satisfied
by the observables f f(t,x, v)qg(v)dv (see in particular [335, Equation (3)]). However
Boltzmann did a considerable job on the interpretation and consequences of this equation,
and also made them widely known in his famous treatise [93], which was to have a lot of
influence on theoretical physics during several decades.
Let us make a few comments about the modelling assumptions. They may look rather
crude, however they can in large part be completely justified, at least in certain particular
cases. By far the deepest assumption is Boltzmann's chaos hypothesis ("Stosszahlansatz")
which is intimately linked to the questions of irreversibility of macroscopic dynamics and
the arrow of time. For a discussion of these subtle topics, the reader may consult the
review paper by Lebowitz [293], or the enlightening treatise by Kac [284], as well as the
textbooks [149,410] for the most technical aspects. We shall say just a few words about the
subject in Sections 2.1 and 2.4.
1.4. Collisionkernels
Since the collision kernel (or equivalently the cross-section) only depends on iv - v, I and
V--V,
on (Iv v,l' a), i.e., the cosine of the deviation angle, throughout the whole text we shall
abuse notations by writing
8(v - v,, ~)= 8(Iv- v,I, cos0),
V--V, >
cos0= Iv-v,l'a" (lO)
Maxwell [335] has shown how the collision kernel should be computed in terms of the
interaction potential r In short, here are his formulas (in three dimensions of space), as
they can be found for instance in Cercignani [141], for a repulsive potential. For given
impact parameter p ~>0 and relative velocity z ~ R 3, let the deviation angle 0 be
fo ~-c~ ds/ s2
0 (p, z) = Jr - 2p V/1 - 7 -
4/-z~P2
~(s)
p/so
- Jr - 2 !
dO
du
where so is the positive root of
p2 ~bl(s0)
- s--~-4 z[ 2 =0.
Then the collision kernel B is implicitly defined by
p dp
B(Izl, cos0) -- sin0 dO
~--Izl. (11)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-85-320.jpg)
![A review of mathematical topics in collisional kinetic theory 83
It can be made explicit in two cases:
9 hard spheres, i.e., particles bounce on each other like billiard balls: in this case
B(lv - v.[, cos0) is just proportional to Iv - v.[ (the cross-section is constant);
9 Coulomb interaction, ~b(r) = 1/r (in adimensional variables and in three dimensions
of space): then B is given by the famous Rutherford formula,
1
B(lv - v,I, cosO) - Iv - v,I 3 sin4(O/2)" (12)
A dimensional factor of (e2/47r eom) should multiply this kernel (e = charge of the particle,
e0 = permittivity of the vacuum, m = mass of the particle). Unfortunately, Coulomb
interactions cannot be modelled by a Boltzmann collision operator; we shall come back
to this soon.
In the important 4 model case of inverse-power law potentials,
~b(r)--rS_l, s>2,
then the collision kernel cannot be computed explicitly, but one can show that
B(Iv - v,I, cos0) -- b(cos0)lv - v,I y,
s - (2N - 1)
• = . (13)
s--1
In particular, in three dimensions of space, g = (s - 5) / (s - 1). As for the function b, it is
only implicitly defined, locally smooth, and has a nonintegrable singularity for 0 --+ 0:
sin N-2 0 b(cos0) ~ KO -1-v v= (N = 3). (14)
s--1
Here we have put the factor sinN-2 0 because it is (up to a constant depending only on the
dimension) the Jacobian determinant of spherical coordinates on the sphere S u-1 .
The nonintegrable singularity in the "angular collision kernel" b is an effect of the huge
amount of grazing collisions, i.e., collisions with a very large impact parameter, so that
colliding particles are hardly deviated. This is not a consequence of the assumption of
inverse-power forces; in fact a nonintegrable singularity appears as soon as the forces are
of infinite range, no matter how fast they decay at infinity. To see this, note that, according
to (11),
f0 f0 f0 Pmax
d__ppdO - Izl p dp - ~ . (15)
B (Izl, cos 0) sin 0 dO - Izl P dO 2
By the way, it seems strange to allow infinite-range forces, while we assumed
interactions to be localized. This problem has never been discussed very clearly, but in
4Inverse power laws are moderately realistic, but very important in physics and in modelling, because they are
simple, often lead to semi-explicit results, and constitute a one-parameter family which can model very different
phenomena. Van der Waals interactions typically correspond to s = 7, ion-neutral interactions to s = 5, Manev
interactions [88,279] to s = 3, Coulomb interactions to s -- 2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-86-320.jpg)
![84 C. Villani
principle there is no contradiction in assuming the range of the interaction to be infinite at a
microscopic scale, and negligible at a macroscopic scale. The fact that the linear Boltzmann
equation can be rigorously derived from some particle dynamics with infinite range [179]
also supports this point of view.
As one sees from formula (13), there is a particular case in which the collision kernel
does not depend on the relative velocity, but only on the deviation angle: particles
interacting via a inverse (2N - 1)-power force (1/r 5 in three dimensions). Such particles
are called Maxwellian molecules. They should be considered as a theoretical model, even if
the interaction between a charged ion and a neutral particle in a plasma may be modelled by
such a law (see, for instance, [164, Theorem 1, p. 149]). However, Maxwell and Boltzmann
used this model a lot,5 because they had noticed that it could lead to many explicit
calculations which, so did they believe, were in agreement with physical observations.
Also they believed that the choice of molecular interaction was not so important, and that
Maxwellian molecules would behave pretty much the same as hard spheres. 6
Since the time of Maxwell and Boltzmann, the need for results or computations has led
generations of mathematicians and physicists to work with more or less artificial variants
of the collision kernels given by physics. Such a procedure can also be justified by the fact
that for many interesting interactions, the collision kernel is not explicit at all: for instance,
in the case of the Debye potential, r = e-r/r. Here are two categories of artificial
collision kernels:
- when one tames the singularity for grazing collisions and replaces the collision kernel
by a locally integrable one, one speaks of cut-off collision kernel;
- collision kernels of the form Iv- v,I • (Y > 0) are called variable hard spheres
collision kernels.
It is a common belief among physicists that the properties of the Boltzmann equation
are quite a bit sensitive to the dependence of B upon the relative velocity, but very little to
its dependence upon the deviation angle. True as it may be for the behavior of macroscopic
quantities, this creed is definitely wrong at the microscopic level, as we shall see.
In all the sequel, we shall consider general collision kernels B(Iv - v,I, cos0), in
arbitrary dimension N, and make various assumptions on the form of B without always
caring if it corresponds to a true interaction between particles (i.e., if there is a r whose
associated collision kernel is B). Our goal, in a lot of situations, will be to understand how
the collision kernel affects the properties of the Boltzmann equation. However, we shall
always keep in mind the collision kernels given by physics, in dimension three, to judge
how satisfactory a mathematical result is.
1.5. Boundary conditions
Of course the Boltzmann equation has to be supplemented with boundary conditions which
model the interaction between the particles and the frontiers of our domain X C R N (wall,
etc.)
5See Boltzmann [93, Chapter 3].
6Further recall that at the time, the "atomic hypothesis" was considered by many to be a superfluous
complication.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-87-320.jpg)
![A reviewof mathematical topics in collisional kinetic theory 85
The most natural boundary condition is the specular reflection:
f (x, Rxv) -- f (x, v), Rxv = v - 2(v. n(x))n(x), x e OX, (16)
where n(x) stands for the outward unit normal vector at x. In the context of optics, this
condition would be called the Snell-Descartes law: particles bounce back on the wall with
an postcollision angle equal to the precollision angle.
However, as soon as one is interested in realistic modelling for practical problems,
Equation (16) is too rough .... In fact, a good boundary condition would have to take
into account the fine details of the gas-surface interaction, and this is in general a very
delicate problem] There are a number of models, cooked up from modelling assumptions
or phenomenological a priori constraints. As good source for these topics, the reader may
consult the books by Cercignani [141,148] and the references therein. In particular, the
author explains the relevant conditions that a scattering kernel K has to satisfy for the
boundary condition
f (x, rout) -- f K (Vin, rout) f (x, Vin)dvin
to be physically plausible. Here we only list a few common examples.
One is the bounce-back condition,
f (x, - v) = f (x, v), x ~ OX. (17)
This condition simply means that particles arriving with a certain velocity on the wall will
bounce back with an opposite velocity. Of course it is not very realistic, however in some
situations (see, for instance, [148, p. 41]) it leads to more relevant conclusions than the
specular reflection, because it allows for some transfer of tangential momentum during
collisions.
Another common boundary condition is the Maxwellian diffusion,
f (x, v) = p-(x)Mw(v), v . n(x) > O, (18)
where p_(x) = fv.n<O f(x, v)dv and Mw is a particular Gaussian distribution depending
only on the wall,
Mw(v) =
Ivl2
e 2Tw
(2g)-~-2 T[ +l"
7Here we assumethat the finedetailsof the surfaceof the wall are invisible at the scaleof the spatial variable,
so thatthe wallis modelled as a smooth surface,but we wishto takethese detailsintoaccountto predictvelocities
aftercollisionwiththe wall. Anotherpossibilityis to assumethat the roughnessof the wallresultsin irregularities
which can be seen at spatial resolution. Thenit is natural in many occasions to assume0X to be very irregular.
For some mathematicalworksaboutthis alternativeapproach, see [33,272,273].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-88-320.jpg)
![86 C Villani
In this model, particles are absorbed by the wall and then re-emitted according to the
distribution Mw, corresponding to a thermodynamical equilibrium between particles and
the wall.
Finally, one can combine the above models. Already Maxwell had understood that a
convex combination of (16) and (18) would certainly be more realistic than just one of
these two equations. No need to say, since the work of Maxwell, much more complicated
models have appeared, for instance the Cercignani-Lampis (CL) model, see [148].
In mathematical discussions, we shall not consider the problem of boundary conditions
except for the most simple case, which is specular reflection. In fact, most of the time we
shall simply avoid this problem by assuming the position space to be the whole of •N, or
the torus qrN. Of course the torus is a mathematical simplification, but it is also used by
physicists and by numerical analysts who want to avoid taking boundary conditions into
account ....
1.6. Variants of the Boltzmann equation
There are many variants. Let us only mention
9 relativistic models, see [50,158,199,200,233-235,14];
9 quantum models, see [196,328,209] and the references therein. They will be discussed
in Section 3 of Chapter 2E. Also we should mention that models of quantum
Boltzmann equation have recently gained a lot of interest in the study of semi-
conductors, see in particular [65,66] and the works by Poupaud and coworkers on
related models [386-388,244,360], also [354,355] (in two dimensions) and [15] (in
three dimensions);
9 linear models, in particular the linear Boltzmann equation,
-- -+-v. Vxf =
Ot N•
dr, dcr B(v - v,, a)[F(v',) f (v') - F(v,) f (v)],
where F is a given probability distribution. As a general rule, such linear equations
model the influence of the environment, or background, on a test-particle (think of
a particle in an environment of random scatterers, like in a random pinball game).
The distribution F is the distribution of the background, and is usually assumed to
be stationary, which means that the environment is in statistical equilibrium. Linear
Boltzmann-like models are used in all areas of physics, most notably in quantum
scattering [206] and in the study of transport phenomena associated with neutrons or
photons [148, pp. 165-172]. A general mathematical introduction to linear transport
equations can be found in [157, Chapter 21];
9 diffusive models, like the Fokker-Planck equation, which is often used in its linear
version,
of
Ot
+ v. Vxf = Vv. (Vvf + fv), (19)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-89-320.jpg)
![A review of mathematical topics in collisional kinetic theory 87
or in its nonlinear form,
0f
+ v. Vxf-- paVv. [TVvf + f(v - u)], (20)
Ot
where o~E [0, 1], and p, u, T are the local density, velocity and temperature defined
by (1). When o~= 1, this model has the same quadratic homogeneity as the Boltzmann
equation. Of course it is also possible to couple the equation only via T and not u,
etc. A classical discussion on the use of the Fokker-Planck equation in physics can be
found in the important review paper by Chandrasekhar [153].
The Landau equation, which is described in detail in the next section, is another
diffusive variant of the Boltzmann equation;
9 energy-dissipating models, describing inelastic collisions. These models are particu-
larly important in the theory of granular materials: see Section 2 of Chapter 2E;
9 model equations, like the (simplistic) BGK model, see, for instance, [141,148]. In this
model one replaces the complicated Boltzmann operator by M f - f, where M f is
the Maxwellian distribution s with the same local density, velocity and temperature
than f. Also variants are possible, for instance multiplying this operator by the local
density p ....
Another very popular model equation for mathematicians is the Kac model [283].
It is a one-dimensional caricature of the Boltzmann equation which retains some of
its interesting features. The unknown is a time-dependent probability measure f on
11~,and the equation reads
Of 1
-
- dr. d0 f'f~, - f, (21)
Ot 27c
where (vt, vt.) is obtained from (v, v.) by a rotation of angle 0 in the plane •2. This
model preserves mass and kinetic energy, but not momentum;
9 discrete-velocity models: these are approximations of the Boltzmann equation where
particles are only allowed a finite number of velocities. They are used in numerical
analysis, but their mathematical study is (or once was) a popular topic. About them
we shall say nothing; references can be found, for instance, in [230,141,148] and also
in the survey papers [384,62]. Among a large number of works, we only mention the
original contributions by Bony [94,95], Tartar [416,417], and the consistency result
by Bobylev et al. [367], for the interesting number-theoretical issues that this paper
has to deal with.
We note [148, p. 265] that discrete-velocity models were once believed by
physicists to provide miraculously efficient numerical codes for simulation of
hydrodynamics. But these hopes have not been materialized ....
Also some completely unrealistic discrete-velocity equations have been studied as
simplified mathematical models, without caring whether they would approximate or
not the true Boltzmann equation. These models sometimes have no more than two
or three velocities! Some well-known examples are the Carleman equation, with two
8SeeSection2.5 below.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-90-320.jpg)
![88 C. Villani
velocities, the Broadwell model, with four velocities in the plane, or the Cabannes
equation with fourteen velocities. For instance, the Carleman equation reads
of 1 ~fl
-S-+ =
Of-1 Of-1 = f?__ f21.
Ot Ox
(22)
For a study of these models see [230,384,141,148,60,61,108,25] and references
included. Of course, these models are so oversimplified9 that they cannot be
considered seriously from the physical point of view, even if one may expect
that they keep some relevant features of the Boltzmann equation. By the way, as
noticed by Uchiyama [432], the Broadwell model cannot be derived from a fictitious
system of deterministic four-velocity particles ("diamonds") in the plane, see [149,
Appendix 4C]. Only at the price of some extra stochasticity assumption can the
derivation be fixed [117].
All the abovementioned models should be considered with appropriate boundary
conditions. These conditions can also be replaced by the effect of a confinement potential
V(x): this means that there is a macroscopic force of the form F(x) = -VV(x) acting on
the system.
Finally, it is important to note that Boltzmann's model is obtained as a result of the
assumption of localized interaction; in particular, it does not take into account a possible
interaction of long (macroscopic) range which would result in a macroscopic mean-field
force, typically
F(x) = -V~(x), qb(x) = p *x ~P,
where 4~is the interaction potential and p the local density. The modelling of the interaction
by such a coupled force is called a Vlasov description.
When should one prefer a Vlasov, or a Boltzmann description? A dimensional analysis
by Bobylev and Illner [88] shows that for inverse-power forces like 1/ rs, and under natural
scaling assumptions,
9 for s > 3, the Boltzmann term should prevail on the mean-field term;
9 for s < 3, the Boltzmann term should be negligible in front of the mean-field term.
The separating case, s = 3, is the so-called Manev interaction [88,279].
There are subtle questions here, which are not yet fully understood, even at a formal
level. Also the uniqueness of the relevant scaling is not clear. From a physicist's point of
view, however, it is generally accepted that a good description is obtained by adding up
the effects of a mean-field term and those of a Boltzmann collision operator, with suitable
dimensional coefficients.
Another way to take into account interactions on a macroscopically significant scale is
to use a description ?~la Povzner. In this model (see, for instance, [389]), particles interact
9As a word of caution, we should add that even if they are so simplified, their mathematical analysis is not
trivial at all, and many problems in the field still remain open.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-91-320.jpg)
![A review of mathematical topics in collisional kinetic theory 89
through delocalized collisions, so that the corresponding Boltzmann operator is integrated
with respect to the position y of the test particle, and reads
fR dy [ dr, B(v - - y)[f (y, v~,)f (x, v') - f (y, v,) f (x, v)].
V,,X
JR
(23)
Note that the kernel B now depends on x - y. On the other hand, there is no collision
parameter cr any more, the outgoing velocities being uniquely determined by the positions
x, y and the ingoing velocities vt, v,.t It was shown by Cercignani [140] that this type of
equations could be retrieved as the limit of a large stochastic system of "soft spheres".
A related model is the Enskog equation for dense gases, which has never been clearly
justified. It resembles Equation (23) but the multiplicity of the integral is 2N - 1 instead
of 2N, because there is no integration over the distance [x - YI-Mathematical studies of the
Enskog model have been performed by Arkeryd, Arkeryd and Cercignani [24,27,28] - in
particular, reference [24] provides well-posedness and regularity under extremely general
assumptions (large data, arbitrary dimension) by a contraction method. See Section 2.1 in
Chapter 2E for an inelastic variant which is popular in the study of granular material.
The study of these models is interesting not only in itself, but also because numerical
schemes always have to perform some delocalization to simulate the effect of collisions.
This explains why the results in [140] are very much related to some of the mathematical
justifications for some numerical schemes, as performed in [453,396].
1.7. Collisions in plasma physics
The importance and complexity of interaction processes in plasma physics justifies that
we devote a special section to this topic. A plasma, generally speaking, is a gas of
(partially or totally) ionized particles. However, this term encompasses a huge variety of
physical situations: the density of a plasma can be extremely low or extremely high, the
pressures can vary considerably, and the proportion of ionized particles can also vary over
several orders of magnitude. Nonelastic collisions, recombination processes may be very
important. We do not at all try to make a precise description here, and refer to classical
textbooks such as Balescu [46], Delcroix and Bers [164], the very nice survey by Decoster
in [160], or the numerous references that can be found therein. All of these sources put a lot
of emphasis on the kinetic point of view, but [160] and [164] are also very much concerned
with fluid descriptions. A point which should be made now, is that the classical collisional
kinetic theory of gases a priori applies when the density is low (we shall make this a little
bit more precise later on) and when nonelastic processes can be neglected.
Even taking into account only elastic interactions, there are a number of processes
going on in a plasma: Maxwell-type interactions between ions and neutral particles, Van
der Waals forces between neutral particles, etc. However, the most important feature,
both from the mathematical and the physical point of view, is the presence of Coulomb](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-92-320.jpg)
![90 C.Villani
interactions between chargedparticles. The basic model for the evolution of the density of
such particles is the Vlasov-Poisson equation
I of
37 + v. Vxf + F(x). Vvf --O,
e 2
F =-VV, V = 4rcsor*x P, p(t, x) = f f (t, x, v) dv.
(24)
For simplicity, here we have written this equation for only one species of particles, and also
we have not included the effect of a magnetic field, which leads to the Vlasov-Maxwell
system (see, for instance, [191]). We have kept the physical parameters e -- charge of the
particle and e0 = permittivity of vacuum for the sake of a short discussion about scales.
Even though this is not the topic of this review paper, let us say just a few words on
the Vlasov-Poisson equation. Its importance in plasma physics (including astrophysics)
cannot be overestimated, and thousands of papers have been devoted to its study. We only
refer to the aforementioned textbooks, together with the famous treatise by Landau and
Lipschitz [304]. From the mathematical point of view, the basic questions of existence,
uniqueness and (partial) regularity of the solution to the Vlasov equation have been solved
at the end of the eighties, see in particular Pfaffelmoser [382], DiPerna and Lions [191],
Lions and Perthame [318], Schaeffer [402]. Reviews can be found in Glassey [233] and
Bouchut [96]. Stability, and (what is more interesting!) instability of several classes of
equilibrium distributions to the Vlasov-Poisson equation have recently been the object of
a lot of studies by Guo and Strauss [264-268]. Several important questions, however, have
not been settled, like the derivation of the Vlasov-Poisson equation from particle systems
(see Spohn [410] and Neunzert [356] for related topics) and the explanation of the famous
and rather mysterious Landau damping effect.
There is no doubt that the Vlasov-Poisson equation is the correct equation to describe a
classical plasma on a short time scale. However, when one wants to consider long periods
of time, it is necessary to take into account collisions between particles. For this it is natural
to introduce a Boltzmann collision operator in the fight-hand side of (24). However, the
Boltzmann equation for Coulomb interactions does not make sense! Indeed, the collision
integral would be infinite even for very smooth (or analytic) distribution functions, l~ This is
due to the very slow decay of the Coulomb potential, and the resulting very strong angular
singularity of the collision kernel given by Rutherford's formula (12). A standard remedy
to this problem is to assume that there is a screening (due to the presence of two species of
particles, for instance), so that the effective interaction potential between charged particles
is not the Coulomb interaction, but the so-called Debye potential
e-r/)~D
~b(r) -- ~ . (25)
4rceor
10More precisely, the natural definition of the collision operator would lead to the following nonsense:
whateverf, Q(f, f) is an elementof {-oo, 0, +oo}, see [450,AnnexI, AppendixA].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-93-320.jpg)
![A review of mathematical topics in collisional kinetic theory 91
Here Xo is the Debye length, i.e., a typical screening distance. In the classical theory of
plasmas,
~eokT
XD-- -~,
where k is Boltzmann's constant, T is the temperature of the plasma (rigorously speaking,
it should depend on x...) and p its mean density (same remark). The resulting collision
kernel is no longer explicit, but at least makes sense, because the very strong angular
singularity in Rutherford's formula (12) is tamed.
The replacement of Coulomb by Debye potential can be justified by half-rigorous, half-
heuristic arguments (see the references already mentioned). However, in most of the cases
of interest, the Debye length is very largewith respect to the characteristic length r0 for
collisions, called the Landaulength:
r0 --
e2
4zceokT
More precisely, in so-called classical plasmas (those for which the classical kinetic
description applies), one has
ro ~ p-l~3 ~ )~D.
This means first that the Landau distance is very small with respect to other scales (so that
collisions can be considered as localized), and secondly that the plasma is so dilute that
the typical distance between particles is very small with respect to the screening distance,
which is usually considered as the relevant space scale.
By a formal procedure, Landau [291] showed that, as the ratio A -- 2XD/r0 --+ ec, the
Boltzmann collision operator for Debye potential behaves as
log A
27rA
QL (f, f),
where QL is the so-called Landaucollisionoperator:
QL(f,f)--Vv.(fRsdv,a(v-v,)[f,(Vf)- f(Vf),]). (26)
Here a(z) is a symmetric (degenerate) nonnegative matrix, proportional to the orthogonal
projection onto z-L:
L[ ziz,]
aij(Z)---~l (~ij iz12 , (27)
and L is a dimensional constant.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-94-320.jpg)
![A reviewof mathematical topics in collisional kinetic theory 93
where the temperature is low, and slowed down in directions where the temperature is high:
this is normal, because in the end the temperature along all directions should be the same.
Formulas like (30) show that in the isotropic case, and under assumption (29), the
nonlinear Landau equation reduces (in a well-chosen orthonormal basis) to the linear
Fokker-Planck equation! By this remark [447] one can construct many explicit solutions
(which generalize the ones in [296]). These considerations explain why the Maxwellian
variant of the Landau equation has become a popular test case in numerical analysis [106].
Let us now review other variants of the Landau equation. First of all, there are relativistic
and quantum versions of it [304]. For a mathematically-oriented presentation, see, for
instance, [298]. There are also other, more sophisticated models for collisions in plasmas:
see, for instance, [164, Section 13.6] for a synthetic presentation. The most famous of
these models is the so-called Balescu-Lenard collision operator, whose complexity is just
frightening for a mathematician. Its expression was established by Bogoljubov [90] via
a so-called BBGKY-type hierarchy, and later put by Lenard under the form that we give
below. On the other hand, Balescu [46,47] derived it as part of his general perturbative
theory of approximation of the Liouville equation for many particles. Just as the Landau
operator, the Balescu-Lenard operator is in the form
Z.. ~ 3 dv, aij(v, v,) f*~vj -- f ~v,j '
tj
(31)
but now the matrix aij depends on f in a strongly nonlinear way:
aij(v, v,) -- ~~R3,1kl~<Km.~
S[k. (V -- V,)] kikj 1
Ikl 4 lel 2
~ dk, (32)
where e is the "longitudinal permittivity" of the plasma,
e=l f 1 k. (V f), dr,.
Ikl~ k. (v - v.) - i0
Here 6 is the Dirac measure at the origin, and
1 = lim 1 _79(1~+i:r8
x - iO s~o+ x - it x]
is a complex-valued distribution on the real line (79 stands for the Cauchy principal part).
Moreover, Kmax is a troncature parameter (whose value is not very clearly determined)
which corresponds to values of the deviation angles beyond which collisions cannot be
considered grazing. This is not a Debye cut!
Contrary to Landau, Balescu and Lenard derived the operator QL (26) as an approxima-
tion of (31). To see the link between these two operators, let us set k = #w,/z ~>0, co E $2;
then one can rewrite (32) as
fS (f0KmaxdjJ~)
dcoS[co. (v - v.)] COicoj 12 9
2 /ZI8 (33)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-96-320.jpg)
![94 C. Villani
But
2 dco6(co, z)coicoj - -~ 6ij
zizj )
izl2 9
So one can replace the operator (31) by the Landau collision operator if one admits that the
integral in d/z in (33) depends very little on v, v., co. Under this assumption, it is natural
to replace f by a Maxwellian, ll and one finds for this integral an expression of the form
1 + (/z2)//z 2, where #D has the same homogeneity as the inverse of a Debye length. Then
one can perform the integration; see Decoster [160] for much more details.
In spite of its supposed accuracy, the interest of the Balescu-Lenard model is not so clear.
Due to its high complexity, its numerical simulation is quite tricky. And except in very
particular situations, apparently one gains almost nothing, in terms of physical accuracy of
the results, by using it as a replacement for the Landau operator. In fact, it seems that the
most important feature of the Balescu-Lenard operator, to this date, is to give a theoretical
basis to the use of the Landau operator!
There exist in plasma physics some even more complicated models, such as those which
take into account magnetic fields (Rostoker operator, see, for instance, [465] and the
references therein). We refrain from writing up the equations here, since this would require
several pages, and they seem definitely out of reach of a mathematical treatment for the
moment .... We also mention the simpler linear Fokker-Planck operator for Coulomb
interaction, derived by Chandrasekhar. This is a linear operator of Landau-type, which
describes the evolution of a test-particle interacting with a "bath" of Coulomb particles
in thermal equilibrium. A formula for it is given in Balescu [46, w in terms of special
functions.
1.8. Physical validity of the Boltzmann equation
Experience has shown that the Boltzmann equation and its variants realistically describe
phenomena which occur in dilute atmosphere, in particular aeronautics at high altitude, or
interactions in dilute plasmas. In many situations, predictions based on the Navier-Stokes
equation are not accurate for low densities; a famous historical example is provided by the
so-called Knudsen minimum effect in a Poiseuille flow [148, p. 99]: if the difference of
pressure between the entrance and the exit of a long, narrow channel is kept fixed, then the
flow rate through a cross-section of this channel is not a monotonic function of the average
pressure, but exhibits a minimum for a certain value of this parameter. This phenomenon,
established experimentally by Knudsen, remained controversial till the 1960's.
Also the Boltzmann equation cannot be replaced by fluid equations when it comes to
the study of boundary layers (Knudsen layer, Sone sublayer due to curvature...) and the
gas-surface interaction.
Nowadays, with the impressive development of computer power, it is possible to perform
very precise numerical simulations which seem to fully corroborate the predictions based
on the Boltzmann equation - within the right range of physical parameters, of course.
11Thisis the naturalstatisticalequilibrium,see Section2.5.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-97-320.jpg)
![A reviewof mathematical topics in collisional kinetictheory 95
All these questions are discussed, together with many numerical simulations and
experiments, in the recent broad-audience survey book by Cercignani [148] (see also the
review paper [136] by the same author in the present volume).
2. Mathematical problems in collisional kinetic theory
In this section, we try to define the most interesting mathematical problems which arise
in the study of Boltzmann-like equations. At this point we should make it clear that the
Boltzmann equation can be studied for the sake of its applications to dilute gases, but also
as one of the most basic and famous models for nonequilibrium statistical mechanics.
2.1. Mathematical validity of the Boltzmann equation
In the last section, we have mentioned that, in the right range of physical parameters, the
physical validity of the Boltzmann equation now seems to be beyond any doubt. On the
other hand, the mathematical validity of the Boltzmann equation poses a more challenging
problem. For the time being, it has been investigated only for the hard-sphere model.
Let us give a short description of the problem in the case of hard spheres. But before
that, we add a word of caution about the meaning of "mathematical validity": it is
not a proof that the model is the right one in a certain range of physical parameters
(whatever this may mean). It is only a rigorous derivation of the model, in a suitable
asymptotic procedure, from another model, which is conceptually simpler but contains
more information (typically: the positions of all the particles, as opposed to the density
of particles). Of course, the Boltzmann equation, just like many models, can be derived
either by mathematical validation, or by direct modelling assumptions, and the second
approach is more arbitrary, less interesting from the mathematical point of view, but not less
"respectable" if properly implemented! This is why, for instance, Truesdell [430] refuses
to consider the problem of mathematical validity of the Boltzmann equation.
The validation approach to be discussed now is due to Grad [249], and it is particularly
striking because the starting point is nothing but the model given by Newton's laws of
classical mechanics. It was not before 1972 that Cercignani [139] showed Grad's approach
to be mathematically consistent, in the sense that it can be rigorously implemented if one
is able to prove some "reasonable" estimates on the solutions. 12
Grad's approach. The starting point is the equation of motion, according to Newton's
laws, for a system of n spherical particles of radius r in Ii~3, bouncing elastically on each
other with billiard reflection laws. The state of the system is described by the positions and
velocities of the centers, Xl, U1, ..., Xn, Vn, and the phase space is the subset of (Rx
3 x R3) n
(or (X x R3) n) such that Ixi - xjl >~r (i :/: j). On this phase space there is a flow (St)t>~o,
well-defined up to a zero-probability set of initial configurations, which is neglected. We
now consider symmetric probability densities fn(xl, Vl ..... Xn, Vn) on the phase space
12This remark is importantbecause many people doubted the possibility of a rigorous derivation, see the
discussionin Section2.4.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-98-320.jpg)
![96 C Villani
(symmetry reflects the physical assumption of undiscernability of particles). Of course, the
flow (St) on the phase space induces a flow on such probability densities, the solution of
which is denoted by (ff)t >1o.Moreover, by integrating ftn over all variables but the k first
position variables and the k first velocity variables, one defines the k-particle distribution
function Pkfn (X l, Vl ..... Xk, Vk) (think of Pk as a projection operator). In probabilistic
terminology, P1fn is the first marginal of fn.
Assume now that
(i) n --+ o0, r --+ 0 (continuum limit) in such a way that nr 2 --+ 1 [the gas is sufficiently
dilute, but not too much, so that only binary interactions play a significant role,
and a typical particle collides about once in a unit of time. This limit is called the
Boltzmann-Grad limit];
(ii) P1 f~ --+ fo, where fo is a given distribution function [this assumption means that
the one-particle function at time 0 can be treated as "continuous" as the number of
particles becomes large];
(iii) P2f~ --+ fo | fo, and more generally, for fixed k,
P, f~ ~ fo | . . . | fo (34)
[this is the chaos assumption at time 0];
the problem is then to prove that P1fP --+ ft, where ft = ft (x, v) is the solution of
the Boltzmann equation with hard-sphere kernel, and with initial datum f0.
This problem seems exceptionally difficult. The main result in the field is the 1973 Lan-
ford's theorem [292]. He proved the result for small time, and under some strong assump-
tions on the initial probability distributions Pkf~: they should be continuous, satisfy ap-
propriate Gaussian-type bounds, and converge uniformly 13 towards their respective limits.
Later his proof was rewritten by Illner and Pulvirenti, and extended to arbitrarily large
time intervals, under a smallness assumption on the initial datum, which enabled to treat
the Boltzmann equation as a kind of perturbation of the free transport equation: see [275,
276] and the nice reviews in [149,394,410]. For sure, one of the outstanding problems in
the theory of the Boltzmann equation is to extend Lanford's result to a more general frame-
work, without smallness assumption. Another considerable progress would be its extension
to long-range interactions, which is not clear even from the formal point of view (see, for
instance, Cercignani [137]).
The Boltzmann-Grad limit is also often called the low-density limit, and presented in
the following manner [149, p. 60]: starting from the equations of Newtonian dynamics,
blow-up the scales of space and time by a factor e-1 (thus e is the ratio of the microscopic
scale by the macroscopic scale), and require the number of particles to be of the order of
e-z, then let e go to 0. In particular, the density will scale as e2/3 since the volume will
scale as e-3, and this explains the terminology of "low density limit".
Remarks about the chaos assumption.
1. Heuristically, the relevance of the chaos assumption in Boltzmann's derivation can be
justified as follows: among all probability distributions fn which have a given marginal
13Oncompactsubsetsof (•3 • i~3)k whichis the setobtainedfrom(It~
3 • R3)k by deletingall configurations
r
withxi = xj for somedistinctindicesi, j.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-99-320.jpg)
![A reviewofmathematicaltopicsincollisionalkinetictheory 97
f = Plf n, the most likely, in some sense which we do not make precise here, 14 is the
tensor product f | @ f. And as n --+ ec, this distribution becomes by far the most likely.
Thus, Boltzmann's chaos assumption may be justified by the fact that we choose the most
likely microscopic probability distribution fn which is compatible to our macroscopic
knowledge (the one-particle distribution, or first marginal, f).
2. In fact, the chaos property is automatically satisfied, in weak sense, by all sequences of
probability measures (fn) on (R 3 x R3) n which are compatible with the density f. More
precisely: let us say that a microscopic configuration z -- (Xl, Vl ..... Xn, Vn) E (R3 X R3) n
is admissible if its empirical density,
09 z --- __ 6(Xi,Vi),
n
i=1
is a good approximation to the density f(x, v)dx dr, and let (fn)neN be a sequence
of symmetric probability densities on (Rx
3 x R3) n respectively, such that the associated
measures/z n give very high probability to configurations which are admissible. In a more
precise writing, we require that for every bounded continuous function q)(x, v) on R 3 x R 3,
and for all s > 0,
fR go(x, v)[o)z(dx dv) - f (t, x, v) dx dr] >s] )0.
n ----~oo
Then, (fn) satisfies the chaos property, in weak sense [149, p. 91]: for any k ~> 1,
pkfn ~ f|
n-----~oo
in weak-, measure sense. This statement expresses the fact that fn is automatically close to
the tensor product f| in the sense of weak convergence of the marginals. However, weak
convergence is not sufficient to derive the Boltzmann equation, because of the problem
of localization of collisions. Therefore, in Lanford's theorem one imposes a stronger
(uniform) convergence of the marginals, or strong chaos property.
3. This is the place where probability enters the Boltzmann equation: via the initial
datum, i.e., the probability density (fn)! According to [149, p. 93], the conclusion of
Lanford's theorem can be reformulated as follows: for all time t > 0, if ftn is obtained
14This is related to the fact that the tensor product f| has minimal entropy among all n-particles probability
distribution functions fn with given first marginal f, and to the fact that the negative of the entropy yields a
measure of "likelihood", see Section2.4 below. [In thisreview,the entropy of a distribution function f is defined
by the formula H (f) = f f log f; note the sign convention.] For hard spheres, a subtlety arises from the fact
that configurations in which two spheres interpenetrate are forbidden, so fn cannot be a tensor product. Since
however the total volume of the spheres, nr3, goes to 0 in the limit, it is natural to assume that this does not
matter. On the other hand, contact points in the n-particle phase space play a crucial role in the way the chaos
property is propagated in time.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-100-320.jpg)
![98 C.Villani
from fff by transportation along the characteristics of the microscopic dynamics, and if/z n
t
is the probability measure on (Nx
3 x N3)n whose density is given by ftn, then for all e > 0,
and for all bounded, continuous function ~o(x,v) on Nx
3 x N3,
3.
, z (R3 xR ) ;
fR ~o(x, dr)- ft(x, v)dv]
V) [(.Oz (dx dx
x~XR3 n-----)(x)
where ft is the solution to the Boltzmann equation with initial datum f0. In terms of/z n"
q)(x, v)[WS,z(dx dr) - ft(x, v)do]
dx
3xR3 n-----)(x)
In words: for most initial configurations, the evolution of the density under the microscopic
dynamics is well approximated by the solution to the Boltzmann equation. Of course, this
does not rule out the existence of "unlikely" initial configurations for which the solution of
the Boltzmann equation is a very bad approximation of the empirical measure.
4. If the chaos property is the crucial point behind the Boltzmann derivation, then one
should expect that it propagates with time, and that
Yt>0, P~fF ~ ft|174 ft. (35)
However, this propagation property only holds in a weak sense. Even if the convergence is
strong (say, uniform convergence of all marginals) in (34), it has to be weaker in (35),
say almost everywhere, see the discussion in Cercignani et al. [149]. The reason for
this weakening is the appearance of microscopic correlations (under evolution by the
microscopic, reversible dynamics). In particular, if the initial microscopic datum is "very
likely", this does not imply at all that the microscopic datum at later times should be very
likely! On the contrary, it should present a lot of correlations ....
5. In fact, one has to be extremely cautious when handling (35). To illustrate this, let us
formally show that for t > 0 the approximation
P2ft (x, v; y, w)"~ ft (x, v)ft (Y, w) (36)
cannot be true in strong sense, uniformly in all variables, 15 as n ~ cx~ (the symbol
here means "approaches, in L ~ norm, uniformly in all variables x, y, v, w, as n ~ ~").
Indeed, assume that (36) holds true uniformly in x, y, v, w, and choose y - x + rtr,
15Constrainedby Ix- Yl~>r.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-101-320.jpg)
![A reviewof mathematical topics in collisionalkinetic theory 99
(v - w, o-) > 0, i.e., an ingoing collisional configuration in the two-particle phase space.
Then, presumably
P2f~(x, v; x + rcr, w) ~_ ft(x, v)ft(x + rcr, w) ~_ ft(x, v)ft(x, w) (37)
as n --+ cx~.But from the specular reflection condition, for any t > 0,
P2f~ (x, v; x + rcr, w) = P2f~ (x, v'; x + rcr, w'),
where v' and w' are post-collisional velocities,
v' = v - (v - w, cr) ~r, w' = w + ( v - w, cr ) cr.
Applying (36) again, this would result in
P2f~(x, v; x -q-rcr, w) = P2f~(x, v'; x nt- rcr, w') ~--ft(x, v')ft(x, w'),
which is not compatible with (37) (unless ft solves Equation (53) below). This contra-
diction illustrates the fact that (36) cannot be propagated by the dynamics of hard spheres. It
is actually property (37), sometimes called one-sided chaos, which is used in the derivation
of the Boltzmann equation, and which should be propagated for positive times: it means
that the velocities of particles which are just about to collide are not correlated. But it is a
very difficult problem to handle Equation (37) properly, because it involves the restriction
of fn to a manifold of codimension 1, and may be violated even for initial data which
satisfy the conditions of Lanford's theorem! So an appropriate generalized sense should
be given to (37). Lanford's argument cleverly avoids any discussion of (37), and only
assumes (36) at time 0, the approximation being uniform outside collisional configurations.
So he plainly avoids discussing one-sided chaos, and does not care what is propagated
for positive times, apart from weak chaos. 16 To sum up: the physical derivation of the
Boltzmann equation is based on the propagation of one-sided chaos, but no one knows
how this property should be expressed mathematically- if meaningful at all.
An easier variant of the validation problem is the derivation of linear transport equations
describing the behavior of a Lorentz gas: a test-particle in a random pinball game, with
scatterers randomly distributed according to (say) a Poisson law. Under a suitable scaling,
the law of this test-particle converges towards the solution of a linear Boltzmann equation,
as was first formalized by Gallavotti [226], before several improvements appeared [409,
91]. See Pulvirenti [394] for a review and introduction of the subject. The convergence
actually holds true for almost all (in the sense of Poisson measure) fixed configuration
of scatterers, but fails for certain specific configurations, for instance a periodic array,
as shown in Bourgain, Golse and Wennberg [102]. We also note that Desvillettes and
Pulvirenti [179] are able to rigorously justify the linear Boltzmann equation for some
interactions with infinite range.
16This is possible because he uses a perturbative proof, based on an iterative Duhamelformula, in which
everythingis expressedin termsofthe initialdatum....](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-102-320.jpg)
![100 C.Villani
Kac's approach. To conclude this section, we mention another line of approach towards
the mathematical justification of the Boltzmann equation. It goes via the construction of
some many-particle stochastic system, such that the first marginal of its law at a given
time t should be an approximation of the solution to the Boltzmann equation if the
initial datum is chaotic. This subject was initiated by Kac 17 [283], and developed by
Sznitman [412] in connection with the problem of propagation of chaos. Recent progress
on this have been achieved by Graham and M616ard [256,344].
The main conceptual difference between both approaches lies in the moment where
probability is introduced, and irreversibility 18 as well. In Lanford's approach, the starting
point is a deterministic particle system; it is only the particular "chaotic" choice of the
initial datum which leads to the macroscopic, irreversible Boltzmann equation in the limit.
On the other hand, for Kac the microscopic particle system is already stochastic and
irreversible from the beginning. Then the main effect of the limit is to turn a linear equation
on a large n-particle phase space, into a nonlinear equation on a reduced, one-particle phase
space.
Of course Kac's approach is less striking than Grad's, because the starting point
contains more elaborate modelling assumptions, since stochasticity is already built in.
Kac formulated his approach in a spatially homogeneous 19 setting, while this would be
meaningless for Grad's approach. In fact, it is as if Kac wanted to treat the positions of
the particles (which, together with ingoing velocities, determine the outgoing velocities)
as hidden probabilistic variables. Then, all the subtleties linked to one-sided chaos can be
forgotten, and it is sufficient to study just propagation of (weak) chaos.
Moreover, Kac's approach becomes important when it comes to make an interpretation
of the Monte Carlo numerical schemes which are often used to compute approximate
solutions of the Boltzmann equation. These schemes are indeed based on large stochastic
particle systems. See Pulvirenti [394,453,396] for references about the study of these
systems, in connection with the validation problem. We do not develop here on the
problem of the rigorous justification of numerical schemes, but this topic is addressed in
the companion review [136] by Cercignani.
2.2. The Cauchyproblem
From the mathematical point of view, the very first problem arising in the study of the
Boltzmann equation is the Cauchy problem: given a distribution function fo(x, v) on
RN x R N (or X x RN), satisfying appropriate and physically realistic assumptions, show
that there exists a (unique) solution of
Of
-+- v . Vx f = Q(f, f),
Ot (38)
f(O,., .)= fo.
17 SeeSection1.5inChapter2E.
18SeeSection2.4.
19SeeSection5.2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-103-320.jpg)
![A reviewof mathematicaltopicsin collisionalkinetictheory 101
Needless to say, the Boltzmann equation seems impossible to solve explicitly,2~except
in some very particular situations: semi-explicit solutions by Bobylev [79], Bobylev and
Cercignani [81]; self-similar solutions of infinite mass by Nikolskii, see [289, p. 286];
particular solutions in a problem of shear flow by Truesdell, see [430, Chapters 14-15],
some simple problems of modelling with a lot of symmetries [148] .... Explicit solutions
are discussed in the review paper [207]. These exact solutions are important in certain
modelling problems, but they are exceptional. This justifies the study of a general Cauchy
problem.
Of course, the question of the Cauchy problem should be considered as a preliminary for
a more detailed study of qualitative properties of solutions of the Boltzmann equation. The
main qualitative properties in which one is interested are: smoothness and singularities,
conservation laws, strict positivity, existence of Lyapunov functionals, long-time behavior,
limit regimes. We shall come back on all of this in the next chapters. As recalled in
Section 3, the properties of the solutions may depend heavily on the form of the collision
kernel.
As of this date, the Cauchy problem has still not received satisfactory answers. As we
shall describe in Section 5, there are several "competing" theories which either concern
(more or less) simplified cases, or are unable to answer the basic questions one may ask
about the solutions. Yet this problem has spectacularly advanced since the end of the
eighties.
Another fundamental problem in many areas of modelling by Boltzmann equation, as
explained, for instance, in Cercignani [148], is the existence of stationary solutions: given
a box X, prove that there exists a (unique?) stationary solution of the Boltzmann equation
in the box:
v. Vxf=Q(f,f), x6X, VE]I~N,
together with well-chosen boundary conditions (ideally, dictated by physical assumptions).
The stationary problem has been the object of a lot of mathematical studies in the past
few years; see, for instance, [31,25,34,36,26,37,38]. We shall not consider it here, except
for a few remarks. This is first because the theory is less developed than the theory of the
Cauchy problem, secondly because we wish to avoid the subtle discussion of boundary
conditions for weak solutions.
2.3. Maxwell's weak formulation, and conservation laws
, ' k) withk--(v-v,)/Iv-v,l, hasunit
The change of variables (v, v,, cr) --+ (vt v,, ,
Jacobian and is involutive. Since cr = (vI f v~
- v,)/I - v,[, one can abuse terminology
by referring to this change of variables as (v, v,) --+ (v~ v,). It will be called the pre-
postcollisional change of variables. As a consequence of microreversibility, it leaves the
collision kernel B invariant.
20Althoughno theoremofnon-solvabilityhas beenproven!](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-104-320.jpg)
![102 C. Villani
The fact that this change of variable has unit Jacobian is not a general feature of
Boltzmann-like equations, actually it is false for energy-dissipating models21 .... Also
the change of variables (v, v,) --+ (v,, v) is clearly involutive and has unit Jacobian. As a
consequence, if q9is an arbitrary continuous function of the velocity v,
fR Q(f' f) qgdv
u
f /.
/ dvdv, ] da B(v- v,, a)(f'ff, - ff,)9 (39)
JRN• N JsN-1
---f~,NxRN dvdv*fsN-1drrB(v-v*'rr)ff*(~~ (40)
1 dvdv, dcr B(v - v,, cr)ff,(qg' + qg, - q9- qg,). (41)
N • N-1
This gives a weak formulation for Boltzmann's collision operator. From the mathematical
point of view, it is interesting because expressions like (40) or (41) may be well-defined in
situations where Q(f, f) is not. From the physical point of view, it expresses the change
in the integral f f(t, x, v)qg(v)dv which is due to the action of collisions. Actually, this
formulation is so natural for a physicist, that Equation (40) was written by Maxwell22 [335,
Equation (3)] before Boltzmann gave the explicit expression of Q(f, f)!
Let f be a solution of the Boltzmann equation (8), set in the whole space ~ff to simplify.
By the conservative properties of the transport operator, v 9Vx,
df f
-~ f (t, x, v)qg(v) dx dv -- Q(f, f)q9 dx dr, (42)
and the right-hand side is just the x-integral of any one of the expressions in formulas (39)-
(41). As an immediate consequence, whenever q9satisfies the functional equation
V(U, V,, O') E ][~N X ]1~N X S N-1 , qg(v') + q3(l/,) -- qg(v) -+-qg(v,) (43)
then, at least formally,
df
dt f (t, x, v)99(v) dx dv - 0
along solutions of the Boltzmann equation. The words "at least formally" of course mean
that the preceding equations must be rigorously justified with the help of some integrability
estimates on the solutions to the Boltzmann equation.
21 See Section2 in Chapter2E.
22Actuallyit is not so easyto recognizethe Boltzmannequationin Maxwell'snotations!](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-105-320.jpg)
![A reviewof mathematical topics in collisionalkinetic theory 103
It can be shown under very weak conditions 23 [142,29], [149, pp. 36-42] that solutions
to (43), as expected, are only linear combinations of the collision invariants:
Ivl 2
qg(v)- 1, Vi, 2 ' 1 <.i <~N.
This leads to the (formal) conservation laws of the Boltzmann equation,
(1)
-~ f (t, x, y) l)i dxdv = O, 1 <.i<.N,
meaning that the total mass, the total momentum and the total energy of the gas are
preserved.
These conservation laws should hold true when there are no boundaries. In presence of
boundaries, conservation laws may be violated: momentum is not preserved by specular
reflection, neither is energy if the gas is in interaction with a wall kept at a fixed
temperature. See Cercignani [141,148] for a discussion of general axioms of the classical
modelling of gas-surface interaction, and resulting laws.
If one disregards this possible influence of boundaries, then the preservation of mass,
momentum and energy under the action of the Boltzmann collision operator is clearly the
least that one can expect from a model which takes into account only elastic collisions.
Yet, to this date, no mathematical theory has been able to justify these simple rules at
a sufficient level of generality. The problem is of course that too little is known about
how well behaved are the solutions to the Boltzmann equation. Conservation of mass and
momentum are no problem, but no one knows how to obtain an a priori estimate which
would imply a little bit more integrability than just finite energy.
Another crucial topic for a fluid description is the validity of local conservation laws,
i.e., continuity equations obtained by integrating the Boltzmann equation with respect to v
only. With notations (1), these equations are
Op
+ Vx. (pu)- o,
o
__~(tOU) .71_Vx " N f v | v d v =0,
o
~--~(plul 2 + NpT) + Vx. g f[v[ 2 ray =0.
(44)
At this moment, only the first of these equations has been proven in full generality [308].
23This problemwas first treated by Gronwall [259,260] and Carleman [119] under stronger conditions. Then
people started to study it under weaker and weaker assumptions. Its interest lies not only in checking that there
are no hiddenconservationlawsin the Boltzmannequation,but alsoin solvingthe importantEquation(53)below,
for which simplermethodsare howeveravailable.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-106-320.jpg)
![104 C. Villani
2.4. Boltzmann's H theorem and irreversibility
In this section, we discuss some of the most famous aspects of the Boltzmann equation.
This will justify a few digressions to make the topic as clear as possible.
Let us symmetrize the integral (39) once more, fully using all the symmetries of the
collision operator. We obtain
f Q(f, f)go dv
1
f dvdv, do- B(v - v,, a)(f'f" ff,)(go' -+-go', go go,) (45)
4
We shall refer to this formula as Boltzmann's weakformulation.
Without caring about integrability issues, we plug go-- log f into this equation, and use
the properties of the logarithm, to find
f• f) log f - -D(f), (46)
Q(f, dv
N
where D is the entropy dissipation functional,
fR f'f*'
1 dvdv, da B(v- v,,a)(f'f',- ff,)log--f~-, ~>0. (47)
D(f) = -~ 2N •
That D(f) >,0just comes from the fact that the function (X, Y) ~ (X-Y)(log X- log Y)
is nonnegative.
Next, we introduce Boltzmann's H functional,
/-
H(f) / f log f. (48)
JRN•
Of course, the transport operator -v. Vx does not contribute in any change of the H
functional in time. 24 As a consequence, if f = f (t, x, v) is a solution of the Boltzmann
equation, then H (f) will evolve in time because of the effects of the collision operator:
d ,(f(,, .))_ O(f(,,x .))dx<.O.
dt ~
(49)
This is the famous Boltzmann's H theorem: the H-functional, or entropy, is nonincreasing
with time. This theorem is "proven" and discussed at length in Boltzmann's treatise [93].
Before commenting on its physical implications, let us give a few analytical remarks:
1. For certain simplified models of the Boltzmann equation, McKean [342] has proven
that the H-functional is, up to multiplicative and additive constants, the only "local" (i.e.,
of the form f A(f)) Lyapunov functional.
24More generally, the transport operator does not contribute to any change of a functional of the form
f a(f)dxdv.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-107-320.jpg)
![A reviewofmathematicaltopicsincollisionalkinetictheory 105
2. There are some versions with boundary conditions; actually, it was emphasized by
Cercignani that the H theorem still holds true for a modified H-functional (including the
temperature of the wall, for instance, if the wall is kept at fixed temperature) as soon as a
certain number of general axioms are satisfied. See [141] for precise statements.
3. The argument above, leading to formula (47), does not work for certain variants of
the Boltzmann equation, like mixtures. Actually Boltzmann had also given a (false) proof
in this case, and once the error was discovered, produced a totally obscure argument to fix
it (see the historical references in [148]). As pointed out by Cercignani and Lampis [150],
the most robust way to prove the H theorem is to use again Maxwell's weak formulation,
and to note that
if f,f~dvdv,da l f ( f, ff,
-~ Bf f, log ff, -~ Bf f, log ff,
f'f~ I-1/~<0, (50)
ff, J
because f Q(f, f) dv = 0 and log X - X + 1 <~0. This line of proof can be generalized to
mixtures [148, w and other models.
4. The Landau equation also satisfies a H theorem. The corresponding entropy
dissipation is (formally)
1 ~ ff, lv/a(v-v,)(V(log f)- [V(logf)],)12dvdv,. (51)
DL (f) = ~ 3xR3
What is the physical sense of Boltzmann's H theorem? First of all, we note that the H
functional should coincide with the usual entropy of physicists up to a change of sign. Also,
it is a dynamical entropy, in the sense that is defined for nonequilibrium systems. Thus
Boltzmann's H theorem is a manifestation of the second law of thermodynamics (Clausius'
law) which states that the physical entropy of an isolated system should not decrease in
time. In particular, it demonstrates that the Boltzmann model has some irreversibility built
in. This achievement (produce an analytical proof of the second law for some specific
model of statistical mechanics) was one of the early goals of Boltzmann, and was later
considered as one of his most important contributions to statistical physics.
But the H theorem immediately raised a number of objections, linked to the fact that
the starting point of the derivation of the Boltzmann equation is just classical, reversible
mechanics- so where does the irreversibility come from? Of course, since the existence
of atoms was controversial at the time, Boltzmann's construction seemed very suspect ....
About the controversy between Boltzmann and opponents, and the way to resolve apparent
contradictions in Boltzmann's approach, one may consult [202,284,149,293]. Here we
shall explain in an informal way the main arguments of the discussion, warning the reader
that the following considerations have not been put on a satisfactory mathematical basis.
Zermelo pointed out that Boltzmann's theorem seemed to contradict the famous
Poincar6 recurrence theorem. 25 Boltzmann replied that the scales of time on which
Poincar6's theorem applied in the present setting were much larger than the age of the
25For almostall choiceofthe initialdatum,a conservativesystemwitha compactphasespacewillalwayscome
arbitrarily closeto its initialconfigurationforlargeenoughtimes.Thistheoremappliesto a systemof n particles
obeyingthe lawsof classicalmechanics,interactingvia elasticcollisions,enclosedin a box.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-108-320.jpg)
![106 C. Villani
universe, and therefore irrelevant. This answer is justified by the fact that the Boltzmann
equation should be a good approximation of the microscopic dynamics, only on a time
scale which depends on the actual number of particles - see the discussion in the end of
Section 2.5.
Then Loschmidt came out with the following paradox. Let be given a gas of particles,
evolving from time 0 to time to > 0. At time to, reverse all velocities and let the gas evolve
freely otherwise. If Boltzmann were fight, the same Boltzmann equation should describe
the behavior of the gas on both time intervals [0, to] and [to, 2t0]. Since the reversal of
velocities does not change the entropy of a distribution function, the entropy at time 2t0
should be strictly less than the entropy at time 0. But, by time-reversibility of classical
mechanics, at time 2t0 the system should be back to its initial configuration, which would
be a contradiction .... To this argument Boltzmann is reported to have replied "Go on,
reverse the velocities !"
The answer to Loschmidt's paradox is subtle and has to do with the probabilistic
content of the Boltzmann equation. Starting with the classical monograph by P. and
T. Ehrenfest [202], it was understood that reversible microdynamics and irreversible
macrodynamics are not contradictory, provided that the right amount of probability is used
in the interpretation of the macroscopic model. This view is very well explained in the
excellent book by Kac [284].
In the case of the Boltzmann derivation, everything seems deterministic: neither the
microscopic model, nor the macroscopic equation are stochastic. But the probabilistic
content is hidden in the choice of the initial datum. As we mentioned earlier,26 with very
high probability the Boltzmann equation gives a good approximation to the evolution of
the density of the gas. And here randomness is in the choice of the microscopic initial
configuration, among all configurations which are compatible with the density f (see
Section 2.1 for more precise formulations). But for exceptional configurations which are
not chaotic, the derivation of the Boltzmann equation fails.
Now comes the tricky point in Loschmidt's argument. As we discussed in Section 2.1,
the chaos property, in strong sense, is not preserved by the microscopic, reversible dynam-
ics. What should be preserved is the one-sided chaos: ingoing collisional configurations
are uncorrelated. More precisely, if the strong chaos assumption holds true at initial time,
then chaos should be true for ingoing configurations at positive times, and for outgoing
configurations at negative times.
So, reversing all velocities as suggested by Loschmidt is an innocent operation at the
level of the limit one-particle distribution (which has forgotten about correlations), but by
no means at the level of the microscopic dynamics (which keeps all correlations in mind).
It will transform a configuration which is chaotic as far as ingoing velocities are concerned,
into a configuration which is chaotic as far as outgoing velocities are concerned27....
Then, the microscopic dynamics will preserve this property that outgoing velocities are
not correlated, and, by repeating the steps of the derivation of the Boltzmann equation, we
find that the correct equation, to describe the gas from time to on, should be the negative
Boltzmann equation (with a minus sign in front of the collision operator). Therefore in
26Recall the remarks about propagation of chaos in Section 2.1.
27In particular, this configuration is very unlikely as an initial distribution. Thus Loschmidt's paradox illustrates
very well the fact that the Boltzmann derivation works for most initial data, but not for all!](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-109-320.jpg)
![A review of mathematical topics in collisional kinetic theory 107
Loschmidt's argument, of course the entropy is unchanged at time to, but then it should
start to increase, and be back at its original value at time 2t0.
At the moment, a fully satisfactorily mathematical discussion of Loschmidt's paradox
is not possible, since we do not know what one-sided chaos should really mean,
mathematically speaking. But one can check, as is done in [149, Section 4.7], that strong
chaos 28 is not propagated in time- so that it will be technically impossible to repeat
Lanford's argument when taking as initial datum the microscopic configuration at time
to, be it before or after reversal of velocities.
With this in mind, one also easily answers to the objection, why would the Boltzmann
equation select a direction of time? Actually, it does not, 29 and this can be seen by the fact
that if strong 3~ chaos is assumed at initial time, then the correct equation should be the
Boltzmann equation for positive times, and the negative Boltzmann equation for negative
times. We have selected a direction of time by assuming the distribution function to be
"very likely" at time 0 and studying the model for positive times. In fact, in Boltzmann's
description, the entropy is maximum at time 0, and decreases for positive times, increases
for negative times. As an amusing probabilistic reformulation: knowing the one-particle
distribution function at some time to, with very high probability the entropy is a maximum
at this time t0! Related considerations can be found in Kac [284, p. 79] for simpler models,
and may explain the cryptic statement by Boltzmann that "the H-functional is always,
almost surely, a local maximum".
Most of the explanations above are already included in Boltzmann's treatise [93], in
physicist's language; in particular Boltzmann was very well aware of the probabilistic
content of his approach. But, since so many objections had been raised against Boltzmann's
theory, many physicists doubted for a long time that a rigorous derivation of the Boltzmann
equation, starting from the laws of classical mechanics, could be possible. This is one of
the reasons why Lanford's theorem was so spectacular.
After this digression about irreversibility, let us now briefly comment on Boltzmann's
H-functional itself. Up to the sign, it coincides with Shannon's entropy (or information)
quantity, which was introduced in communication theory at the end of the forties. 31 In
the theory of Shannon, the entropy measures the redundancy of a language, and the
maximal compression rate which is applicable to a message without (almost) any loss
of information: see [156] and the many references therein. In this survey, we shall make
precise some links between information theory and the kinetic theory of gases, in particular
via some variants of famous information-theoretical inequalities first proven by Stam [165].
28Roughly speaking, in the sense of uniform convergence of the marginals towards the tensor product
distributions,recall Section2.1.
29Bythe way,Boltzmannhimselfbelievedthatthe directionof positivetimes shouldbe defined as the direction
in whichthe H-functionalhas a decreasingbehavior....
30"Double-sided" shouldbe the fightconditionhere!
31Hereis a quotationby Shannon,extractedfrom[331],whichwe learntin [16]. "My greatestconcernwas how
to callit. I thoughtof callingit 'information'. But the wordwas overlyused, so I decidedto callit 'uncertainty'.
When I discussedit with Johnvon Neumann,he had a better idea. He told me: "Youshouldcall it entropy,for
two reasons.In firstplace youruncertaintyhas beenusedin statisticalmechanicsunderthat name, so it already
has a name. In secondplace, and moreimportant,no one knowswhat entropyreally is, so in a debate you will
always havethe advantage.""](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-110-320.jpg)
![108 C. Villani
From the physical point of view, the entropy measures the volume of microstates
associated to a given macroscopic configuration. 32 This is suggested by the following
computation, due to Boltzmann (see [93]). Let us consider n particles taking p possible
different states; think of a state as a small "box" in the phase space •N • ii~N.Assume that
the only information to which we have access is the number ni of particles in each state i. In
other words, we are unable to distinguish particles with different states; or, in a probabilistic
description, we only have access to the one-particle marginal. This macroscopic description
is in contrast with the microscopic description, in which we can distinguish all the particles,
and know the state of each of them. Given a macroscopic configuration (nl .... , n p), the
number of compatible microscopic configurations is
n!
nl!" .np!
Let us set j5 = ni In, and let all ni's go to infinity. By Stirling formula (or other methods,
see [156, p. 282]), one shows that, up to an additive constant which is independent of the
ft" 'S,
1 P
- log ~2 --+ - E fi log j~.
n
i=1
This result explains the link between the H-functional and the original definition of the
entropy by Boltzmann, as the logarithm of the volume of microstates. 33 So we see that it
is the exponential of the negative of the entropy, which plays the role of a "volume" in
infinite dimension. Up to a normalization, this quantity is known in information theory as
the entropy power34:
./V'(f) = exp(-2H(f)/N). (52)
More remarks about the physical content of entropy, or rather entropies, are formulated
in Grad [251 ].
In the discussion of Boltzmann's derivation and irreversibility, we have seen two distinct
entropies: the macroscopic entropy H(f), which is fixed by the experimenter at initial
time, and then wants to decrease as time goes by; and the microscopic entropy, H(fn),
which is more or less assumed to be minimal at initial time (among the class of microscopic
distributions fn which are compatible with f), and is then kept constant in time by the
microscopic dynamics. There is no contradiction between the fact that H (ft) is decreasing
32Here, the one-particle probability distribution f is the macroscopic description of the system, while the many-
particle probability distribution fn is the microscopic state.
33This is the famous formula S = k In ~2, which was written on Boltzmann's grave.
34The analogy between power entropy and volume can be pushed so far that, for instance, the Shannon-Stam
entropy power inequality, .A/'(f 9 g) ~>.A/'(f) + A/'(g), can be seen as a consequence of the Brunn-Minkowski
inequality on the volume of Minkowski sums of compact sets IX + YI1/d >1IX[1/d + IF[1/d. This (very)
nontrivial remark was brought to our attention by E Barthe.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-111-320.jpg)
![A reviewof mathematical topicsin collisionalkinetictheory 109
and H(ff) is constant, because there is no link between both objects 35 if there are
correlations at the level of ftn.
As a last comment, the decrease of the entropy is a fundamental property of the
Boltzmann equation, but the H-functional is far from containing all the information about
the Boltzmann equation. This is in contrast with so-called gradient flows, which are partial
differential equations of the form Of/Ot = -grad E(f), for some "entropy" functional E
and some gradient structure. For such an equation, in some sense the entropy functional
encodes all the properties of the flow ....
The main, deep reason for the fact that the Boltzmann equation cannot be seen as a
gradient flow, is the fact that the collision operator depends only on the velocity space; but
even if we restrict ourselves to solutions which do not depend on space, then the Boltzmann
equation is not (to the best of our knowledge) a gradient flow. Typical gradient flows, in a
sense which will be made more precise later, are the linear Fokker-Planck equation (19),
or the McNamara-Young model for granular media (see [70] and Section 2 in Chapter 2E).
The lack of gradient flow structure contributes to the mathematical difficulty of the
Boltzmann equation.
2.5. Long-time behavior
Assume that B(v - v,, cr) > 0 for almost all (v, v,, cr), which is always the case in
applications of interest. Then equality in Boltzmann's H theorem occurs if and only if
for almost all x, v, v,, cr,
f' f~ -- f f.. (53)
Under extremely weak assumptions on f [149,307,377], this functional equality forces f
to be a local Maxwellian, 36 i.e., a probability distribution function of the form
f (x, V) = p(X)
e-lV-U(X)12/2T(x)
(2:rrT (x)) N/2
Thus it is natural to guess that the effect of collisions is to bring f (t, .) closer and closer
to a local Maxwellian, as time goes by. This is compatible with Gibbs' lemma: among all
distributions on R N with given mass, momentum and energy, the minimum of the entropy
is achieved by the corresponding Maxwellian distribution.
35Except the inequality H(ft) <~liminfH(f~)/n .... By the way [149, pp. 99-100], the function ~b(X)=
X logX is, up to multiplication by a constant or addition of an affine function, the only continuous function
4~which satisfies the inequality f 4~(P1fn) ~ f dp(fn)/n for all fn,s, with equality for the tensor product
distribution.
36The result follows from the characterization of solutions to (43), but can also be shown directly by Fourier
transform as in [377]. However,no proof is moreenlighteningthanthe one due to Boltzmann (see Section4.3 in
Chapter 2C). His proof requires C1 smoothness,but Lions [307,p. 423] gavea beautiful proof that L1 solutions
to (53) have to be smooth. And anyway, one can always easily reduce to the case of smooth densities by the
remarks in Section4.7 of Chapter 2C.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-112-320.jpg)
![110 C. Villani
Of course, Equation (53) implies Q(f, f) = 0. As a corollary, we see that solutions of
the functional equation
Q(f, f) = o, (54)
where the unknown f(v) is a distribution function on ~N with finite mass and energy, are
precisely Maxwellian distributions.
As we have seen, local Maxwellian states are precisely those distribution functions for
which the dissipation of entropy vanishes. But if the position space X is a bounded domain
on R N (with suitable boundary conditions, like specular reflection, or Maxwellian re-
emission), then one can show that there are very few time-dependent local Maxwellian
distributions which satisfy the Boltzmann equation. Except in particular cases (domains
with symmetries, see [254]), such a solution has to take the form
e-lv-ulZ/2T
f (t, x, v) = p (2JrT)N/2 , (55)
for some parameters p, u, T which depend neither on t nor on x. A state like (55) is called a
global Maxwellian, or global equilibrium state. It is uniquely determined by its total mass,
momentum and energy.
The problem of the trend to equilibrium consists in proving that the solution of the
Cauchy problem (38) converges towards the corresponding global equilibrium as t --+ +c~,
and to estimate, in terms of the initial datum, the speed of this convergence. On this subject,
the (now outdated) paper by Desvillettes [168] accurately surveys existing methods up to
the beginning of the nineties. Since that time, new trends have emerged, with the research
for constructive estimates and the development of entropy dissipation techniques. To briefly
summarize recent trends, we should say that the problem of trend to equilibrium has
received rather satisfactory answers in situations where the Cauchy problem is known to
have well-behaved solutions. We shall make a detailed review in Chapter 2C.
On the other hand, if more complicated boundary conditions are considered, it may
happen that the global equilibrium be no longer Maxwellian; and that the mere existence
of a global equilibrium already be a very difficult problem. 37
Also, when the gas gets dispersed in the whole space, then things become complicated:
in certain situations, and contrary to what was previously believed by many authors,
solutions to the Boltzmann equation never get close to a local Maxwellian state [383,419,
327]. The main physical idea behind this phenomenon is that the dispersive effects of the
transport operator may prevent particles to undergo a sufficient number of collisions. In the
whole space setting, the relevant problem is therefore not trend to equilibrium, but rather
dispersion: find estimates on the speed at which the gas is dispersed at infinity. We shall
not develop on this problem; to get information the reader may consult Perthame [376] (see
also [277] for similar estimates in the context of the Vlasov-Poisson equation). Dispersion
estimates play a fundamental role in the modern theory of the Schr6dinger equations, and
there is a strong analogy with the estimates appearing in this field [135].
37 Seethe referencesin Section2.2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-113-320.jpg)
![112 C. Villani
If f solves the Boltzmann equation, then f~ solves the rescaled Boltzmann equation
0fe 1
~ + V . Vx fe = -Q(fe, fE).
Ot s
Hence the role of the macroscopic parameter s is to considerably enhance the role of
collisions. In view of the H theorem, one expects fs to resemble more and more a local
Maxwellian when s --+ 0: this is the assumption of local thermodynamical equilibrium,
whose mathematical justification is in general a delicate, still open problem.
Here is an equivalent, nonrigorous way of seeing the limit: the time scale of trend to
local equilibrium should be of the order of the mean time between collisions, which should
be much smaller than the macroscopic time.
For fixed s, the macroscopic quantities (density, momentum, temperature) associated to
fe via (1) satisfy the equations
Ops
-ST + v~. (p~u~) = o,
3t +Vx. u f~v | vdv =0,
-~(p~lu, + NpsTe) + Vx . felvl 2 ray =0.
N
The assumption of local thermodynamical equilibrium enables one to close this system in
the limit s ~ 0, and to formally obtain
Op
-~ + v~ . (pu)- o,
O(pu) + 7x 9(pu | u + pTIN) = 0
Ot
0
~(plul 2 + Npr) + Vx . (plulZu + (N + 2)pru) = 0
(56)
with IN standing for the identity N • N matrix. System (56) is nothing but the system of
the compressible Euler equations, when the pressure is given by the law of perfect gases,
p = pT. See [431].
Other scalings are possible, and starting from the Boltzmann equation one can get many
other equations in fluid mechanics [56]. In particular, by looking at perturbations of a
global equilibrium, it is possible to recover Navier-Stokes-type equations. This is one of
several possible ways of interpreting the Navier-Stokes equation, see [467] for remarks
about other interpretations. From a physicist's point of view, the interesting aspect of this
limit is the appearance of the viscosity from molecular dynamics. From a mathematician's
perspective, another interesting thing is that there are some well-developed mathematical
theories for the Navier-Stokes equation, for instance the famous theory of weak solutions
by Leray [299-301], see Lions [313,314] for the most recent developments- so one can
hope to prove theorems!](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-115-320.jpg)
![A review of mathematical topics in collisional kinetic theory 113
An interesting remark, due to Sone and coworkers [408,407], shows that sometimes a
hydrodynamic equation which looks natural is actually misleading because some kinetic
effects should have an influence even at vanishing Knudsen number; this phenomenon was
called "ghost effect".
There are formal procedures for "solving" the Boltzmann equation in terms of a
series expansion in a small parameter e (like our e above), which are known as Hilbert
and Chapman-Enskog expansions. These procedures have never received a satisfactory
mathematical justification in general, but have become very popular tools for deriving
hydrodynamical equations. This approach is described in reference textbooks such as [154,
430,141,148,48], and particularly [250, Section 22 and following]. But it also underlies
dozens of papers on formal hydrodynamical limits, which we do not try to review. By the
way, it should be pointed out that equations obtained by keeping "too many" (meaning 3
or 4) terms of the Hilbert or Chapman-Enskog series, like the so-called Burnett or super-
Burnett equations, seem to be irrelevant (a discussion of this matter, an ad hoc recipe to fix
this problem, and further references, can be found in Jin and Slemrod [282]). Also, these
expansions are not expected to be convergent, but only "asymptotic". In fact, a solution of
the Boltzmann equation which could be represented as the sum of such a series would be a
very particular one4~ it would be entirely determined by the fields of local density, mean
velocity and pressure associated with it.
All these problems illustrate the fact that the Hilbert and Chapman-Enskog methods rely
on very sloppy grounds. Rather violent attacks on their principles are to be found in [430].
In spite of this, these methods are still widely used. Without any doubt, their popularity lies
in their systematic character, which enables one to formally derive the correct equations in
a number of situations, without having to make any guess.
An alternative approach, which is conceptually simpler, and apparently more effective
for theoretical purposes, is a moment-based procedure first proposed by Grad [249]. For
Maxwell molecules, this procedure leads to hydrodynamic equations whose accuracy could
be expected to be of high order; the method has been developed in particular by Truesdell
and collaborators [274,430] under the name of "Maxwellian iteration".
The problem of rigorous hydrodynamical limits has been studied at length in the
literature, but most of these results have been obtained in a perturbative setting. Recently,
some more satisfactory results (in the large) have been obtained after the development of a
spectacular machinery (see [441] for a presentation). We shall discuss a few references
on both lines of approach in Section 5, as we shall go along the presentation of the
mathematical theories for the Cauchy problem.
On the other hand, the Boltzmann equation, as a model, does not capture the full range
of hydrodynamical equations that one would expect from dynamical systems of interacting
particles. At the level of the Euler equation above, this can be seen by the fact that the
pressure law is of the form p = p T. See [363,397,467] for more general equations and
partial results on the problem of the direct derivation of hydrodynamical equations from
particle systems. This however does not mean that it is not worth working at the level of
the Boltzmann equation; first because the limit should be simpler to rigorously perform,
than the limit for "raw" particle systems; secondly because the Boltzmann equation is one
40"Normal"in the terminologyofGrad[250].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-116-320.jpg)
![114 c. Villani
of the very few models of statistical physics which have been derived from mechanical
first principles. This kind of preoccupations meets those expressed by Hilbert in the
formulation of his sixth problem41 about the axiomatization of physics" can one put the
equations of fluid mechanics on a completely rigorous basis, starting from Newton's laws
of microscopic motion?
2.7. The Landau approximation
This problem occurs in the kinetic theory of dilute plasmas, as briefly described in
Section 1.7. Starting from the Boltzmann equation for screened Coulomb potential (Debye
potential), can one justify the replacement of this operator by the Landau operator (26) as
the Debye length becomes large in comparison with the space length scale? This problem
can easily be generalized to an arbitrary dimension of space.
A few remarks are in order as regards the precise meaning of this problem. First of all,
this is not a derivation of the Landau equation from particle systems (such a result would be
an outstanding breakthrough in the field). Instead, one considers the Boltzmann equation
for Debye potential as the starting point. Secondly, if one wants to stick to the classical
theory of plasmas, then
- either one neglects the effects of the mean-field interaction; in this case the problem
has essentially been solved recently, as we shall discuss in Section 5 of Chapter 2B;
- or one takes into account this effect, and then the Landau equation should only
appear as a long-time correction to the Vlasov-Poisson equation. To the best of our
knowledge, no such result has ever been obtained even in simplified regimes.
2.8. Numerical simulations
The literature about numerical simulation for the Boltzmann equation is considerable. All
methods used to this day consider separately the effects of transport and collision. This
splitting has been theoretically justified by Desvillettes and Mischler [178], for instance,
but a thorough discussion of the best way to implement it seems to be still lacking.
Dominant methods are based on Monte Carlo simulation, and introduce "particles"
interacting by collisions. Of course, transport is no problem for a particles-based method:
just follow the characteristics, i.e., the trajectories of particles in phase space. Then one
has to implement the effect of collisions, and many variants are possible. Sometimes the
dynamics of particles obey the Newton laws of collision only on the average, in such a
way that their probability density still comply with the Boltzmann dynamics. We do not
try to review the literature on Monte Carlo simulation, and refer to the very neat survey in
Cercignani [148, Chapter 7], or to the review paper [136] by the same author. Elements of
the theoretical justification of Monte Carlo simulation, in connection with chaos issues,
are reviewed in Cercignani et al. [149, Chapter 10], Pulvirenti [394] and Graham and
Mrlrard [257].
41Directlyinspiredby Boltzmann'streatise,amongotherthings.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-117-320.jpg)
![A reviewof mathematicaltopicsin collisionalkinetictheory 115
Let us however say a few words about deterministic methods which have emerged
recently, thanks to the increase of computational capacity.42 For a long time these methods
were just unaffordable because of the high computational cost of the (2N- 1)-fold
integral in the Boltzmann collision operator, but they are now becoming more and more
competitive.
Deterministic schemes based on conservation laws have been devised in the last years
by Buet, Cordier, Degond, Lemou, Lucquin [106,297,105]. In these works, the simulated
distribution function is constrained to satisfy conservation of mass, momentum and energy,
as well as decreasing of entropy. This approach implies very clever procedures, in particular
to handle the discretization of the spheres appearing in the Boltzmann representation
(this problem is pretty much the same than the consistency problem for discrete velocity
models). For variants such as the Landau equation, these difficulties are less important, but
then one has to hunt for possible undesirable symmetries, which may introduce spurious
conservation laws, etc.
Another deterministic approach is based on Fourier transform, and has been developed
by Bobylev and Rjasanow, Pareschi, Perthame, Russo, Toscani [82,89,370,371,373,372].
We shall say a little bit more on these schemes in Section 4.8, when introducing Fourier
transform tools.
At the moment, both methods have been competing, especially in the framework of the
Landau equation [107,215]. It seems that spectral schemes are useful to give extremely
accurate results, but cannot beat conservation-based schemes in terms of speed and
efficiency. Certainly more is to be expected on the subject.
2.9. Miscellaneous
We gather here a few other basic questions about the Boltzmann equation, which, even
though less important than the ones we have already presented, do have mathematical
interest.
Phenomenological derivation of the Boltzmann equation. We have seen two ways to
introduce the Boltzmann equation: either by direct modelling assumptions (dilute gas,
chaos, etc.) or by rigorous theorems starting from particle systems. There is a third way
towards it, which is by making some "natural" phenomenological assumptions on the form
of the collision operator, and try to prove that these assumptions uniquely determine the
form of the collision operator. A classical discussion by Bobylev [84] exemplifies this point
of view. To roughly sum up the very recent work by Desvillettes and Salvarani [180], it is
shown that (essentially) the only smooth quadratic forms Q acting on probability densities
such that
(1) evolution by Otf = Q(f, f) preserves nonnegativity,
(2) Q has Galilean invariance,
(3) Q(M, M) = 0 for any Maxwellian M,
42Thesameincreaseofcomputationalcapacityseemsto makediscrete-velocitymodelslessandlessattractive.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-118-320.jpg)
![116 C. Villani
have to be linear combinations of a Boltzmann and a Landau operator. Note the inversion
of the point of view: in Section 2.4 we have checked that the form of Boltzmann's operator
imposed a Maxwellian form to an equilibrium distribution, while here we see that the
assumption of Maxwellian equilibrium states43 contributes to determine the structure of
the Boltzmann operator.
Image of Q. This problem is very simply stated: for any function f(v) with enough
integrability, we know that f Q(f, f)go(v)dv = 0 whenever qg(v) is a linear combination
of 1, vi (1 ~< i ~< N), Ivl2. Conversely, let h be a function satisfying f hq9 = 0 for the
same functions qg,is it sufficient to imply the existence of a probability distribution f such
that Q (f, f) = h ? This problem apparently has never received even a partial answer for a
restricted subclass of functions h.
Divergence form of Boltzmann's collision operator. Unlike several other operators,
particularly Landau or Fokker-Planck, the Boltzmann operator is not written in divergence
form, even though it is conservative. Physicists consider this not surprising, in as much as
Boltzmann's operator models sudden (as opposed to continuous) changes of velocities.
However, generally speaking, any function Q whose integral vanishes can be written as
the divergence of something. This writing is in general purely mathematical and improper
to physical interpretation. But a nice feature of Boltzmann's operator, as we have shown
in [448], is that an explicit and (relatively) simple expression exists. We obtained it by
going back to the physical interpretation in terms of collisions, as in Landau [291 ]. Since
there is no well-defined "flux", one is led to introduce fictitious trajectories, linked to the
parametrization of the pre-collisional variables. Here is one possible expression for the
flux, which is a priori not unique: Q(f, f) = -Vv 9J (f, f), where
J(f' f) = - fv-v,,o~)>o
dr, do9B(v - v,, o9)
fo
(V-V*,~
x dr f(v + ro9)f (v, + ro9)o9 (57)
,-v,vo-v)<O
dvo dr, f(vo)f(v,)
V -- Vo ) V -- Vo
xB Vo-V,, IV-Vol Iv-vo[ N" (58)
The Boltzmann operator can even be written as a double divergence,
02
Q(f' f) = Z.. OVi OVj
tj
Aij (f, f), Z Aii (f, f) = -Vv" JE (f, f),
i
for some explicit functionals Aij and jE, 1 <<.i, j ~ N. Each of the "fluxes" J, A j, jE is
related to a conservation law.
43Which can be taken as physically reasonable from outer considerations, like central limit theorem, etc.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-119-320.jpg)
![A reviewof mathematicaltopicsin collisionalkinetictheory 117
Besides casting some more light on the structure of Boltzmann's operator and its
relations with divergence operators, these formulas may be used for devising conservative
numerical schemes via the "velocity diffusion" method (implemented in works by Mas-
Gallic and others).
Eternal solutions. Let us introduce this problem with a quotation from Truesdell and
Muncaster [430, p. 191] about irreversibility: "In a much more concrete way than
Boltzmann's H theorem, [the time-behavior of the moments of order 2 and 3 for the
Boltzmann equation with Maxwellian44 molecules] illustrate[s] the irreversibility of the
behavior of the kinetic gas. This irreversibility is particularly striking if we attempt to trace
the origin of a grossly homogeneous condition by considering past times instead of future
ones. Indeed the magnitude of each component of [the pressure tensor] and [the tensor of
the moments of order 3] that is not 0 at t = 0 tends to cxz as t --+ -cxz. Thus any present
departure from kinetic equilibrium must be the outcome of still greater departure in the
past."
Appealing as this image may be, it is our conviction that it is actually impossible45 to
let t --+ -o~. More precisely, heuristic arguments in [450, Annex II, Appendix] make the
following conjecture plausible: let f = f (t, v) be a solution of the Boltzmann equation for
v 6 R N, t e •, with finite kinetic energy. Then f is a Maxwellian, stationary distribution.
In the case of the spatially inhomogeneous Boltzmann equation, the conjecture should
be reformulated by saying that f is a local Maxwellian (which would allow "travelling
Maxwellians", of the form M (x - vt, v), where M (x, v) is a global Maxwellian in the x
and v variables).
This problem, which is reminiscent of Liouville-type results for parabolic equa-
tions [468], may seem academic. But in [450, Annex II, Appendix] we were able to con-
nect it to the important problem of the uniformity of trend to equilibrium, and to prove
this conjecture for some simplified models; Cabannes [109,110] also was able to prove it
for some simplified discrete-velocity approximations of the Boltzmann equation. At this
stage we mention that the nonnegativity of the solution is fundamental, since nontrivial,
partially negative solutions may exist. By the way, the nonnegativity of the solutions to the
Boltzmann equation is also very important from the mathematical point of view: if it is not
imposed, then irrelevant blow-up of the distribution function may occur.
Very recently, Bobylev and Cercignani [81] made a considerable progress in this
problem, proving the conjecture for so-called Maxwellian collision kernels, under the
additional requirement that the solution has all of its moments finite for all time.
Simultaneously, they discovered some very interesting self-similar eternal solutions of
the Boltzmann equation with infinite kinetic energy, which may describe the asymptotic
behavior of the Boltzmann equation in some physical regimes where the energy is very
large.
44SeeSection3.
45Whichmaybe anevenmorestrikingmanifestationofirreversibility?](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-120-320.jpg)
![118 C. Villani
3. Taxonomy
After having introduced the main mathematical problems in the area, we are prepared for a
more precise discussion of Boltzmann-like models. This section may look daunting to the
non-specialist, and we shall try to spare him. The point is that there are, truly speaking, as
many different Boltzmann equations as there are collision kernels. Hence it is absolutely
necessary, before discussing rigorous results, to sketch a "zoological" classification of
these. For simplicity, and because this case is much better understood, we limit the
presentation to the classical, elastic case. For background about relativistic, quantum,
nonelastic collision kernels, see Section 1.6, Chapter 2E, or the survey papers [209]
and [146].
We shall conclude this section with a brief description of the main mathematical and
physical effects that the choice of the collision kernel should have on the properties of the
Boltzmann equation.
3.1. Kinetic and angular collision kernel
Recall that the collision kernel B = B(v - v,, ~r) = B(lv - v,I, cos0) [by abuse of
notations] only depends on Iv - v,I and cos0, the cosine of the deviation angle. Also
recall that for a gas of hard spheres,
B(Iv - v,l, cosO) -- glv - v,I, g>0,
while for inverse s-power forces, B factors up like
8(1 - o,I, coso) = - I)b(cosO), (59)
where ~(Izl) = Izl y, y = (s - 5)/(s - 1) in dimension N = 3, and b(cos0)sinN-2 0 ~,
KO -(l+v), v = 2/(s - 1) in dimension N = 3 also.
Exactly what range of values of s should be considered is by no means clear in the
existing literature. Many authors [111,18,170] have restricted their discussion to s > 3.
Klaus [288, p. 895] even explains this restriction by the impossibility of defining the
Boltzmann linearized collision operator for s ~<3. However, as we shall explain, at least a
weak theory of the Boltzmann equation can be constructed for any exponent s 6 (2, +cx~).
The limit value s = 2 corresponds to the Coulomb interaction, which strictly speaking does
not fit into the framework of the Boltzmann equation, as we have discussed in Section 1.7.
REMARK. What may possibly be true, and anyway requires clarification, is that the
derivation of the Boltzmann equation from particle systems may fail for s ~< 3, because
of the importance of the mean-field interaction. But even in this case, the Boltzmann
description of collisions should be rehabilitated in the investigation of the long-time
behavior.
Even though one is naturally led to deal with much more general collision kernels,
products like (59) are the basic examples that one should keep in mind when discussing](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-121-320.jpg)
![A review of mathematical topics in collisional kinetic theory 119
assumptions. By convention, we shall call q0 the kinetic collision kernel, and b the angular
collision kernel. We shall discuss both quantities separately.
From the mathematical point of view, the control of Boltzmann's collision operator is all
the more delicate that the collision kernel is "large" (in terms of singularities, or behavior
as Iv- v,I ~ c~). On the contrary, when one is interested in such topics as trend to
equilibrium, it is good to have a strictly positive kernel because this means more collisions;
then the difficulties often come from the vanishings of the collision kernel. In short, one
should keep in mind the heuristic rule that the mathematical difficulties encountered in the
study of the Cauchy problem often come from large values of the collision kernel, those
encountered in the study of the trend to equilibrium often come from small values of the
collision kernel.
3.2. The kinetic collision kernel
It is a well-established custom to consider the cases q~(Iv - v,I) = Iv - v,I • and to
distinguish them according to
9 9/> 0: hard potentials;
9 Y = 0: Maxwellian potentials;
9 Y < 0: soft potentials.
For inverse-power forces in dimension 3, hard potentials correspond to s > 5, soft
potentials to s < 5.
We shall stick to this convention, but insist that it is quite misleading. First of all, "hard
potentials" are not necessarily associated to an interaction potential! It would be better to
speak of "hard kinetic collision kernel". But even this would not be a neat classification,
because it involves at the same time the behavior of the collision kernel for large and
for small values of the relative velocity, which makes it often difficult to appreciate the
assumptions really needed in a theorem. Sometimes a theorem which is stated for hard
potentials, would in fact hold true for all kinetic collision kernels which are bounded below
at infinity, etc. As typical examples, trouble for the study of the Cauchy problem may arise
due to large relative velocities for hard potentials, or due to small relative velocities for soft
potentials ....
How positive may 9/be? For hard spheres, Y - 1, hence a satisfactory theory should be
able to encompass this case. In many cases one is able to treat Y < 2 or 9/~< 2, or even less
stringent assumptions.
Conversely, how negative may 9/be? Contrarily to what one could think, critical values
of the exponent s do not, in general, correspond to critical values of Y- As a striking
example, think of Coulomb potential (s = N - 1), which normally should correspond to
a power law Y = N/(N - 2) in dimension N. Besides the fact that this is meaningless
when N - 2, this exponent is less and less negative as the dimension increases; hence the
associated Cauchy problem is more and more easy because of the weaker singularity.
The following particular values appear to be most critical: 9/ = -2, Y = -N. The
appearance of the limit exponent -2 in the study of several mathematical properties [437,
247,248,446] has led us in [446] to suggest the distinction between moderately soft
potentials (-2 < 9/< 0) and very soft potentials (Y < -2). It is however not clear whether](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-122-320.jpg)
![120 C. Villani
the border corresponds to a change of mathematical properties, or just an increase in
difficulty.
Note that dimension 3 is the only one in which the Coulomb potential coincides with the
limit exponent -N, which makes its study quite delicate46 !
3.3. The angular collision kernel
We now turn to the angular collision kernel b(cos0) = b(k. or), k = (v - v.)/Iv - v.I.
First of all, without loss of generality one may restrict the deviation angle to the range
[0, zr/2], replacing if necessary b by its "symmetrized" version, [b(cos0) + b(cos(zr -
0))]10~<0~<~r/2. From the mathematical point of view, this is because the product ftf~
appearing in the Boltzmann collision operator is invariant under the change of variables
~r ~ -or; from the physical point of view this reflects the undiscernability of particles.
As mentioned above, for inverse-power law forces, the angular collision kernel presents
a nonintegrable singularity as 0 --+ 0, and is smooth otherwise. The fact that the collision
kernel presents a nonintegrable singularity with respect to the angular variable is not a
consequence of the choice of inverse-power forces, recall (15).
By analogy with the examples of inverse-power forces, one would like to treat the
following situations:
b(cos0) sinN-2 0 ~ KO -(l+v)
0~<v<2.
as 0 ~ 0, (60)
Grad's angular cut-off [250,141 ] simply consists in postulating that the collision kernel
is integrable with respect to the angular variable. In our model case, this means
fo
N_lb(k.cr)dff "-[sN-21 b(cosO)sinN-2OdO < 00. (61)
The utmost majority of mathematical works about the Boltzmann equation crucially rely
on Grad's cut-off assumption, from the physical point of view this could be considered as
a short-range assumption.
3.4. The cross-section for momentum transfer
Let M(lv - v,I) be defined by
fs 1
U-, dcr B(v - v,, o')[v- v'] = -~(v- v,)M([v- v,[). (62)
46All the more that y = -3 also seems to have some special, bad properties independently of the dimension,
see [429].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-123-320.jpg)
![A review of mathematical topics in collisional kinetic theory 121
Indeed, by symmetry, the left-hand side is parallel to v - v,. The quantity M is called the
collision kernel for momentum transfer. In our model case, M(Iv - v,[) -- #q0(lv - v,]),
where
f0 f
/Z = [sN-2 [ b(cos0)(1 - cos0) sinN-2 0 d0. (63)
From the physical point of view, the cross-section for momentum transfer is one of the
basic quantities in the theory of binary collisions (see, for instance, [405,164]) and its
computation via experimental measurements is a well-developed topic.
On the other hand, the mathematical importance of the cross-section for momentum
transfer has not been explicitly pointed out until very recently. From the mathematical
point of view, finiteness of M (for almost all v - v,) is a necessary condition for the
Boltzmann equation to make sense [450, Annex I, Appendix A], in the sense that if it is
not satisfied, then Q(f, f) should only take values in {-oc, 0, +oc}. On the other hand,
together with Alexandre we have recently shown [12] that suitable assumptions on M are
essentially what one needs to develop a coherent theory for the Boltzmann equation without
Grad's angular cut-off assumption. Note that in our model case, finiteness of M requires
that b(cos0)(1 - cos 0), or equivalently b(cosO)O 2, be integrable on S N-1 . This precisely
corresponds to the range of admissible singularities v ~ [0, 2).
When v - 2, the integral (63) diverges logarithmically for small values of 0: this is
one reason for the failure of the Boltzmann model to describe Coulomb collisions. Very
roughly, the Debye truncature yields a finite/z, which behaves like the logarithm of the
Debye length; this is what physicists call the Coulomb logarithm. And due to the fact that
the divergence is only logarithmic, they expect the cross-section for momentum transfer to
depend very little on the precise value of the Debye length.
3.5. The asymptotics of grazing collisions
The mathematical links between the Boltzmann and the Landau collision operator can be
made precise in many ways. As indicated for instance by Degond and Lucquin [162], for
a fixed, smooth f, one can consider Q L(f, f) as the limit when e ~ 0 of a Boltzmann
collision operator for Coulomb potential, with a truncated angular collision kernel,
b~(cos0) -- log S-1 b(cos0) 10>e.
The factor log e-1 compensates for the logarithmic divergence of the Coulomb cross-
section for momentum transfer. As for the parameter e, it is proportional to the "plasma
parameter", which is very small for classical plasmas, and actually goes to 0 as the Debye
length goes to infinity.
Also in the case of non-Coulomb potentials can one define an asymptotic regime in
which the Boltzmann equation turns into a Landau equation. Such a formal study was](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-124-320.jpg)
![122 C.Villani
performed by Desvillettes [169]: he considered the limit of the Boltzmann operator when
the collision kernel is of the form
,
sinN-2 0 be (cos 0) -- ~ sinn-2 - b cos
6
These two asymptotic procedures seem very different both from the mathematical and
the physical point of view. Also, the truncation in the Coulomb case does not correspond
to the Debye cut: indeed, for the Debye interaction potential, the collision kernel does
not factor up. However, all limits of the type Boltzmann --+ Landau can be put into a
unified formalism, first sketched in our work [446], then extended and made more precise
in Alexandre and Villani [12]. The idea is that all that matters is that (1) all collisions
become grazing in the limit, (2) the cross-section for momentum transfer keep a finite
value in the limit. For simplicity we state precise conditions only in our model case where
the collision kernel factors up as q~(Iv - v,l)be(cosO):
u > 0, be(cos0) > 0 uniformly in 0 ~>00,
6-+0
f0 2"
lZe ~ [sN-21 be(cos0) (1- cos0) sinN-2 0d0 >/z.
e---~0
(64)
This limit will be referred to as the asymptotics of grazing collisions. It can be shown that
in this limit, the Boltzmann operator turns into the general Landau operator
QL(f,f)--Vv'(fRNdV, a(v--v,)[f,(Vf)-- f(Vf),]), (65)
ZiZj]
aij(z) -- tit(Izl) 6ij izl2 , (66)
taking into account the identity
Izl2~(Izl) (67)
q/(Izl) = 4(N- 1
-
-
-
-
-
-
-
-
-
-
-
~
According to (67), we shall use the terminology of hard, Maxwellian, or soft potentials
for the Landau operator depending on whether q'(Izl) in (66) is proportional to Izl•
with y > 0, y = 0, y < 0 respectively. Let us insist that the most relevant situation is
the particular case introduced in (26)-(27) to describe Coulomb collisions. The general
Landau operator (65) can be considered in several ways,
- either like an approximation of the Landau equation with Coulomb potential: from
the (theoretical or numerical) study of the corresponding equations one establishes
results which might be extrapolated to the Coulomb case;
- either as an approximation of the effect of grazing collisions in the Boltzmann
equation without cut-off: one could postulate that such an operator may be well
approximated by Q1 + Q2, where Q1 is a Boltzmann operator satisfying Grad's cut-
off assumption and Q2 is a Landau operator (see [141] for similar considerations);](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-125-320.jpg)
![A reviewof mathematical topicsin collisionalkinetictheory 123
- or like a mathematical auxiliary in the study of the Boltzmann equation. This point of
view will prove useful in Chapter 2C.
To conclude this section, we mention that limits quite similar to the asymptotics of
grazing collisions appear in the study of kinetic equations modelling quantum effects such
as the Bose-Einstein condensation (like the Kompaneets equation, see Chapter 2E), as can
be seen in the recent work of Escobedo and Mischler [209].
3.6. What do we care about collision kernels ?
We shall now try to explain in a non-rigorous manner the influence of the collision kernel
(or equivalently, of the cross-section) on solutions to the Boltzmann equation. By the
way, it is a common belief among physicists that the precise structure of the collision
kernel (especially the angular collision kernel) has hardly any influence on the behavior of
solutions. Fortunately for us mathematicians, this belief has proven to be wrong in several
respects. We make it clear that the effects to be discussed only reflect the influence of the
collision kernel, but may possibly come into conflict with boundary condition effects, 47 for
instance. We also point out that although some illustrations of these effects are known in
many regimes, none of them has been shown to hold at a satisfactory degree of generality.
Distribution tails. First of all, let us be interested in the behavior of the distribution
function for large velocities: how fast does it decay? can large distribution tails occur?
The important feature here is the behavior of the kinetic collision kernel as the relative
velocity Iv - v, I goes to infinity. If the collision kernel becomes unbounded at infinity,
then solutions should be well-localized, and automatically possess finite moments of very
high order, even if the initial datum is badly localized. 48 On the other hand, if the collision
kernel decreases at infinity, then a slow decay at infinity should be preserved as time goes
by.
This effect is best illustrated by the dichotomy between hard and soft potentials. In
certain situations, it has been proven that for hard potentials, no matter how slowly the
initial datum decays, then at later times the solution has finite moments of all orders. For
soft potentials on the other hand, a badly localized initial datum leads to a badly localized
solution at later times. Precise statements and references will be given in Chapter 2B.
As a consequence, it is also expected that the trend to equilibrium be faster for hard
potentials than for soft potentials. We shall come back to this in Chapter 2C.
Regularization effects. We now turn to the smoothness issue. Two basic questions are in
order: if the initial datum is smooth, does it imply that the solution remains smooth? If
the initial datum is nonsmooth, can however the solution become smooth? This time the
answer seems strongly dependent on the angular collision kernel. If the angular collision
kernel is integrable (Grad's cut-off assumption), then one expects that smoothness and
47In particular, it is not clear for us whether specular reflection in a non-convex domain would not entail
appearanceof singularities.Thetroublescausedby non-convexdomainshavebeenwell-studiedin lineartransport
theory,but not, to our knowledge,in the contextof the Boltzmannequation.
48See the heuristic explanationsin Section2.2 of Chapter 2B.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-126-320.jpg)
![124 C. Villani
lack of smoothness are propagated in time. In other words, the solution at positive times
should have precisely the same smoothness as the initial datum. The understanding of this
property has made very significant progress in the past years.
On the other hand, when the collision kernel presents a nonintegrable singularity,
then the solution should become infinitely smooth for positive times, as it does for
solutions to the heat equation. This idea emerged only in the last few years, and its very
first mathematical implementation was done by Desvillettes [171] for a one-dimensional
caricature of the Boltzmann equation. Since then this area has been very active, and
nowadays the regularization effect begins to be very well understood as well. Yet much
more is to be expected in this direction.
Effects of kinetic singularity. If we summarize our classification of collision kernels, there
are only three situations in which they can become very large: small deviation angles (as
illustrated by the cut-off vs. non-cutoff assumption), large relative velocities (as illustrated
by hard vs. soft potentials), and small relative velocities (as illustrated by hard vs. soft
potential, but in the reverse way). If the influence of the former two is now fairly well
understood, it is not so at all for the latter. It is known that a singularity of the collision
kernel at small relative velocities is compatible with propagation of some smoothness, but
no one knows if it preserves all the smoothness or if it entails blow-up effects in certain
norms, or conversely regularization phenomena. In Chapter 2E, we shall say a little bit
more on this issue, so far inexistent, and which we believe may lead to very interesting
developments in the future.
4. Basic surgery tools for the Boltzmann operator
Here we describe some of the most basic, but most important tools which one often needs
for a fine study of the Boltzmann operator,
Q(f, f) = ~ dv. fs
N N-1
da B(v - v,, a)(f' ff, - f f,).
Later on, we shall describe more sophisticated ingredients which apply in specific
situations.
4.1. Symmetr&ation of the collision kernel
In view of formulas (5), the quantity
f'f',-ff,
is invariant under the change of variables cr --~ -cr. Thus one can replace (from the very
beginning, if necessary) B by its "symmetrized" version
-B(z, or)- [B(z, or) + B(z,-a)] lz.~>0.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-127-320.jpg)
![A review of mathematical topics in collisional kinetic theory 125
In other words, one can always assume the deviation angle 0 to be at most zr/2 in
absolute value. This is why all spherical integrals could be written with an angular variable
going from 0 to zr/2, instead of zr. From the physical point of view, this constatation rests
on the undiscernability of particles (and this principle does not hold for mixtures). From
the mathematical point of view, this trick is very cheap, but quite convenient when one
wants to get rid offrontal collisions (deviation angle close to zr, which almost amounts to
an exchange between the velocities).
4.2. Symmetric and asymmetric point of view
There are (at least) two entirely different ways to look at the Boltzmann operator Q(f, f).
The first is the symmetric point of view: the important object is the "tensor product"
f f, - f | f, and the Boltzmann operator is obtained by integrating (f @ f)(v', v~,)-
(f | f)(v, v,) with respect to the variable v, and the parameter or. This point of view is
often the most efficient in problems which have to do with the trend to equilibrium, because
the H theorem rests on this symmetry.
On the other hand, one can consider Q(f, f) as the action upon f of a linear operator
which depends on f: Q(f, f)= l~f(f). This introduces an asymmetry between f,
(defining the operator) and f (the object on which the operator acts). This point of view
turns out to be almost always the most effective in a priori estimates on the Boltzmann
equation.
For many asymmetric estimates, it is important, be it for the clarity of proofs or for the
methodology, to work with the bilinear (but not symmetric!) Boltzmann operator
Q(g' f) -- f•u dv*fsu-, dcr B(v - v,, cr)(g',f' - g, f). (68)
Note that we have reversed the natural order of the arguments to make it clear that Q(g, f)
should be understood as s (f) ....
4.3. Differentiation of the collision operator
The following simple identities were proven in Villani [445] (but certainly someone had
noticed them before):
V Q+ (g, f) = Q+ (Vg, f) + Q+ (g, v f). (69)
These formulas enable one to differentiate the collision operator at arbitrary order via a
Leibniz-type formula.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-128-320.jpg)
![126 C. Villani
4.4. Joint convexity of the entropy dissipation
Remarkably, Boltzmann's entropy dissipation functional
if
D(f)- -4 dvdv, da B(v - v,,a)(f' f~ - ff,) log ~
f'f',
ff,
is a convex functional of the tensor product f f, - but not a convex functional of f!
This property also holds for Landau's entropy dissipation, which can be rewritten as
if
DL(f)- -~ dvdv, q~(Iv- v,I)I/7(v- v,)(V- V,)(ff,) [2
ff,
so that convexity of DL results from convexity of the function (x, y)~ [xl2/y in
~N X]~+.
Such convexity properties may be very interesting in the study of some weak limit
process, because weak convergence is preserved by tensor product. But beware! f f, is a
tensor product only with respect to the velocity variable, not with respect to the x variable.
4.5. Pre-postcollisional change of variables
A universal tool in the Boltzmann theory is the involutive change of variables with unit
Jacobian49
' k)
(v, v,, a) ~ (v', v,, , (70)
where k is the unit vector along v - v,,
k
l) m U,
[v - - V,[
Since a (vt I vt
= - v,)/[ - v,[, the change of variables (70) formally amounts to the
exchange of (v, v,) and (vt, v~,).As a consequence, under suitable integrability conditions
on the measurable function F,
f F(v, v,, v', v',)B(lv- v,l,k.a)dvdv, da
-f
-f
F(v, v,, v', vt,)B(lv - v,[, k. a)dr' dye,dk
!
F(v', v,, v, v,)B(Iv- v,l,k.a)dvdv, da.
49A way to see that this change of variables has unit Jacobian is to use the w-representation of Section 4.6. In this
representation, very clearly the pre-postcollisional change of variables has unit Jacobian; and the Jacobian from
the a-representation to the co-representation is the same for pre-collisional and for post-collisional velocities.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-129-320.jpg)
![A review of mathematical topics in collisional kinetic theory 127
Here we have used Iv' '
- v,I- Iv- v,I, a .k -- k.a to keep the arguments of B unchanged;
also recall the abuse of notations B(v - v,, a) = B(Iv - v,I, k. a).
!
Note that the change of variables (v, v,) --->(v~, v,) for given a is illegal V
4.6. Alternative representations
There are other possible parametrizations of pre- and post-collisional velocities. A very
popular one is the co-representation,
v' = v - (v - v,, o9)o9,
!
V, -- V, + (V -- V,, CO) CO, COE S N-1 . (71)
In this representation, the bilinear collision operator5~ reads
1/
Q(g, f) - -~ dr, do) B(v - v,, co)(g~,f' - g, f), (72)
where
- B (z, co) --
2
N-2
B(z,a).
!
We have kept a factor 1/2 in front of B to recall that each pair (v~, v,) corresponds to two
distinct values of co.
One of the advantages of the co-representation is that it is possible to change variables
l
(v, v,) +, (v~, v,) for fixed co and this is again an involutive transformation with unit
Jacobian. Another advantage is that it is a linear change of variables. Yet, as soon as one
is interested in fine questions where the symmetries of the Boltzmann operator play an
important role, the o--representation is usually more convenient.
A third representation is the one introduced by Carleman [119], particularly useful for
the study of the gain operator Q+ when the collision kernel satisfies Grad's angular cut-
the
off. The principle of Carleman's representation is to choose as new variables v~and v,,
I
pre-collisional velocities. Of course, not all values of v~and v, are admissible. If v and v~
is the hyperplane Evv,, orthogonal to
are given, then the set of admissible velocities v,
v - v~and going through v. Using the identity v - v, -- 2v - v~- v,,~one gets
Q(g, f)
1 , , -v,
- dv ~ dye, B 2v-v -v,,
N IV l)tl N- , IVt t
vv t -- __ V,I
x [g(v~,)f(v ') - g(v' + v~, - v)f(v)]. (73)
50All the representations formulas below for Q also work just the same for its gain and loss terms (Q + and Q- )
separately, with obvious changes.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-130-320.jpg)
![128 C. Villani
To conclude this section, we mention Tanaka's representation [415], which is equivalent
to Maxwell's weak formulation:
Q+ (g, f) = f dv dr, g, f (Fl~v,v, - 17v,v,), (74)
where Fly,v, (resp./7v~,v,) is a measure on the sphere S N-l, l~v,v, = B(v - v,, o")do"6v
(resp. Fl~,v, = B(v - v,, o") do" av,).
4.7. Monotonicity
Each time one has to handle an expression involving a nonnegative integrand and the
collision kernel, it may be useful to consider it as a monotone function of the collision
kernel. This point of view is particularly interesting for the entropy dissipation (47), which
obviously is an increasing function of the collision kernel. Therefore, to bound (47) from
below for a given collision kernel B, it is sufficient to bound it below for an auxiliary,
simplified collision kernel B0 such that B ~>B0.
Most of the time, the "simplified" collision kernel will be a Maxwellian one. As we shall
see in Chapter 2D, Maxwellian collision kernel have specific properties.
4.8. Bobylev's identities
We now turn to more intricate tools introduced by Bobylev. Even though the Boltzmann
operator has a nice weak formulation (Maxwell's formula in Section 2.3), it is a priori quite
painful to find out a representation in Fourier space. It turns out that such a representation
is not so intricate, at least when the collision kernel is Maxwellian! This fact was first
brought to the attention of the mathematical community by Bobylev, who was to make
Fourier transform an extremely powerful tool in the study of the Boltzmann operator with
Maxwellian collision kernel (see the review in [79]).
Here is Bobylev's identity: let b(cos 0) be a collision kernel depending only on the cosine
of the deviation angle, and let
Q(f' f)-- ~u •
dr, do" b(cosO)[f' f~, - f f,]
be the associated Boltzmann operator. Then its Fourier transform is
:)d: (75)
where f stands for the Fourier transform of f, ~ is the Fourier variable, and
~i_ ~ + I~1~
- --------~--- 9 (76)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-131-320.jpg)
![A reviewof mathematicaltopicsin collisionalkinetictheory 129
Note that I~+ 124- 1~-12 -- 1.
A remarkable feature about (75) is that the integral is now (N - 1)-fold, instead of
(2N - 1)-fold. This formula is actually a particular case of a more general one which does
not assume Maxwellian collision kernel [10, Appendix]"
1 , )
f'[Q(g, f)](,e) _ (2n') N/2 N• , ]-~ 9O" [~(,e- 4- ~,)f(~+ _ ~,)
-- i(~,)f(~ -- ~,)] d~, dcr, (77)
A
where the Fourier transform B of B = B(Iz[, cos0) is with respect to the variable z
only. Of course, in the particular case B(Izl, cos0) = b(cos0), we have B(I~.I, cos0) --
(27r)N/26[~. -- 0] b(cos0), and this entails formula (75). Thus we see that the reduction of
the multiplicity in the integral is directly linked to the assumption of Maxwellian collision
kernel.
As a consequence of (75), results Bobylev's lemma 51" if Q is a Boltzmann operator with
Maxwellian collision kernel, then, whatever the Maxwellian probability distribution M,
Q(g 9 M, f 9 M) -- Q(g, f) 9 M.
This is a very useful regularization lemma when dealing with Maxwellian collision kernels.
4.9. Application of Fourier transform to spectral schemes
Here we digress a little bit to briefly discuss numerical schemes based on Fourier transform,
which are related to Bobylev's ideas. Here are the main ideas of these "spectral schemes":
(1) truncate the support of the distribution function f, then extend f into a periodic
function on ]I~N;
(2) expand f in Fourier series, and compute the expression of the collision operator
Q(f, f) in terms of the Fourier coefficients of f.
Special attention must be given to the way the support is truncated! As explained
in [372], for instance, if the support of f is reduced to a compact set with diameter R,
then it should be extended by periodicity with period T ~> (2 4- ~/2)R, in order to avoid
overlap problems in the computation of the collision integral. Assume, for instance, T -- Jr,
R = Mr, )~= 2/(3 + ~/2).
After passing in Fourier representation,
m
(27r)N ~,Yr] N f (v)e -ik'v dr,
51Thislemmawas actuallyproven,for aconstantcollisionkernel,byMorgenstern[351,Section 10] in the fifties!
However,Bobylevwas the author who devised a generalproof, madethis lemmawidely known and linked it to
otherpropertiesof Maxwellian collisionkernels.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-132-320.jpg)
![130 C. Villani
and truncating high Fourier modes, a very simple expression is obtained for the k-th mode
Q(k) of Q(f, f)"
Q'(k) = Z fg J~/~(g' m), Ikl ~<K,
g§ Igl,lml <<.
K
where
/~(g, m) -- fzl~2)vTr
dzfSN-1do"B(z, o")[e i~'z++im'z- - eiz'm],
zi z+lzl~
= ~ . (78)
2
Of course, this formula is very much reminiscent of (77). Spectral schemes have several
advantages: once in Fourier space, the numerical simulation is immediate Oust a simple
first-order system of coupled ODE's). Moreover, all coefficients/~(g, m) can be computed
once for all with extreme precision; this may demand some memory space, but is quite
a gain in the speed of the computation. They are able to give extremely precise results.
However, these schemes are rather rigid and do not allow for all the numerical tricks which
can be used by other methods to reduce computational time. Moreover, since conservation
laws are not built in the schemes, numerical simulations have to be conducted with a certain
minimal precision in order to get realistic results.
From the mathematical point of view, this method presents very interesting features,
think that it simulates the Boltzmann equation in weak formulation. In particular, the
method works just the same with or without Grad's angular cut-off assumption. This can
be used in theoretical works like [374] (analysis of the behavior of the method in the
asymptotics of grazing collisions).
5. Mathematical theories for the Cauchy problem
In this section, we try to survey existing mathematical frameworks dealing with the Cauchy
problem (and also more quantitative issues) for Boltzmann-like equations. These theories
are all connected, but more or less tightly. Before this, in the first section we describe the
most apparent problems in trying to construct a general, good theory.
5.1. What minimalfunctional space ?
In the full, general situation, known a priori estimates for the Boltzmann equation are only
those which are associated to the basic physical laws:
9 formal conservation of mass and energy,
9 formal decrease of the entropy.
Of course, the latter means two estimates! an estimate on the entropy, and another one on
the dissipation of entropy.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-133-320.jpg)
![A reviewof mathematicaltopicsin collisionalkinetictheory 131
When the position space is unbounded, say the whole of ~N, then the properties
of the transport operator also make it rather easy to get local (in time) estimates on
f f (t, x, v)Ix 12dx all3. For this it suffices to use the identity
df
dt f (t, x, v)lx - vtl 2 dx dv - O.
On the whole, one gets for free the a priori estimates
finN • RN
f(t,x, v)[1 + Ivl2 -t-Ixl 2 -t-logf(t,x, v)]dxdv
fots
+ D(f(r, x, .)) dr dx
N
fRN •
f(O,x, v)[1 + 21xl2 + (2t 2 + 1)lvl 2 + log f(0, x, v)]dxdv. (79)
Apart from the term in D(f), estimate (79) does not depend on the collision kernel B.
Disregarding this entropy dissipation estimate which is difficult to translate in terms of size
and smoothness of the distribution function, all that we know for the moment about f is
f E L~ T]; L~(IRN x NN) n LlogL(R N x IKf)).
Here
IIf IIL~ - fRN ~RN f(x, v)(1 + Ixl2 -+-Ivl2) dx dr.
It is easily seen that the estimate in L log L, combined with the moment estimate, entails
an L 1 control of f (log f)+.
REMARK. One may be interested in situations where the total mass is infinite (gas in
the whole space). The estimates can then be adapted to this situation: see in particular
Lions [310].
By the Dunford-Pettis compactness criterion, sequences of solutions to the Boltz-
mann equation with a uniform entropy estimate will be weakly compact, say in
N
L p ([0, T]; L l(lt{x
N x 11~
v )), and this ensures that their cluster points cannot be singular
measures.
This looks like a mathematical convenience: after all, why not try to handle the
Boltzmann equation in a genuine measure setting, which seems to best reflect physical
intuition? It turns out that the space of measures is not stable 52 - and thus irrelevant- for
the study of the Boltzmann equation. This can be seen by the following remark. Consider
52On the other hand, the space of functions with bounded entropy is stable, as shown by the works of DiPerna
and Lions discussed below.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-134-320.jpg)
![132 C. Villani
as initial datum a linear combination of Dirac masses approximating a continuous profile
(think of each of these Dirac masses as clusters of particles with a common velocity, each
cluster being located at a different position). Even though the meaning of the Boltzmann
equation for so singular data is not a priori clear, it is easy to figure out what weak
solution we should expect: each cluster should keep moving with constant velocity, until
two clusters happen to be in the same position and collide. Should this eventuality occur,
particles in each cluster would be scattered in all directions, with respective probabilities
given by the Boltzmann collision kernel. But the point is, by modifying very slightly
the initial positions of the clusters, one can always make sure (in dimension N/> 2)
that no collision ever occurs! Therefore, this example shows that one can construct a
sequence of weak solutions of the Boltzmann equation, converging in weak measure sense
towards a continuous solution of the free transport equation. Thus there is no stability in
measure space .... It would be tedious, but quite interesting, to extend this counterexample
to absolutely continuous initial data, say linear combinations of very sharply peaked
Maxwellians.
The aforegoing discussion is strongly linked to the fact that both x and v variables are
present in the Boltzmann equation. If one is interested in solutions which do not depend on
the space variable, then it is perfectly possible to construct a meaningful theory of measure
solutions, as first noticed by Povzner [389].
To come back to our original discussion, we have just seen that it is in general impossible
to deal with arbitrary measures, and therefore the entropy estimate is welcome to prevent
concentration. But is it enough to handle the Boltzmann operator? Very very roughly, it
seems that to control the Boltzmann operator, we would like to have a control of f dr, ff,,
which is just IIf IILLf. The L lx,v norm of this quantity is just the norm of f in L2(L 1). This
is the kind of spaces in which we would like to have estimates on f. But such estimates
are a major open problem! It seems that the entropy dissipation estimate is not sufficient
for that purpose, one of the troubles being caused by local Maxwellian states, which make
the entropy dissipation vanish but can have arbitrarily large macroscopic densities.
Thus the only a priori estimates which seem to hold in full generality do not even allow
us to give a meaningful sense to the equation we wish to study.., this major obstruction
is one of the reasons why the Cauchy problem for the Boltzmann equation is so tricky -
another reason being the intricate nature of the Boltzmann operator. Due to this difficulty,
most theories only deal with more or less simplified situations. On the other hand, for
simpler models like the BGK equation, in which the collision operator is homogeneous of
degree 1, the Cauchy problem is much easier [375,378,398].
By the way, the lack of a priori estimates for the Boltzmann equation is also very
cumbersome when dealing with boundary conditions, especially in the case of Maxwellian
diffusion. Indeed, the treatment of boundary conditions in a classical sense requires the
trace of the solution to be well-defined on the boundary, which is not trivial [271,149].
Weak formulations are available (see in particular Mischler [347]), but rather delicate.
We distinguish six main theories, inequally represented and inequally active - and by
no means hermetically independent, especially in recent research. By order of historical
appearance: spatially homogeneous theory, theory of Maxwellian molecules, perturbative
theory, solutions in the small, renormalized solutions, one-dimensional problems.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-135-320.jpg)
![A review of mathematical topics in collisional kinetic theory 133
5.2. The spatially homogeneous theory
In the spatially homogeneous theory one is interested in solutions f(t, x, v) which do
not depend on the x variable. This approach is rather common in physics, when it comes
to problems centered on the collision operator: this looks reasonable, since the collision
operator only acts on the velocity dependence. Moreover, the spatially homogeneous theory
naturally arises in numerical analysis, since all numerical schemes achieve a splitting of the
transport operator and the collision operator. Finally, it is expected that spatial homogeneity
is a stable property, in the sense that a weakly inhomogeneous initial datum leads to a
weakly inhomogeneous solution of the Boltzmann equation. Under some ad hoc smallness
assumptions, this guess has been mathematically justified by Arkeryd et al. [32], who
developed a theory for weakly inhomogeneous solutions of the full Boltzmann equation.
Thus, the spatially homogeneous Boltzmann equation reads
af = Q(f, f), (80)
ot
and the unknown f = f (t, v) is defined on IK+ • ]~N. Note that in this situation, the only
stationary states are Maxwellian distributions, by the discussion about Equation (54) in
Section 2.5.
The spatially homogeneous theory was the very first to be developed, thanks to the
pioneering works by Carleman [118] in the thirties. Carleman proved existence and
uniqueness of a solution to the spatially homogeneous problem for a gas of hard spheres,
and an initial datum f0 which was assumed to be radially symmetric, continuous and
decaying in O(1/Ivl 6) as Ivl --+ +c~. He was also able to prove Boltzmann's H theorem,
and convergence towards equilibrium in large time. Later he improved his results and
introduced new techniques in his famous treatise [119]. Then in the sixties, Povzner [389]
extended the mathematical framework of Carleman and relaxed the assumptions.
In the past twenty years, the theory of the Cauchy problem for spatially homogeneous
Boltzmann equation for hard potentials with angular cut-off was completely rewritten
and extensively developed, first by Arkeryd, then by DiBlasio, Elmroth, Gustafsson,
Desvillettes, Wennberg, A. Pulvirenti, Mischler [17,20,23,186,187,204,269,270,170,458,
456,457,392,393,349]. It is now in a very mature state, with statements of existence,
uniqueness, propagation of smoothness, moments, positivity .... Optimal conditions for
existence and uniqueness have been identified by Mischler and Wennberg [349,463].
Recent works by Toscani and the author [428] have led to almost satisfactory results about
the H theorem and trend to equilibrium, even though some questions remain unsettled.
Also, one can work in a genuine measure framework.
The study of singular kernels (be it soft potentials or potentials with nonintegrable
angular singularities) is much more recent [18,172,171,173,446,449,10]. This area is still
under construction, but currently very active.
In view of the last advances, it is quite likely that very soon, we shall have a fairly
complete picture of the spatially homogeneous theory, with or without cut-off, at least when
the kinetic collision kernel is not too singular for small relative velocities. A review on the
state of the art for the spatially homogeneous theory is performed in Desvillettes [174].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-136-320.jpg)
![134 C. Villani
A task which should be undertaken is to systematically extend all of these achievements to
the framework of weakly inhomogeneous solutions [32].
5.3. Maxwellian molecules
After spatial homogeneity, a further simplification is the assumption that the collision
kernel be Maxwellian, i.e., do not depend on the relative velocity but only on the (cosine
of the) deviation angle. The corresponding theory is of course a particular case of the
preceding one, but allows for a finer description and presents specific features. It was first
developed by Wild and Morgenstern in the fifties [464,350,351]. Then Truesdell [274]
showed that all spherical moments of the solutions could be "explicitly" computed. Simple
explicit solutions, important for numerical simulations, were produced independently by
Bobylev [78] and Krook and Wu.
Later, Bobylev set up and completed an ambitious program based on the Fourier
transform (see the survey in [79]). Included were the classification of several families of
semi-explicit solutions, and a fine study of trend to equilibrium. Key tools in this program
were the identities of Section 4.8. As of now, the theory can be considered as complete,
with the exception of some non-standard problems like the identification of the image of
Q or the classification of all eternal solutions53 ....
5.4. Perturbation theory
Another regime which has been extensively studied is the case when the distribution
function is assumed to be very close to a global Maxwellian equilibrium, say the centered,
unit-temperature Maxwellian M.
Under this assumption, it is natural to try to linearize the problem, in such a way that
quadratic terms in Boltzmann's operator become negligible. In order to have a self-adjoint
linearized Boltzmann operator, the relevant change of unknown is f = M(1 4- h). Then,
one can expand the Boltzmann operator Q(f, f) by using its bilinearity, and the identity
Q(M, M) = 0. This leads to the definition of the linearized Boltzmann operator,
Lh -- M-I[Q(M, Mh) 4- Q(Mh, M)].
L is a symmetric, nonpositive operator in L2(M) (endowed with the scalar product
(hi, h2)M -- f hlh2M), nonpositivity being nothing but the linearized version of the H
theorem. Moreover, the remainder in the nonlinear Boltzmann operator, Q(Mh, Mh), can
be considered small if h is very close to 0 in an appropriate sense. Note that smallness of
h in LZ(M) really amounts to smallness of f - M in L2(M-1), which, by the standards
of all other existing theories, is an extremely strong assumption!
The linearized Boltzmann equations also have their own interest, of course, and the
spectral properties of the linearized Boltzmann operator have been addressed carefully;
53Theseproblemshavebeenexplainedin Section2.9.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-137-320.jpg)
![A review of mathematical topics in collisional kinetic theory 135
see [141] for discussion and references. The different analysis required by the linearized
Landau equation is performed in [161,298]. A remarkable feature is that when the collision
kernel is Maxwellian, then the spectrum can be computed explicitly (and eigenfunctions
too: they are Hermite polynomials). This calculation was first performed in a classical
paper by Wang Chang and Uhlenbeck [454], then simplified by Bobylev [79, p. 135] thanks
to the use of Fourier transform.
Grad has set up the foundations for a systematic study of the linearized Boltzmann
equation, see [250,252,255]. At the same time he initiated in this perturbative framework
a "rigorous" study of hydrodynamical equations based on Chapman-Enskog or moments
expansion [249,253]. Later came the pioneering works of Ukai [434,435] on the nonlinear
perturbation [434,435], followed by a huge literature, among which we quote [333,359,
111,437,403,436,286]. This theory is now in an advanced stage, with existence and
uniqueness theorems, and results of trend to equilibrium. The proofs often rely on the
theory of linear operators, abstract theory of semigroups, abstract Cauchy-Kowalewski
theorems.
As we just said, with the development of this branch of the Cauchy problem
for the Boltzmann equation, came the first rigorous discussions on the transition to
hydrodynamical equations, in a perturbative framework, after a precise spectral analysis of
the linearized operator was performed. On this approach we quote, for instance, [112,113,
159,203,358,438,44,334,58,43,212,213 ]. Actually, in the above we have mixed references
dealing with stationary and with evolutionary problems, for which the setting is rather
similar .... An up-to-date account of the present theory can be found in [214]. However, all
known results in this direction deal with smooth solutions of the hydrodynamic equations.
Also related to this linearized setting is a large literature addressing more qualitative
issues, like half-space problems, to be understood as a modelization for kinetic layers,54
or the description of simple shock waves. Among many works, here are a few such papers:
[42,357,115,258,52,51,332,155,151,188,214] (the work [37] is an exception, in the sense
that it deals with the Milne problem in the fully nonlinear case).
It is certainly mathematically and physically justified to work in a perturbative setting,
as long as one keeps in mind that this only covers situations where the distribution function
is extremely close to equilibrium. Thus this theory can in no way be considered as a general
answer to the Cauchy problem.
Even taking this into account, some criticisms can be formulated, for instance, the use
of abstract spectral theory which often leads to nonexplicit results. Also, we note that the
great majority of these works is only concerned with hard potentials with cut-off, and most
especially hard spheres. Early (confusing) remarks on the Boltzmann operator without cut-
off can be found in Pao [369] and Klaus [288], but this is all. Soft potentials with cut-off
have been studied by Caflisch [111], Ukai and Asano [437].
54A kinetic layer describes the transitionbetween a domainof space where a hydrodynamicdescriptionis
relevant, and anotherdomainin which a moreprecisekinetic descriptionis in order;for generalbackground
see [148].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-138-320.jpg)
![136 C. Villani
Theoretical research in the area of linearized or perturbative Boltzmann equation is not
so intensive as it used to be.* We shall not come back to this theory and for more details
we address the reader to the aforementioned references.
5.5. Theories in the small
Another line of approach deals with short-time results. This may seem awkward for
statistical equations which are mostly interesting in the long-time limit! but may be
interesting when it comes to validation issues. Conversely- but this is more delicate-,
it is possible to trade the assumption of small time for an assumption of small initial datum
expanding in the vacuum. This case could be described as perturbation of the vacuum, and
exploits the good "dispersive" properties of the transport operator. The modern approach
starts with the classical papers of Kaniel and Shinbrot for the small-time result [285], Illner
and Shinbrot for the small-datum result [278].
Also it should be clear that Lanford's theorem55 contains a proof of local in time
existence under rather stringent assumptions on the initial datum. It was computed that
Lanford's bounds allowed about 15% of the particles to collide at least once! The
extensions of Lanford's results by Illner and Pulvirenti [275,276] were also adaptations
of small-datum existence theorems.
Theories in the small were further developed by Toscani and Bellomo [64,425,418-
420,368], some early mistakes being corrected by Polewczak [385] who also proved
smoothness in the x variable on this occasion. See the book of Bellomo, Palczewski and
Toscani [63] for a featured survey of known techniques at the end of the eighties. Then the
results were improved by Goudon [246,247] (introducing some new monotonicity ideas),
and also Mischler and Perthame [348] in the context of solutions with infinite total kinetic
energy.
One of the main ideas is that if the initial datum is bounded from above by a well-chosen
Maxwellian (or squeezed between two Maxwellians), then this property remains true for
all times, by some monotonicity argument. Therefore, solutions built by these methods
usually satisfy Gaussian-type bounds. To get an idea of the method, a very pedagogical
reference is the short proof in Lions [308] which covers smooth, fast-decaying collision
kernels.
Bellomo and Toscani have also studied cases where the decay of the solution is not
Gaussian, but only polynomial. It is in this framework that Toscani [419] was able to
construct solutions of the Boltzmann equation in the whole space, which do not approach
local equilibrium as time becomes large.56
From the mathematical point of view, these theories cannot really be considered in a
mature stage, due to a certain rigidity. For instance, it is apparently an open problem to
treat boundary conditions: only the whole space seems to be allowed. Also, since the
*Note addedinproof: Aftercompletionofthisreview,twoimportantworksbyGuo,abouttheLandauequation
and the Boltzmannequation for soft potentials, in a close-to-equilibrium, periodic setting, popped up just to
contradictthisstatement.
55See Section2.1.
56SeeSection2.5.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-139-320.jpg)
![A review of mathematical topics in collisional kinetic theory 137
proofs strongly rely on Grad's splitting between the gain and loss part, the treatment of
non-cutoff potentials is open. Finally, the limitations of small time, or small initial datum,
are quite restrictive, even though not so much as for the linearized theory (the bounds
here have the noticeable advantage to be explicit). However, as mentioned above, theories
in the small have led to two quite interesting, and physically controversial, results: the
validation of the Boltzmann equation for hard-spheres via the Boltzmann-Grad limit, at
least in certain cases; and a simple construction of solutions of the Boltzmann equation
which do not approach local equilibrium as time becomes large. Recent works by Boudin
and Desvillettes [101] in this framework also resulted in interesting proofs of propagation
of regularity and singularities, which were implicitly conjectured by physicists [148].
We shall not develop further on this line of approach, and address the reader to the above
references for more.
5.6. The theory of renormalized solutions
Introduced by DiPerna and Lions at the end of the eighties, this theory is at the moment
the only framework where existence results for the full Boltzmann equation, without
simplifying assumptions, can be proven [190,192,194,167,307-309,311,310,306,316,12,
13]. Apart from a high technical level, this theory mainly relies on two ingredients:
9 the velocity-averaging lemmas: under appropriate conditions, these lemmas, initiated
by Golse, Perthame and Sentis [243] and further developed in [242,232,195,379,
99,312,315], yield partial regularity (or rather regularization) results for velocity-
averages f g(t, x, v)qg(v) dv of solutions of transport equations Og/Ot + v. Vxg = S.
The regularity is of course with respect to the time-space variables, and thus the
physical meaning of these lemmas is that macroscopic observables enjoy some
smoothness properties even when the distribution function itself does not.
9 the renormalization: this trick allows one to give a distributional sense to the
Boltzmann equation even though there does not seem to be enough a priori estimates
for that. It consists in formally multiplying (8) by the nonlinear function of the density,
fll (f), where 13belongs to a well-chosen class of admissible nonlinearities. By chain-
rule, the resulting equation reads
Off(f)
Ot
+ v. Vxfl (f) = fl' (f) Q (f, f). (81)
Assume now that Ifi'(f)l ~ C/(1 -+-f), for some C > 0. Then, since the Boltzmann
operator Q(f, f) is quadratic, one may expect fl'(f)Q(f, f) to be a sublinear
operator of f... in which case the basic a priori estimates of mass, energy
and entropy57 would be enough to make sense of (81). Distribution functions
satisfying (81) in distributional sense are called renormalized solutions. Strictly
speaking, these solutions are neither weaker, nor stronger than distributional solutions.
Typical choices for/3 are fl (f) = 3-1 log(1 + 6f) (3 > 0) or/3 (f) = f~ (1 + 6f).
57See Section5.1.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-140-320.jpg)
![138 C. Villani
Apart from the study of the Boltzmann equation, renormalization and velocity-averaging
lemmas have become popular tools for the study of various kinetic equations [191,
375,318,176,229], ordinary differential equations with nonsmooth (Sobolev-regular)
coefficients [193,97], or the reformulation of some hyperbolic systems of conservation
laws as kinetic systems [319,320,280,381,380].* The idea of renormalizafion has even been
exported to such areas of partial differential equations as nonlinear parabolic equations
(see [76,77] and the references therein). In fact, renormalization is a general tool which
can be applied outside the field of renormalized solutions; in this respect see the remark at
the level of formula (130).
As regards the Boltzmann equation, many fundamental questions are still unsolved: in
particular uniqueness, propagation of smoothness, energy conservation, moment estimates,
positivity, trend to equilibrium .... Therefore, as of this date, this theory cannot be
considered as a satisfactory answer to the Cauchy problem. However, it provides a
remarkable answer to the stability problem.
The techniques are robust enough to adapt to boundary-value problems [271,144,
30,347,346] (be careful that some of the proofs in [271] are wrong and have been
corrected in [30]; the best results are those of Mischler [347]). As an important application
of the theory of renormalized solutions, Levermore [302] proved the validity of the
linearizafion approximation if the initial datum is very close to a global Maxwellian. Also
the hydrodynamical transition towards some models of fluid mechanics can be justified
without assumption of smoothness of the limit hydrodynamic equations: see, in particular,
Bardos, Golse and Levermore [57,55,54,53], Golse [239], Golse et al. [241], Golse and
Levermore [240], Lions and Masmoudi [317], Golse and Saint-Raymond [245], Saint-
Raymond [401]. The high point of this program is certainly the rigorous limit from the
DiPerna-Lions renormalized solutions to Leray's weak solutions of the incompressible
Navier-Stokes equation, which was performed very recently in [245]; see [441] for a
review.
The original theory of DiPerna and Lions heavily relied on Grad's cut-off assumption,
but recent progress have extended it to cover the full range of physically realistic collision
kernels [12]. This extended theory has set a framework for the study of very general effects
of propagation of "regularity", in the form of propagation of strong compactness [308], or
"smoothing", in the form of appearance of strong compactness [311,316,12,13].
Moreover, even if a uniqueness result is not available, it appears that renormalized
solutions are strong enough to prove some results of weak-strong uniqueness [308,324]:
under certain assumptions on the collision kernel, if we know that there exists a strong
solution to the Boltzmann equation, then there exists a unique renormalized solution, and
it coincides with the strong solution. On the occasion of this study, Lions [308] pointed
out the possibility to construct very weak solutions, called "dissipative solutions", which
are of very limited physical value, but have been used in various areas as a powerful tool
for treating some limit regimes, be it in fluid mechanics for such degenerate equations
as the three-dimensional Euler equation [313], in hydrodynamical limits [239,241,401 ] or
stochastic fully nonlinear partial differential equations [321,322]. Thus ramifications of the
DiPerna-Lions theory have been a source of inspiration for problems outside the field.
*Note added in proof: Recently,Bouchuthas shownhow to use velocity-averaginglemmasto studyclassical
hypoellipticityin certainkineticequations.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-141-320.jpg)
![A review of mathematical topics in collisional kinetic theory 139
In view of these achievements and of the current vitality of the theory of renormalized
solutions, we shall come back to it in more detail in the next chapters.
5.7. Monodimensional problems
It is of course impossible to speak of a monodimensional Boltzmann equation, since elastic
collisions are meaningless in dimension 1. But in many problems of modelling [148],
symmetry assumptions enable one to consider solutions depending on the position in space,
x, through only one variable. From the mathematical point of view, such problems seem to
present specific features, one of the reasons being that the dispersive power of the transport
operator is very strong in dimension 1, so that dispersion estimates can be used to (almost)
control the collision operator.
In the end of the eighties, Arkeryd [22] was able to apply a contraction method similar to
the one in [24] in order to get existence results for the Boltzmann equation in one dimension
of space, however he needed a physically unrealistic damping in the collision operator for
small relative velocities in the space direction.
Then, building on original works by Beale [61] and especially Bony [94,95] on
discrete-velocity Boltzmann equations, Cercignani [145,147] was able to extract some new
estimates in this one-dimensional situation, and prove existence of "strong" solutions to
the Boltzmann equation, under rather stringent assumptions on the collision kernel. Here
"strong" means that Q+(f, f) E L~oc(RX • RN). For some time this line of research was
quite promising, but it now seems to be stalled ....](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-142-320.jpg)
![Preface
The motion of fluids has intrigued scientists since antiquity and we may say that the
field of mathematical fluid dynamics originated more than two centuries ago. In 1755
Euler [2] gave a mathematical formulation of the principle of conservation of mass in
terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear
and angular momentum by PDEs. Material symmetry and frame invariance were used by
Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of
internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint-
Venant [11] and Stokes [12].
In the 19th century no sharp distinction was drawn between mathematicians and
physicists as we sometime see in more recent times. The formulation of the equations
of fluid motion could be considered as either mathematics or physics. The first work
in fluid dynamics that has a "modern" mathematical flavor may have been done by
Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous
problem. Riemann recognized the mathematical nature of the entropy. This notion led
him to his duality method for solving the non-characteristic Cauchy problem for linear
hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his
contemporaries. What we now call the Cauchy problem for a PDE and the search for its
solution did not have the significance that it is accorded nowadays. Even Poincar6 did not
raise that kind of question in his Th~orie des tourbillons.
For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly
accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the
Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28.
Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of
the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much
as Riemann, Leray developed new mathematical tools which proved to have independent
interest: e.g., weak solutions (that we now call Leray's solutions in this context) and
topological degree (a joint work with Schauder [4]).
Since the 1930s, the interest that mathematicians devote to fluid dynamics has
unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von
Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a
Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on
the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic
displays such a variety of models and questions that thousands of scientists, among them
many mathematicians, focus their research on fluid dynamics.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-144-320.jpg)
![A review of mathematical topics in collisional kinetic theory 143
The meaning of"Cauchy problem" in this chapter is to be understood in an extended sense:
we shall not only be concerned in existence and uniqueness of solutions, but also in a priori
estimates. Three main issues will be addressed: decay of the solutions at large velocities
(and also at large positions, but large velocities are the main concern), smoothness, and
strict positivity. As we explained above, the decay of the solutions mainly depends on
the behavior of the kinetic collision kernel, while their smoothness heavily relies on the
angular collision kernel. As for the strict positivity, the matter is not very clear yet.
We have adopted the following presentation: first, we recall a bit about velocity-
averaging lemmas, which have become a universal tool in the study of transport equations,
and we shall comment on their use in the particular context of the Boltzmann equation. In
Section 2, we address moment estimates, and discuss the influence of the kinetic collision
kernel. Then in Section 3, we first enter the core of the study of Boltzmann's operator, and
we discuss issues of propagation of smoothness and propagation of singularities when the
angular collision kernel is integrable (Grad's angular cut-off). Conversely, in Section 4, we
explain the structure of Boltzmann's operator when the angular collision kernel presents a
nonintegrable singularity for grazing collisions, and associated theorems of regularization.
Since the Landau equation is linked to the Boltzmann equation via the emphasis on grazing
collisions, this will lead us to discuss the Landau approximation in Section 5. We conclude
in Section 6 with lower bound estimates.
In many places the picture is incomplete, especially in the full, spatially inhomogeneous
situation.
Our discussion is mainly based on a priori estimates. We have chosen not to discuss
existence proofs, strictly speaking. Sometimes these proofs follow from the a priori
estimates by rather standard PDE arguments (fixed point, monotonicity, compactness),
sometimes they are very, very complicated. In any case they are unlikely to be of much
interest to the non-specialist reader, and we shall skip them all. Complete proofs of the
most famous results can be found in [149].
Also, in this review we insist that a priori estimates should be explicit, but we do not care
whether solutions are built by a constructive or non-constructive method. This is because
we are mainly concerned with qualitative statements to be made about the solutions, and
their physical relevance. If we were more concerned about practical aspects like numerical
simulation, then it would be important that existence results be obtained by constructive
methods.
As a last remark, we note that we have excluded from the discussion all references
which include nonstandard analysis [19,21,25] -just because we are not familiar with
these techniques.
1. Use of velocity-averaging lemmas
1.1. Reminders
Velocity-averaging lemmas express the local smoothness in macroscopic variables (t, x) of
averages of the distribution function with respect to the microscopic variable (the velocity).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-145-320.jpg)
![144 C. Villani
Here is a basic, important example: assume that f satisfies
af
at
--+v.Vxf --S, f EL 2 SEL 2 (Hv s)
t,x,v, t,x 9 (82)
Then, for any q99 C~ (RN),
1
/4 2(l+s)
f (t, x, v)qg(v) dv 9 "'t,x 9
N
Here H c~is the Sobolev space of order c~, and when we write "e", this really means "lies
in a bounded subset of".
From the physical point of view, averaging lemmas express the fact that observables
(typically, the local density) are smoother than the distribution function f itself. From the
mathematical point of view, they are consequences of a "geometric" fact which we shall
describe briefly. Consider the Fourier transform of f with respect to the variables t and x,
write (r, ~) for the conjugate variables, then (82) becomes
so that
ill 2 -
i 12
Ir+v.~l 2"
Since the numerator vanishes for well-chosen values of v, this does not tell us much about
the decay of f as r and ~ go to infinity. But when v varies in a compact set of NN, the
set of values of v such that r + v 9~ is small will itself be very small; this is why on the
average Ifl will decay at infinity faster than IS'l.
Many variants are possible, see in particular [242,195]. A pedagogical introduction
about velocity-averaging lemmas is provided by Bouchut [96]. Let us make a few
comments:
1. The L 2 a priori bound for f may be replaced by a L p bound, p > 1 (then the
regularization holds in some W ~'q Sobolev space), but not by an L 1 bound. Some
replacements with L 1 estimates can be found, e.g., in Saint-Raymond [400].*
2. It is possible to cover cases in which the fight-hand side also lies in a negative Sobolev
space with respect to the x variable, provided that the exponent of differentiation be less
than 1. Obviously, if the exponent is greater than 1, then the transport operator, which is
first-order differential in x, cannot regularize .... The case where the exponent is exactly 1
is critical, see Perthame and Souganidis [379].
3. The above theorem considers time and space variables (t, x) E R • R N, but there are
local variants, see in particular [99].
4. The transport operator v 9Vx may be replaced by a(v) 9Vx under various conditions
on a.
9Note added in proof: See also a recent note by Golse and Saint-Raymond.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-146-320.jpg)
![A review of mathematical topics in collisional kinetic theory 145
5. Some vector-valued variants show that convolution products of the form f *v q9 are
smooth in all t, x, v variables.
A remarkable aspect of averaging lemmas is that they do not rely on the explicit solution
to the linear transport equation (82) (at least nobody knows how to use the explicit solution
for that purpose !). Instead, they are usually based on Fourier transform, or more generally
harmonic analysis.
There are variants of averaging lemmas which do not lead to smoothness but to a gain
of integrability, with estimates in L p (L q (Lr)), and sometimes apply in a larger range of
exponents. Developed by Castella and Perthame [135] with a view of applications to the
Vlasov-Poisson equation, these estimates are analogous to a famous family of inequalities
due to Strichartz for the Schrrdinger equation. Even though these estimates also give more
information about the transport operator (which appears to be much more complex than
it would seem!), it is still not very clear what to do with them. A discussion of the links
between these Strichartz-like estimates and velocity-averaging lemmas can be found in
Bouchut [96].
In the next two sections, we briefly describe the interest of averaging lemmas in the
context of the theory of renormalized solutions for the Boltzmann equation.
1.2. How to use velocity-averaging lemmas in the Boltzmann context?
No need to say, it would be very useful to get regularity results on averages of solutions
of the Boltzmann equation. Since the Boltzmann collision operator looks a little bit like
a convolution operator with respect to the v variable, we could hope to recover partial
smoothness for it, etc.
However, if we try to rewrite the Boltzmann equation like (82), with S = Q(f, f), we
run into unsurmountable difficulties. First of all, we do not have the slightest a priori
estimate on S! Something like integrability would be sufficient, since measures can be
looked as elements of negative Sobolev spaces, but even this is not known in general. 1
Next, we only have f E L log L, and this seems to be a limit case where averaging
lemmas do not apply 2 .... As pointed out to us by E Bouchut, L logl+e L would be
feasible, although extremely technical, but for L log L this seems to be linked with deep
unsolved questions of harmonic analysis in Hardy spaces.
This is the place where the clever DiPerna-Lions renormalization trick will save the
game. After rewriting the Boltzmann equation in renormalized formulation,
Off(f)
Ot
+ v. Vxfl(f)= fl'(f)Q(f, f),
we see an opportunity to apply averaging lemmas to the function fl(f), which lies, for
instance, in L 1 N L ~ as soon as fl(f) <~Cf/(1 + f). If we take it for granted that we
shall find a meaningful definition of fl'(f)Q(f, f), i.e., a renormalizedformulation of the
1Except when the x-variable is one-dimensional, see Section 5.7 in Chapter 2A.
2Note however the following result by Golse and Saint-Raymond [245]. If Otfn + v. Vx fn = Sn with (fn)
weakly compact in L 1 and (Sn) bounded in L 1, then f ~o(v)fn dv is strongly compact in (t,x).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-147-320.jpg)
![146 C. Villani
Boltzmann operator, withbounds like fl'(f)Q(f, f) ~ L1 ([0, T] x IRN; H-c~(IRN)), then
we shall get a smoothness bound for f # (f)~ do, or t~(f) *~ ~ ....
Of course, smoothness for averages of fl(f) does not mean smoothness for averages
of f. But since fl may be chosen to vary over a large range of admissible nonlinearities,
by using approximation arguments combined with the a priori bounds on mass, energy,
entropy, it is not difficult to show that averages like f f (v) q)(v) dr, f 9 q9 (g) ~ L ~ will
lie in a uniformly strongly compact set of L 1.
The interest of such approximation arguments is that they are robust and easy, and
retain the softest information, which is gain of compactness. However, their nonexplicit
nature is one of the drawbacks of the theory, and one can expect that important efforts
will be devoted in the future to turn them into quantitative statements (as in [101,176], for
instance).
1.3. Stability/propagation/regularization
Some of the main principles in the theory of renormalized solutions are (1) as a starting
point, try to work with just the basic known a priori estimates of mass, energy, entropy,
entropy dissipation, (2) treat the Cauchy problem as a stability problem, and (3) replace
smoothness by strong compactness.
Point (2) means the following: consider a sequence fn(t, x, V)n~r~ of solutions, or
approximate solutions, satisfying uniform a priori estimates of mass, energy, entropy,
entropy dissipation. Without loss of generality, fn ~ f weakly in L p ([0, T]; L I(]~N •
N
R v )), 1 ~< p < (x~. Then one would like to prove that f also solves the Boltzmann
equation. As a corollary, this will yield a result of existence and stability of solutions.
Since a priori estimates are so poor, this is a very bad, unsatisfactory existence result. But
for the same reason, this is an extremely good stability result.
Now for point (3), it consists in replacing the statement "f is a smooth function", which
is meaningless in a framework where so little information is available, by the statement
,,fn lies in a strongly compact set of L l''. For instance,
9 "smoothness propagates in time" is replaced by "if fn(o,., .) lies in a strongly
N
compact set of L I(~ N • R v ), then for all t > 0, so does fn (t,., .)";
9 "singularities propagate in time" is replaced by "if fn (0,., .) does not lie in a strongly
N ,,
compact set of L l(R u x I~v ), then for all t > 0, neither does fn (t,., .) ;
9 "there is an immediate regularization of the solution" is replaced by "for (almost all)
N ,,
t > O, fn (t,., .) lies in a strongly compact set of L 1(RN x IRv ) .
Note that the second item in the list can be rephrased as "smoothness propagates backwards
in time".
One of the nice features of the theory of renormalized solutions is that, with the help of
averaging lemmas, these goals can be achieved by a good understanding of the structure
of the Boltzmann operator alone. This approach has been developed by Lions, especially
in [307] and in [311]. As a typical example, if we suspect some regularization effect due
to collisions and wish to prove appearance of strong compactness, then, it essentially
suffices to derive some smoothness estimate in the velocity variable, coming from an
a priori estimate where the effect of collisions would be properly used,3 together with
3Most typically, the entropy dissipation estimate.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-148-320.jpg)
![A review of mathematical topics in collisional kinetic theory 147
a meaningful renormalized formulation. Indeed, the velocity-smoothness estimate would
imply that whenever q9is an approximation of a Dirac mass, then fn *v q9should be close
to fn, in strong sense. On the other hand, from the use of averaging lemmas one would
expect something like: fn , 99is "smooth" in t, x, v. Then the strong compactness would
follow. This strategy was introduced by Lions [311 ].
Of course, the technical implementation of these fuzzy considerations turns out to be
very intricate. All of the statements of the previous lines are only approximately satisfied:
for instance we will not know that fn *v ~Pis close to fn, but rather that y(f n) *v q9is
close to y(fn), and we will not know whether this holds for almost all t, x, but only for
those t, x at which the local mass, energy, entropy are not too high, etc. In all the sequel
we shall conscientiously wipe out all of these difficulties and address the reader to the
references above for details. On the other hand, we shall carefully describe the structure of
the Boltzmann operator, its renormalized formulation, and how these properties relate to
statements of propagation of smoothness or regularization.
2. Moment estimates
Moment estimates are the first and most basic estimate for the Boltzmann equation. Since
one wants to control the energy (= second moment), it is natural to ask for bounds on
moments higher than 2. In fact, if one wants to rigorously justify the identity
f• Q(f, f)lv[ 2 dv - 0,
N
and if the kinetic collision kernel in the Boltzmann operator behaves like Iv - v,I • , then it
is natural to ask for bounds on the moments of order 2 + y.
Of course, once the question of local (in time) estimates is settled, one would like to
have information on the long-time behavior of moments.
In the spatially homogeneous situation, moment estimates are very well understood,
and constitute the first step in the theory. In the case of the full, spatially-inhomogeneous
Boltzmann equation there is absolutely no clue of how to get such estimates. This would
be a major breakthrough in the theory. As for perturbative theories, they are not really
concerned with moment estimates: by construction, solutions have a very strong (typically,
Gaussian) decay at infinity.
As for the long-time behavior of moments, it is also well controlled in the spatially
homogeneous case. In the full setting, even for much simpler, linear variants of the
Boltzmann equation, the problem becomes much trickier, and satisfactory answers are only
beginning to pop out now.
In all the sequel, we shall only discuss the spatially homogeneous situation. The starting
point of most estimates [389,204,170,460,349] is the weak formulation
J~ Q(f , f)qg(v) dv
u
_1 ::,
2N N-1](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-149-320.jpg)
![148 C. Villani
applied to r = Ivls, s >2, or more generally to ~r(lvl2), where ~ is an increasing
convex function.
2.1. Maxwelliancollisionkernels
The most simple situation is when the collision kernel is Maxwellian. As noticed by Trues-
dell [274], integral expressions like fsN_~b(cosO)~o(v')dcrcan be explicitly computed
when ~0is a homogeneous polynomial of the velocity variable. As a consequence, the in-
tegral in (83) can be expressed in terms of moments of f and angular integrals depending
on b. This makes it possible to establish a closed system of differential equations for all
"homogeneous" moments. So in principle, the exact values of all moments can be explic-
itly computed for any time. Then, Truesdell showed that all moments which are bounded
at initial time converge exponentially fast to their equilibrium values. Moreover, if some
moment is infinite at initial time, then it can never become finite.
2.2. Hardpotentials
In the case of hard potentials, or more generally when the kinetic collision kernel grows
unbounded at infinity, then the solution to the Boltzmann equation is expected to be well-
localized at infinity, even if the decay at initial time is relatively slow. Heuristically, this can
be understood as follows: if the collision kernel diverges for very large relative velocity,
then very fast particles have a very high probability to collide with rather slow particles,
which always constitute the majority of the gas. Thus, these fast particles will certainly be
slowed down very quickly.
At the level of weak formulations like (83), this means that the "dominant" part
will be negative (as soon as q9 is a convex function of Ivl2). More precisely, if, say,
B(lv- v.I, cos0)= Iv- v.lZb(cosO),y > 0, then, for some constants K > 0, C < +cx~
depending only on s, N, y and b,
f•u Q(f' f)lvlSdv ~ --K(fRN f dv) (fRNflvlS+• dv)
+c(fRu flvl• dv) (fRu flvlSdv)" (84)
This inequality is just one example among several possible ones. It easily follows from
the Povznerinequalitiesor their variants, introduced in [389] and made more precise
by Elmroth [204], Wennberg [460], Bobylev [85], Lu [328]. Here is a typical Povzner
inequality from [328]: for any s > 2, and y ~<min(s/2, 2), [01 ~<zr/2,
,i s s s is
Iv'l s + Iv, -Ivl -Iv, I ~<-x~(0)lv + Cs(IvlS-ylv, Iy + Ivl• lS-y),
where xs(O)is an explicit function of 0, strictly positive for 0 < 0 ~<rr/2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-150-320.jpg)
![A review of mathematical topics in collisional kinetic theory 149
Let us now look at the applications to the solutions of the spatially homogeneous
Boltzmann equation. As a consequence of (84) and the conservation of mass (f f = 1),
Elmroth [204] proved uniform boundedness of all moments which are finite at initial time.
This kind of estimates has been simplified and lies at the basis of spatially homogeneous
theory. Let us explain the argument without entering into details, or looking for best
possible constants. Multiplying the inequality above by Iv - v, I• (0 < y < 2, say), and
integrating against the angular collision kernel, one easily gets
1 is b(cosO)[v v,[ • s -q-[v~,[s Iv]s Iv, Is) do-
4 N-1
<<-girl s+• + C(IvlSlv, Iy + Iv, lSlol • (85)
(additional terms, like Ivls-y Iv, I2y, are easily absorbed into the last term in the right-hand
side by Young's inequality). Then, let us integrate (85) against ff,: after application of
Fubini's identity, we find (84), which can be rewritten as
d f• f(t, Olvl s dv (86)
dt N
The last integral is bounded because of the energy bound and y < 2. Since f f = 1, one
finds that the s-order moments,
Ms(t) -- f f (t, Olvl s dr,
satisfy some system of differential inequalities
d
dtMs <~-KsMs+• + CsMs. (87)
Now, by HOlder's inequality and f f = 1 again,
Ms+• <<.M~+• (88)
Since solutions of dX/dt <<,--CS l+~ .qt_CX, ol > 0, are uniformly bounded, Elmroth's
theorem follows.
Desvillettes [170] pointed out that the conclusion is much stronger: if at least one
moment of order s > 2 is finite at initial time, then all moments immediately become finite
for positive times - and then of course, remain uniformly bounded as time goes to infinity.
The result was further extended by Wennberg [460], Wennberg and Mischler [349], in
particular the assumption of finite moment of order s > 2 at initial time can be dispended
with. Moreover, these results hold for cut-off or non-cutoff angular collision kernels.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-151-320.jpg)
![150 C. Villani
Bobylev [85] has given a particularly clear discussion of such moment estimates,
with various explicit bounds of Ms(t) in terms of Ms(O) and s. A very interesting by-
product of this study was the proof of Gaussian tail estimates. By precise estimates of
the growth of the bounds on Ms (t), he was able to prove that if the initial datum satisfies
f exp(c~olvl2) fo(v)dv < +c~ for some oto > O, then, at least when g = 1 (hard spheres),
there exists some ot > 0 such that
sup L exp(~ f (t, v) dv < +cx~.
t/>0 N
Anticipating a little bit on precise results for the Cauchy problem, we can say that
moment estimates have been a key tool in the race for optimal uniqueness results in the
context of hard potentials with cut-off In fact, progress in this uniqueness problem can
be measured by the number of finite moments required for the initial datum: Carleman
needed 6, Arkeryd [17] only 4, Sznitman [412] was content with 3, Gustafsson [270] with
2 + V, Wennberg [458] needed only 2 + e (e > 0). Finally, Mischler and Wennberg [349]
proved uniqueness under the sole assumption of finite energy. On this occasion they
introduced "reversed" forms of Povzner inequalities, which show that the kinetic energy
of weak solutions to the Boltzmann equation can only increase or stay constant; hence
the uniqueness result holds in the class of weak solutions whose kinetic energy is
nonincreasing.
More surprisingly, these moment estimates can also be used for proving nonuniqueness
results! in the class of weak solutions whose kinetic energy is not necessarily constant, of
course. The idea, due to Wennberg [463], is quite simple: consider a sequence (f~)neN
of initial data, made up of a Maxwellian (equilibrium) distribution, plus a small bump
centered near larger and larger velocities as n ~ cx~.The bump is chosen in such a way
that its contribution to the total mass is negligible as n --+ cx~,but not its contribution to
the kinetic energy; so that the total kinetic energy of fn is, say, twice the energy E of the
Maxwellian. For each n, one can solve the corresponding Boltzmann equation with hard
potentials, and it has energy 2E. One can check that, as n ~ cx~,this sequence of solutions
converges, up to extraction of a subsequence, to a weak solution of the Boltzmann equation,
with Maxwellian initial datum. But, by means of some precise uniform moment bounds,
one can prove that for positive times, the kinetic energy passes to the limit:
u lim fR fn(t, v)lvl2 dv = fR f (t, v)lvl2 dv.
n---+~ N N
Hence this weak solution f of the Boltzmann equation has energy E at time 0, and energy
2E for any time t > 0, in particular it is not the stationary solution ....
2.3. Soft potentials
When the kinetic collision kernel decays as Iv - v,I--+ c~, or more generally when it
is uniformly bounded, then local in time moment estimates are much easier to get. On](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-152-320.jpg)
![A review of mathematical topics in collisional kinetic theory 151
the other hand, the result is much weaker, since only those moments which are initially
bounded, can be bounded at later times.
When the collision kernel presents a singularity for zero relative velocity, say Iv - v,] •
with 7 < 0, then additional technical difficulties may arise. When ~, <~ -1, and even more
when ~, < -2, it is not a priori clear that power laws ]vis (or their mollified versions,
(1 4- IVI2)s/2) are admissible test-functions for the spatially homogeneous Boltzmann
equation. This difficulty is overcome for instance by the method in Villani [446], or the
remarks in [10].
In all cases, anyway, one proves local in time propagation of moments. But now, due
to the decay of the collision kernel at infinity, it becomes considerably more difficult to
find good long time estimates. Obtaining polynomial bounds is quite easy, but this is not
always a satisfactory answer. The main result on this problem is due to Desvillettes [170]"
he showed that when the kinetic collision kernel behaves at infinity like Iv - v,] •
-1 < 7 < 0, then all those moments which are initially bounded, can be bounded by
C (14- t), as the time t goes to infinity. By interpolation, if the initial datum has a very good
decay at infinity (many finite moments), then the growth of "low"-order moments will be
very slow. Thus, even if uniform boundedness is not proven, the "escape of moments at
infinity" has to be slow.
The results of Desvillettes can be extended to the case where 7 > -2, though there is no
precise reference for that (at the time where Desvillettes proved his result, weak solutions
were not known to exist for 7 <~ -1). On the other hand, when 7 ~< -2, then it is still
possible to prove local in time propagation of moments of arbitrary order, but the bounds
are in general polynomial and quite bad ....
In the case of the Landau equation where things are less intricate, one can derive the
following estimate [429] when the decay at infinity is like Iv - v,[ • -3 ~< y ~< -2" then
the moment of order s grows no faster than O((1 + t);~), with X -- (s - 2)/3.
2.4. Summary
We now sum up all the preceding discussion in a single theorem. As we said above, for the
time being it is only in the spatially homogeneous setting that relevant moment estimates
have been obtained. The conditions of the following theorem are enough to guarantee
existence of weak solutions, but not necessarily uniqueness.
THEOREM 1. Let B(iv-v,], cos0)= ]v-v,[• be acollisionkernel, -3 <. }, <. 1,
fo b(cos0)(1 - cos0) sinN-20 dO < +cx~. Let fo E L~(R N) be an initial datum with
finite mass and energy, and let f (t, v) be a weak solution of the Boltzmann equation,
f (O, .)- fo, whose kinetic energy f f (t, v)iv] 2 dv is nonincreasing. Then this kinetic
energy is automatically constant in time. Moreover, if
Ms(t) =--f f (t, v)lvl s dr,
then,](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-153-320.jpg)
![152 C. Villani
(i) If y = O, then for any s > 2,
Vt >0, [Ms(t) < +o0 ~ Ms(O) < +~],
Ms(O) < +~------> sup Ms(t) < +cx~.
t />o
Moreover, under the sole assumption M2 < +cx~, there exists a convex increasing function
4~, 4~(Ivl) ~ c~ as Ivl ~ ~, such that
supf f (t, v)qb(lvl)lv[ 2dr < +c~.
t />o
(ii) If F > O, then for any s > 2,
Vto > O, sup Ms (t) < +cx~.
t/>to
Moreover, if y = 1, and the initial datum fo satisfies f exp(c~olvl 2) fo(v) dv < +c~ for
some oto > O, then there exists some ot > 0 such that
sup f
] e~ (t, v) dv < +c~.
t/>o J
(iii) If Y < O, then for any s > 2,
vt > o, [Ms(t) < +~ ~ M~(O) < +~].
Moreover,
(a) if F > -2, then
Ms(O) < +cx~ ~ 3C > O, Ms(t) <. C(1 + t);
in particular, for any e > O,
Ms/s < +cxz----> 3C > O, Ms(t) <~C(1 + t)s;
(b) if y < - 2, then
Ms(O) < +cx~ ~ 3C > O, 3X > O, Ms(t) <. C(1 + t) x
(X = 1for s <~4).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-154-320.jpg)
![A review of mathematical topics in collisional kinetic theory 153
REMARKS. (1) All the constants in this theorem are explicit.
(2) The statement about finiteness of f f(t, p)r 2 dv in point (i) is interesting
because it implies
lim sup fly
e--+ cx)t>/0 I>~R
f(t, v)lvl 2 dv = 0;
in other words, no energy leaks at infinity. Such an estimate is obvious in situation (ii); it
is a seemingly difficult open problem4 in situation (iii).
(3) The range y 6 [-3, 1] has been chosen for convenience; it would be possible to
adapt most of the proofs to larger values of y, maybe at the expense of slight changes
in the assumptions. Values of y which would be less than -3 pose a more challenging
problem, but do not correspond to any physical example of interest.
The first part of point (i) is due to Truesdell [274], while the statement about point
(ii) is mainly due to Desvillettes [170] and improved by Wennberg [458], Wennberg and
Mischler [349]; the estimate for exponential moments is due to Bobylev [85]. As for point
(iii), it is proven in Desvillettes [170] for y > - 1, and elements of the proof of the rest can
be found in [446,444].
For the Landau equation (with q/(Iv- v,I) = KIv- v,I • the very same theorem
holds, with the following modification: point (ii) is known to hold only if there exists
so > 2 such that Ms0(0) < +oo (see [182]). As for point (iii)(c), the more precise estimate
)~= (s - 2)/3 holds [429], at least if the collision kernel is replaced by a mollification which
decreases at infinity like Iv - v,I • but does not present a singularity for Iv - v,I --~0.
3. The Grad's cut-off toolbox
We now present several tools which are useful to the study of Boltzmann's equation when
the collision kernel satisfies Grad's angular cut-off assumption. This means at least that
whenever Iv - v, I # 0,
A(lv - v,]) = f B(v - v,, a)do- < +cx~. (89)
Typical examples are Iv - v, I• b(cos0), where
f0 rr b(cos0) sinN-2 0 dO < +~.
We shall mainly insist on two ingredients: the important Q+ regularity theorem, and the
DiPerna-Lions renormalization.
4See Section 5.3 in Chapter 2C for more, and some results.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-155-320.jpg)
![154 C. Villani
3.1. Splitting
When Grad's assumption holds true, then one can split the Boltzmann collision operator
into the so-called "gain" and "loss" terms, and the loss term is then particularly simple. We
give this splitting in asymmetric form:
Q(g, f) = Q+ (g, f) - Q-(g, f) -- Q+ (g, f) - f (A 9 g). (90)
Clearly, the delicate part in the study is to understand well enough the structure of the
complicated integral operator
L /,
Q+ (g, f)- dr, da B(v - v,, a)g(v,) f (v').
N N-1
As early as in the thirties, this problem led Carleman to the altemative representation
u1) !
[1) -- Vii N-1
Vt t )
-- V, t
B 2v- v'- v~,, [v' , g(v,)f(v'),
(91)
with Evv, standing for the hyperplane going through v, orthogonal to vI - v.
In Sections 3.3 and 3.4, we shall expand a little bit on the structure of the Q+ operator.
Before that, we give an easy lemma about the control of Q+ by means of the entropy
dissipation.
3.2. Control of Q+ by Q- and entropy dissipation
Using the elementary identity
1 X
(X-Y) log K>I
X ~ K Y + log K -Y' '
with X = flf~, and Y = ff,, we find, after integration against B dr, da,
Q+(f, f) ~<K Q-(f, f)+
4
d(f),
log K
(92)
where
1s
d(f) = -~ N•
(f,ft, _ ff,) log f'ff* B dr, da
ff,
is a nonnegative operator satisfying f d(f) do = D(f), the entropy dissipation functional.
Inequality (92) was first used by Arkeryd [21], and has proven very useful in the
DiPema-Lions theory [192].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-156-320.jpg)
![A review of mathematical topics in collisional kinetic theory 155
3.3. Dual estimates
Many estimates for Q+ are best performed in dual formulation, with the help of the
pre-postcollisional change of variables. For instance, to bound IIQ+ (g, f)IIL,(RN), it is
sufficient to bound
LN Q+(g' f)q)dv- fiR2,,,,dvdv*g*f(LN-1B(Iv-v,l,cosO)q)(v')do')
uniformly for I1~oIILp' ~< 1. So the meaningful object is the linear operator
~o~ LN-' B(Iv -- v,I, cos0)q)(v')do'. (93)
Pushing the method a little bit, one easily arrives at the following abstract result: let X, Y
be two Banach spaces of distributions, equipped with a translation-invariant norm. Assume
that the linear operator
~T""(D1""~LN_I B(Ivl, cos 0)~o( v+ 21
vI~r) dCr
is bounded (as a linear map) from Y to X. Then, the following estimate on Q+ holds,
IIQ+':.f, g)llw CIIgIIL,Ilfllx'.
Actually, Y' (resp. X') does not really need to be the dual of Y (resp. X), it suffices that
IIQ+llr, = sup{f Q+g; Ilgllr = 1} (resp. f fO <~Ilfllxllgtllx,).
As an example of application of this result, consider the simple situation B(v - v., o.) =
4"(Iv - v.l)b(cosO), where 4, is bounded and b(cos0) sinN-2 0 is integrable with support
in [0, re/2]. Obviously, 7" is bounded L ~ ~ L ~ and by the change of variables v ~ v'
(which is valid for fixed o. because we have restricted ourselves to 0 6 [0, 7r/2]), one
can prove that 7" is bounded L 1 --+ L 1. By interpolation, 7" is bounded L p ~ L p for
1 ~<p ~<oc, and therefore we obtain the estimate
][Q+(g,f)ncp ~CIIgllLlllfll zp, 1 ~p~. (94)
A variant of the argument when B(v - v,, o.) = Iv - v,l• y > 0, leads to
[1Q+(g' f)[[Lp <-CIIgllL~IlfllL~, 1 ~ p ~ ~, (95)
where we use the notation
(L
IIflIL,P -- NfP(1 4-IriS) p dv (96)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-157-320.jpg)
![156 C. Villani
More sophisticated variants of estimates like (95) have been studied by more compli-
cated means in Gustafsson [269,270]. They constitute a first step in the L p theory for the
spatially homogeneous Boltzmann equation with hard potentials and cut-off. These esti-
mates show that, in first approximation, the Q+ operator resembles a convolution operator.
But we shall see in the next paragraph that a stronger property holds.
To conclude this paragraph, we mention that the case where there is a singularity in the
kinetic collision kernel for Iv - v, I -~ 0 (soft potentials ...) has never been studied very
precisely from the point of view of LP integrability.
3.4. Lions' theorem: the Q+ regularity
One of the main ideas of Grad [252], when developing his linear theory and making clear
his assumptions, was that the Q+ term should be considered as a "perturbation". This may
sound strange, but think that the linear counterpart of Q+ is likely to be an integral operator
with some nice kernel, while the linear counterpart of Q- will contain a multiplicative,
noncompact operator. Generally speaking, since Q+ is more "mixing" than Q-, we could
expect it to have a smoothing effect.
In a nonlinear context, the idea that Q+ should be smoother than its arguments was
made precise by Lions [307]. In this paper he proved the following estimate.
PROPOSITION 2. Let B be a C ~ collision kernel, compactly supported as a function
of Iv - v, l, vanishing for Iv - v, I small enough, compactly supported as a function of
0 9 (0, Jr). Then, there is afinite constant C depending only on B and N such that for any
g 9 L1 (RN), f 9 L2(RN),
IIa+<g II.--
, N~I ~<Cllgllc, Ilfllc2, (97)
and similarly
[IQ+(g,f ll,, CllgllLzllfllL '' (98)
The proof was based on a duality method quite similar to the one above, and very
sophisticated tools about Fourier integral operators; this estimate is actually linked to the
regularity theory of the Radon transform. In fact, as a general rule [406], operators of the
form
7"qg(x) = fsx b(x, y)qg(y) dcrx(y),
where b is a smooth kernel and Sx is a hypersurface varying smoothly with x 9 ]I~N , satisfy
an estimate like IIT~ollH~N-,)/2 <<.CIl~ollL2, under some nondegeneracy condition 5 about the
way Sx varies with x.
5In the case of operator (93), Sx is the sphere with diameter [0, x] and crx is just the uniform measure on Sx.
The nondegeneracy condition is not satisfied at y = 0 (which is fixed), and this is why Lions' theorem does not](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-158-320.jpg)
![A review of mathematical topics in collisional kinetic theory 157
Lions' theorem has become one of the most powerful tools in the study of fine properties
of the Boltzmann equation with cut-off (see [101,307,308,349,429,393]). Wennberg [459]
gave a simplified proof of this result, with explicit(able) constants, by using Carleman's
representation (91). Then, Bouchut and Desvillettes [98], and, independently, Lu [325]
devised an elementary proof, with simple constants, of a slightly weaker result:
IIa+<g f)I1
, N21 <~Cllgllc2 II/11c2. (99)
However, the qualitative difference between (97) and (99) is significant in some applica-
tions: see, for instance, our a priori estimates in LP norms for collision kernels which decay
at infinity [429].
Also a relativistic variant of this result has been established by Andrrasson [14]. In a
more recent paper, Wennberg [462] has put both the classical and the relativistic estimates
in a unified context of known theorems for the regularity of the Radon transform. He
noticed that these two cases are the only ones, for a whole range of parameters, where
these theorems apply.
Of course, by Sobolev embedding (and interpolation), Lions' theorem yields refinements
of (94) when the collision kernel is smooth. A task which should be undertaken is to use a
precise version of Lions' theorem, like the one by Wennberg, to derive nice, improved
weighted Lp bounds for realistic collision kernels. This, combined with the methods
in [429], should enable one to recover the main results of Gustafsson in a much more
elegant and explicit way.6
As a last remark, the fact that the collision kernel vanish at v - v. = 0 is essential in
Lions' theorem (not in the weaker version (99)). If this is not the case, then smoothness
results similar to the ones obtained by (97) require additional integrability conditions
on g. In the case of hard potentials however, it is still possible to prove a weaker gain
of smoothness or integrability with respect to f, without further assumptions on g.
3.5. Duhamelformulas andpropagation of smoothness
The idea to consider Q+ as a perturbation can be made precise by the use of Duhamel-type
formulas. For instance, in the spatially homogeneous case, the Boltzmann equation can be
rewritten as
af
Ot
+ (Lf) f = Q+ (f, f), f(O, .) = fo,
where we use the notation A 9 f = Lf. Then the solution can be represented as
fot
f(t, v)-- fo(v)e -foLf(r'v)dr + e-ftLf(r'v)drQ+(f, f)(s, v)ds. (100)
apply in presenceof frontalcollisions,and needs the collisionkernel to be vanishingcloseto 0 = Jr. Another
degeneracyariseswhen v - v. goesto 0, so the Q+ smoothnessalsoneedsvanishingof the collisionkernelat
zerorelativevelocity.
6At the timeof writing,thistaskhasjust beenperformedby C. Mouhotwiththe helpof the author'sadvice.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-159-320.jpg)
![158 C. Villani
In the spatially inhomogeneous case, one can write similarly
( 0 ) tLf(r,x-(t-r)v,v)dr _ Q+(f, f)ef oLf(r,x-(t-r)v v)dr
-~ + v" Vx fefo -- '
or
f (t,x, v) = fo(x -tv, v)e- fd Lf(r,x-(t-r)v,v)dr
fo
+ Q+(Z, f)(s,x - (t - s)v, v)e-f's LI(~,x-(,-~)~,~)d~. (101)
By these formulas, one can understand, at least heuristically, the phenomenon of
propagation of regularity and singularities. Let us illustrate this in the case of the spatially
homogeneous Boltzmann equation, starting with formula (100). If f0 E L2, then, at least
for collision kernels which become very large at infinity, f(t, .) is uniformly bounded
in L 2. Now, assume for simplicity that the collision kernel B is sufficiently smooth that
estimate (97) applies, then it becomes clear that the second term in the right-hand side
of (100) is H 1-smooth (when N = 3). Indeed, thanks to the convolution structure, Lf is
also smooth, say C ~176
if B is C ~176
On the other hand, the first term on the right-hand side has
exactly the same smoothness as f0. Thus both regularity and singularities are propagated
in time. More precise theorems of this kind, for some realistic collision kernels, e.g., hard
spheres, are to be found in Wennberg [459]. In principle one could actually prove that
f (t, v) = G(t, v) fo(v) + H (t, v), (102)
where, at least if B is very smooth, G is a positive C ~176
function of t, v and H is smoother
than f0. This would follow by an iteration of Duhamel's formula, in the spirit of [349].
Alternatively, one expects that f (t, .) can be decomposed into the sum of a part which is
smooth (at arbitrarily high order) and a part which is just as singular as f0, but decays
exponentially fast.
In the spatially inhomogeneous situation, the same kind of results is expected. In
particular, in view of (101) it is believed that singularities of the initial datum are
propagated by the characteristics of free transport, (x, v) ~ (x + tv, v). Such a result
was recently proven by Boudin and Desvillettes for small initial data: they showed the
following generalization of (102),
f (t, x, v) = G(t, x, v) fo(x - tv, v) + H (t, x, v),
where G and H are not necessarily smooth, but at least possess a fractional Sobolev
regularity. Their proof is based on a combination of the Q+ smoothness and averaging
lemmas.
We should make an important point here about our statement above about propagation of
singularities. If we consider an initial datum which is very smooth apart from some singular
set S, then it is not expected that the solution stay very smooth apart from the image St
of this singular set by the characteristic trajectories. It is possible that the smoothness of](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-160-320.jpg)
![A review of mathematical topics in collisional kinetic theory 159
the solution deteriorate, even very far from St. But one expects that the worst singularities
always lie on St. And in any case, one always expects that the singular part of the solution
be decaying very fast as time goes by.
3.6. The DiPerna-Lions renormalization
The Q+ smoothing effect cannot be applied directly to the spatially inhomogeneous
equation,
~f
Ot
+ v. Vxf = Q(f, f), (103)
because of the lack of nice a priori estimates.
As explained above, the idea of renormalization consists in giving a meaningful
definition of fl'(f)Q(f, f) under very weak a priori estimates. Solutions are then defined
as follows:
N
DEFINITION 1. Let f e C([0, T]; Ll(Ii~x
N x ~U))A L~(R+, L~_(Rx
N x II~v )). It is said
to be a renormalized solution of the Boltzmann equation if for any nonlinearity fl
CI(~+, R+), such that fl(0) =0, Ifl'(f)l ~<C/(1 + f), one has
3fl(f)
Ot
-b v. Vxfl (f) = fl' (f) Q (f, f), (104)
in distributional sense.
The DiPerna-Lions renormalization [190,192] achieves this goal by using the split-
ting (90). First of all,
fl' (f) Q- (f, f) = ffl' (f)(A 9 f). (105)
Following Section 5.1, let us assume that
sup f•zu f(t,x, V)(1 + IV[2 + IX[2)dxdv < -q-cx~,
O~t~r
and that A(z) = o(Izl2). Then
A 9 I L (t0, L' (Rf; L?oc(Rf))).
Further assume that the nonlinearity fl satisfies
0 <~fl'(f) ~<
C
l+f](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-161-320.jpg)
![160 C.Villani
Then, ffl'(f) 9 L~, and as a consequence (105) is well-defined in L~oc.
As for the renormalized gain term, it can be handled easily because it is nonnegative. As
a general "rule", when one manages to give sense to all terms but one in some equation, and
when this last term has a sign, then the equation automatically yields an a priori estimate.
In our situation this is accomplished by integrating Equation (104) in all variables on
[0, T] x IR~ x IR~; using the bounds on mass and energy, one gets
foTf• fl'(f) Q+(f, f) dt dx dv
2N
<~ fl(fo) dx do + (f) f Lf dt dx dv < +cx~,
2N 2N
whence
fl'(f) Q+ (f, f) e L1 ([0, T] x JRN x NN).
By the way, there is another, more widely known, version of this estimate, based on the
entropy dissipation [192]; but this variant is more complicated and has the disadvantage to
use a symmetric estimate, while the renormalization procedure can also be done from an
asymmetric point of view:
fl'(f)Q(g, f) = fl'(f)Q+(g, f) - fl'(f)fLg.
On the other hand, the entropy dissipation has been very useful, both in the proof of
stability of renormalized solutions, and in certain refinements of the theory by Lions [308].
Also, by using the entropy dissipation one can define a renormalized formulation with a
stronger nonlinearity, namely fl(f) = ~/1 + f - 1. This is implicit in [308], and has been
shown by an elementary method in [444, Part IV, Chapter 3], as an outgrow of the idea of
H-solutions explained in Section 4.1.
Renormalization and averaging lemmas were the two basic tools in the DiPema-Lions
theorem, which was the very first existence/stability result in the large for the spatially
inhomogeneous Boltzmann equation. More precisely, these authors have proven [192] that
the renormalized Boltzmann equation (104) is stable under weak convergence (a priori
estimates of mass, energy and entropy being used). This theorem has remained a singular
point in the field, due to the complexity of the proof and the use of technicalities which
have no real counterpart for the rest of the theory ....
Since then, Lions [307] found a simpler proof of existence, using the Q+ regularity.
In fact, he proved that strong compactness propagates in time for the Boltzmann equation
with cut-off: this is the analogue of the spatially homogeneous results discussed in the
section above. Once strong compactness is established, passing to the limit is almost
straightforward.
By the way, it is rather easy to prove the converse property, namely that the sequence
of initial data has to be strongly compact if the sequence of solutions is. This theorem is a
(very weak) illustration of the principle of propagation of singularities.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-162-320.jpg)
![A review of mathematical topics in collisional kinetic theory 161
3.7. Summary
To conclude this section, we give explicit theorems illustrating the discussion above. As of
this date, all of them are best in their category, but not optimal.
We begin with the spatially homogeneous situation.
THEOREM 3. Let B(v- v,,rr)= Iv- v,l• be a collision kernel for hard
potentials, satisfying Grad's cut-off assumption:
fo rr b(cos0) sinN-2 0 dO < +cx~, y>0,
and let fo be a probability distribution function with finite second moment,
fR f~ < +cx~.
N
Then,
(i) there exists a unique energy-preserving solution to the Cauchy problem associated
to B and fo. This solution is unique in the class of weak solutions whose energy does not
increase;
(ii) if fo ~ L~1 (q LPs for some p E [1,+cxz], Sl > 2, 1 < s <. Sl - V/P, then
supt>~0 Ilf(t, ")ILL; < +~;
(iii) ifmoreover y > 1/2, s >~2 and fo ~ HI(RN), N -- 3, then supt,>0 Ilf(t, ")lln~ <
+cx~;
(iv) if on the other hand, F > 1/2, s ~ 2and fo q~HI(~N), N -- 3, then, for all t ~ 0,
f (t, .) ~ HI(RN). But f (t, .) = g(t, .) + h(t, .) where Ilg(t, ")llts~ = O(e-~t) for some
lZ > O, and supt,>0 IIh(t, ")lln~ < +~.
Point (i) is due to Mischler and Wennberg [349]; point (ii) to Gustafsson [270] for
p < cx~ and to Arkeryd [20] for p = cx~; as for point (iii), is is due to Wennberg [459],
point (iv) being an immediate consequence of the proof. A recent work by Mouhot and
the author recovers the conclusion of (ii) under slightly different assumptions on sl, with
the advantage of getting explicit constants; we are working on extending the allowed range
of exponents sl. Further work is in progress to extend also the range of validity of the
conclusion of (iii) (arbitrary dimension, more general collision kernels) as well as to treat
propagation of H ~ smoothness for arbitrary k.
We now turn to the inhomogeneous theory in the small.
THEOREM 4. Let B = B(z, rr) be a collision kernel, B E L ~ (S N-1 , W l'cx~(~N)), N = 3.
Let fo(x, v) be a nonnegative initial datum satisfying the Maxwellian bound
fo(x v) <. Coexp(-Ixl2-+- Ivl2)
' 2 '
where Co = 1/(81[IBIIL~(~N;LI(SN-1))). Then,](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-163-320.jpg)
![162 C. Villani
(i) there exists a global solution to the spatially inhomogeneous Boltzmann equation
with collision kernel B and initial datum fo, and for all t ~ [0, T] (T > 0), it satisfies a
Maxwellian bound of the form
f (t x v) <. C;r exp(- lx - vtlZ + lvl2 )
' ' 2 '
(106)
with C7~depending only on T and Co;
(ii) there exist "smooth"functions R and S in//lo
c~(I~+ x IRxNx I~vN)for any ot < 1/25,
such that
f (t,x, v) = fo(x - vt)R(t,x, v) + S(t,x, v);
(iii) if moreover fo ~ W k'c~for some k ~ N (or k = oo), then
I W,
o'c (R+ • Rx •
This theorem is extracted from Boudin and Desvillettes [101]. Part (i), inspired from
Mischler and Perthame [348], is actually an easy variation of more general theorems by
Illner and Shinbrot [278]. One may of course expect the smoothness of R and S to be
better than what this theorem shows! But this result already displays the phenomenon
of propagation of singularities along characteristic trajectories. Moreover, it conveys the
feeling that it will be possible to treat propagation of smoothness and singularity for very
general situations, as soon as we have solved the open problem of finding nice integrability
a priori estimates for large data.
The main idea behind the bound (106) is the following: the left-hand side satisfies the
differential inequality
~f
Ot
+ v. Vxf = Q(f, f) ~< Q+ (f, f),
while one can devise a fight-hand side of the form
g(t,x, v)-C(t)exp(-Ix- vtl2 +
which satisfies the differential inequality
Og
Ot
+ v. Vxg ~ Q+(g, g),
so that it is natural to expect f ~< g if f0 ~< go. Note that since g is a Maxwellian,
Q+ (g, g) = Q- (g, g).
Finally, we consider the DiPerna-Lions theory of renormalized solutions.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-164-320.jpg)
![A reviewofmathematicaltopicsin collisionalkinetictheory 163
THEOREM 5. Let B(lv - v,I, cos0) = r - v,l)b(cosO) be a collision kernel satisfying
Grad's angular cut-off together with a growth condition at infinity:
f0 zrb(cos0) sinN-2 0 dO < +c~,
qS(lz[)=o(lzl 2) aslzl--+cx~.
Let (f~)nel~ be a sequence of initial data with uniformly bounded mass, energy, entropy,
supfR
nr N•
f~(x, v)[1 +Ix] 2 -+-Iv[2 + log f~(x, v)]dxdv < +oo.
Let fn (t, x, v) be a sequence of solutions7 of the Boltzmann equation
of'+
Ot v. Vxfn = Q(fn fn),
fn (0,., .) = f~) .
t ~0, X e]I~N, VE]t~N,
(107)
Assume that the fn's satisfy uniform bounds of mass, energy, entropy and entropy
dissipation:
sup sup f~
n~Nt~[O,T] Nx•N
fn(t,x,v)[1 + [x[2 + Iv[2 +logfn(t,x,v)]dxdv < +oo,
(lo8)
fo T
sup D(fn(t,x, .)) dx dt < +c~.
hEN
(109)
Without loss of generality, assume that fn __+f in L p ([0, T]; L ~(~N • ~N)), 1 ~<p < ~,
T < oo. Then,
(i) f is a renormalized solution of the Boltzmann equation. It satisfies global
conservation of mass and momentum, and the continuity estimate f ~ C ([0, T]; L 1(i~u x
N
R~));
(ii) moreover, for all t > O, fn __+f strongly in L 1 if and only if f~ --+ fo strongly; in
this case the convergence actually holds in C ([0, T], L I(I~u x I~u));
(iii) if moreover there exists a strong, classical solution of the Boltzmann equation with
initial datum fo, then f coincides with fo.
COROLLARY 5.1. Let fo be an initial datum with finite mass, energy and entropy:
f~N•
fo(x, v)[1 + Iv[2 + Ix[2 + log fo(x, v)] dx dv < +c~.
7Renormalizedsolutions,or strongsolutions,or approximatestrongsolutions.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-165-320.jpg)
![164 C. Villani
Then there exists a renormalized solution f (t,x, v) of the Boltzmann equation, with
f (O, ., .)-- fo.
REMARKS. (1) A typical way of constructing approximate solutions is to solve the
equation
ofn Q(fn, fn)
+ V.Vxf n--
1 fn
Ot 1 + n f dv
which is much easier than the "true" Boltzmann equation because the collision operator is
sublinear.
(2) If the fn's are strong, approximate solutions, then the bounds (108)-(109)
automatically hold, provided that the initial data have sufficient regularity. This remark,
combined with the preceding, explains why the corollary follows from the theorem.
(3) Point (iii) as stated above is slightly incorrect: for this point it is actually necessary
to assume that the fn's are strong, approximate solutions, or are constructed as limits of
strong, approximate solutions.
Point (i) is due to DiPerna and Lions [192], points (ii) to Lions [307,308]. For the sake
of simplicity, we have stated unnecessarily restrictive assumptions on the collision kernel
in points (i) and (ii). Point (iii) was first proven by Lions under an assumption which
essentially implies q~ ~ L~, then extended by Lu [324]. We have not made precise what
"classical" means in point (iii): in Lions' version, g should satisfy the Boltzmann equation
almost everywhere on [0, T] • RN • RN, and also satisfy the dissipative inequalities
introduced in Lions [308]. The discussion of dissipative inequalities is subtle and we
preferred to skip it; let us only mention that this concept is based on the entropy dissipation
inequality, and that it led Lions to a clean proof of local conservation of mass, as well as to
the concept of dissipative solutions. In Lu's theorem, much more general collision kernels
are included, at the price of slightly more restrictive (but quite realistic) assumptions
imposed on the strong solution g. Lu also uses results from the theory of solutions in
the small [63] to show existence of such strong solutions when q'(Izl) = O(Izl• Z > -1,
and the initial datum is bounded by a well-chosen, small function. So these results bridge
together the theory of renormalized solutions and the theory of solutions in the small.
At the moment, point (ii) is the most direct way towards the corollary. The scheme of
the proof is as follows. In a first step, one uses the uniform bounds and the Dunford-Pettis
criterion to get weak compactness of the sequence of solutions in L 1. This, combined
with the renormalized formulation and averaging lemmas, implies the strong compactness
of velocity-averages of fn. Since the operators L = A. and Q+ are velocity averaging
operators in some sense (remember the Q+ regularity theorem), one can then prove the
strong compactness of Lf n and Q+ (fn, fn). This is combined with a very clever use of
Duhamel formulas to prove the strong compactness of the sequence fn itself, provided that
the sequence of initial data f~ is strongly compact. Finally, one easily passes to the limit
in the renormalized Q- operator, and then in the renormalized Q+ operator by a variant
of the dominated convergence theorem which involves the domination of Q+ by Q- and
(a little bit of) the entropy dissipation, as in Section 3.2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-166-320.jpg)
![A review of mathematical topics in collisional kinetic theory 165
4. The singularity-hunter's toolbox
In this section, we now examine the situation when the collision kernel presents a
nonintegrable angularsingularity. This branch of kinetic theory, very obscure for quite
a time, has undergone spectacular progress in the past few years, which is why we shall
make a slightly more detailed exposition than in the case of Grad's angular cut-off. The
starting point for recent progress was the work by Desvillettes [171] on a variant of the
Kac model, which was devised to keep some of the structure of the Boltzmann equation
without cut-off. The study of these regularizing effects was first developed in the spatially
homogeneous theory, and later in the theory of renormalized solutions, in the form of strong
compactification.
As explained previously, the main qualitative difference with respect to the situation
where Grad's assumption holds, is that one expects immediate regularization of the
solution. From the mathematical point of view, the first clear difference is that the splitting
Q(f, f)= Q+ (f, f)- Q-(f, f)
is impossible: both terms should be infinite. From the physical point of view, one can argue
that when particles collide, there is an overwhelming probability for the change in velocity
to be extremely small, hence the density in probability space should spread out, like it does
in a diffusion process.
The main analytical idea behind the regularization effect is that Q(f, f) should look
like a singular integral operator. As we shall see, it resembles a fractional diffusion
operator; this illustrates the physically nontrivial fact that collision processes for long-
range interactions are neither purely collisional in the usual sense, nor purely diffusive, but
somewhat in between.
There is an important body of work due to Alexandre about the study of the non-cutoff
Boltzmann operator, with the help of pseudo-differential formalism [2-9], and on which
we shall say almost nothing, the main reason being that most of the results there (some of
which have been very important advances at the time of their appearance) can be recovered
and considerably generalized by means of the simpler techniques described below.
Generally speaking, there are two faces to singular operators in partial differential
equations. On one hand one would like to control them, which means (i) find weak
formulations, or (ii) find if, in some situations, they induce compensations due to
symmetries. On the other hand, we would like to have (iii) simple estimates expressing the
fact that they really are unbounded operators, and that the associated evolution equation
does have a regularizing effect. To illustrate these fuzzy considerations, think of the
Laplace operator, and the formulas (i) f Aft0 = f fA~0, (ii) (Af) 9 ~0 = f 9 (A~0),
(iii) f f(Af) -- -IIf[121 . Keeping this discussion in mind may help understanding the
interest of the weak formulations in Section 4.1, the cancellation lemma in Section 4.2,
and the entropy dissipation estimate in Section 4.3, respectively.
Finally, as we already mentioned several times, another singularity problem will come
into play: when one is interested in soft potentials, then the kineticcollision kernel presents
a singularity for Iv - v, I -~ 0. When the strength of this singularity is high, this will entail
additional technical difficulties, but it is not clear at the moment that this feature is related
to physically relevant considerations.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-167-320.jpg)
![166 C.Villani
4.1. Weakformulations
In presence of a nonintegrable singularity, Boltzmann's collision operator is not a bounded
operator between weigthed L1 spaces; it is not even clear that it makes sense almost
everywhere. Thus one should look for a distributional definition. The most natural way
towards such a definition (both from the mathematical and the physical points of view) is
via Maxwell's weak formulations"
f Q(f, f) do = f dvdv*ff*[f su-, B(v-v,,a)(qg'-qg)da]. (110)
As pointed out by Arkeryd [18], if ~0 is a smooth test-function, then ~ot - q9 will vanish
when 0 ~ 0 (because then v~_~ v), and this may compensate for a singularity in B. This
circumstance actually explains why one is able to compute relevant physical quantities,
such as the cross-section for momentum transfer, even for non-cutoff potentials ....
For the sake of discussion, we still consider the model case B(v - v., a) = Iv -
v,l• Using moment estimates, and the formula ~0' - ~o= O([v - v,10) (which
holds true when q9is smooth), Arkeryd [18] was able to prove existence of weak solutions
for the spatially homogeneous Boltzmann equation as soon as
y~-l, f b(cosO)O sinN-2 0 dO < 00.
The use of the more symmetric form obtained by the exchange of variables v +-~ v, does
not a priori seem to help a lot, since one has only
Iqg'+ qgt,- q9- go,[ ~ C(go)lv - 0,120,
so there is no gain on the angular singularity.
But, as noted independently by several authors (see, for instance, [248,446]), an extra
order of 0 can be gained by integrating in spherical coordinates. More precisely, use the
standard parametrization of cr in terms of 0, r (r 6 SN-2), then
fs (~o'+ ~o~,- ~p- ~p,)de
N-2
C(99)1v- 0,120 2. (111)
This simple remark enables one to extend Arkeryd's results to
y ~>-2, f b(cosO)(1 - cos0) sinN-2 0 dO < ~.
In dimension 3, these assumptions are fulfilled by inverse-power forces 1/r s when s ~>7/3
(to compare with s > 3 for Arkeryd's original result.., the reader may feel that the gain is
infinitesimal, but remember that s = 2 should be the truly interesting limit exponent!).
However, this point of view, which relies on the v +-~ v. symmetry, is in part misleading.
The same control on the angular singularity (but worse in the kinetic singularity) can be](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-168-320.jpg)
![A reviewof mathematical topics in collisional kinetic theory 167
obtained without using the symmetry v ~ v,, as shown in Alexandre and Villani [12] by
the use of more precise computations. When one is only interested in weak solutions in a
spatially homogeneous problem, this remark is of no interest, but it becomes a crucial point
in spatially inhomogeneous situations, or in the study of fine regularization properties.
Here is a precise bound from [12]. Introduce the cross-section for momentum transfer,
formulas (62) or (63). Then
Q(g, f)q9 dv
u
~<~ll~0llw2,~ 2N dvdv, g,f[v - v,[(1 + Iv - v,I)M(Iv - v,I). (112)
To treat values of y below -2 with the help of formula (111) and others, it seems that
one should require nontrivial a priori estimates like
f~, f(v)f(v,)
2N ]U- U,[ -(y+2)
dv dr, < +oc
(the exponent of Iv - v,I in the denominator is positive!). As we shall see in Section 4.3,
such estimates are indeed available in most cases of interest. But they are by no means
easy!
At the time when these extra estimates were not yet available, the search for a treatment
of values of y below -2 led the author [446] to introduce a new weak formulation (H-
solutions), based on the a priori bound
f Q(f, f)q9 ~< ~D(f) 1 Bff,(go' nt- g)', - q) - g0,) 2
Here D is Boltzmann's entropy dissipation functional (47). This new bound was based on
Boltzmann's weak formulation (45), and the elementary estimate
D(f) >~s • xsN-1
B(v - v,,o)(v/f'ff, - fx~,)edvdv, da (113)
(which follows from (X - Y)(log X - log Y) >~4(~/-X - ~/-~)2). It enabled the author to
prove existence of weak solutions under the assumptions
y > -4, f b(cosO)(1 - cos0) sinN-2 0 dO < e~,
which allow the three-dimensional Coulomb potential as a limit (excluded) case. A main
application was the first proof of the Landau approximation 8 for realistic potentials in a
spatially homogeneous setting.
8See Section 5.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-169-320.jpg)
![168 C. Villani
This use of the entropy dissipation for the study of grazing collisions had the merit
to display some interesting feature: a partial regularity estimate associated to the entropy
dissipation. More precisely, finiteness of the entropy dissipation, when the collision kernel
is singular, implies a partial smoothness estimate for f f. in the tensor velocity space
NN x R N. This effect is best seen at the level of the Landau equation: Landau's entropy
dissipation can be rewritten as
DL(f) = 2 f~,NxRN IFI(v -- v*)(V - V*)v/ ff*O(Iv - v*l)12dvdv*" (114)
Recall that II(v- v,) is the orthogonal projection on (v- v,) • Equation (114) is a
regularity estimate on the function ff,, but only in the variable v- v,, and only in
directions which are orthogonal to v- v,. On the whole, this means N- 1 directions
out of 2N. At the level of Boltzmann's entropy dissipation, for each point (v, v,) E ]1~2N,
these N - 1 directions are precisely the tangent plane to the (N - 1)-dimensional manifold
Svv, = {(v', v',)satisfying (5)}.
One may conclude to the simple heuristic rule: entropy dissipation yields a smoothness
estimate along collisions.
These entropy dissipation bounds have a lot of robustness in a spatially homogeneous
context, due to the tensorial structure of the entropy dissipation functional. For instance,
one can prove that if De is the entropy dissipation functional associated to a Boltzmann
operator QE "converging" in a suitable sense to Landau's operator, and DL is Landau's
entropy disipation, and fe ~ f in weak L l, then
D L(f) ~<lim inf DE(f~).
e--+O
On the other hand, precisely because they rely so much on the symmetry v +-~ v, and the
tensor product structure, these methods turn out to be inadapted to more general problems.
More efficient approaches will be presented in the sequel.
4.2. Cancellation lemma
In this and the next two sections, we shall introduce more sophisticated tools for fine
surgery on Boltzmann's operator.
As discussed above, integrals such as
f• dv.da B(v - v.,a)(g' - g) (115)
NxSN-1
are well-defined for a smooth function g, at least if the collision kernel is not too much
singular. When g is not smooth (say L1), it is not clear at all that such an integral should
converge. This is however true with great generality, due to symmetry effects. A precise](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-170-320.jpg)
![A reviewofmathematicaltopicsin collisionalkinetictheory 169
quantitative version was introduced by the author in [449] (related estimates are to be
found in Desvillettes [173] and Alexandre [6]). The estimate in [449] shows that when
the collision kernel B in (115) depends smoothly on v - v, and presents an nonintegrable
angular singularity of order v < 2, then the integral (115) converges.
The need to cover more singular situations motivated further refinement of this estimate;
here we present the sharp version which is proven in [12]. It only requires finiteness of
the cross-section for momentum transfer, M, and a very weak regularity assumption with
respect to the relative velocity variable.
PROPOSITION 6. Let B(Izl, cos0) be a collision kernel with support in 0 E [0, re~2], and
let S be defined by
Sg =--fi[~N• N-I
dr, do" B(v - v,, a)(g~, - g,).
Then, for any g E L 1 (]t~vN),
Sg=g*vS,
where the convolution kernel S is given by
f zr/2
dO sinu-2 0
S(Izl)-- IsN- l o
E ' ( 'z' ) ,]
x B ~ , c o s 0 - B(lzl,cos0 . (116)
COSN (0/2) cos(0/2)
Recall from Section 4.1 that the assumption about the deviation angle is no loss of
generality. The proof of this lemma is rather elementary and relies on the change of
' which for fixed a E S N-1 is allowed if the integration domain avoids
variables v, ~ v,,
frontal collisions (0 ~_ -+-jr).
Here is an easy corollary:
COROLLARY 6.1. With the notations z = v - v,, k = (v - v,)/lv - v,I, let
B'(z, or) -
IB()~z, a) - n(z, a)l
sup
l<X~x/2 (Z- 1)[Z[
fs B'(z,a)(1 -k.a)da.
M'(Iz[) = n-,
(117)
Then
Is(Izl)L CN[M(Izt)+ tzIM'(Izl)],](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-171-320.jpg)
![170 C. Villani
where CN is a constant depending only on N. In particular, if
IzIM'(Izl) L' (118)
then S E t~oc(]l~N).
What typical collision kernels are allowed by this lemma? It turns out that the quantity
M I measures the regularity of B with respect to the relative velocity variable in a very
weak sense. For instance, assumption (118) is satisfied for all potentials of the form
B(v - v,, a) = Iv - v,l• y > -N.
Of course, this excludes the borderline case where y = -N, which may be the most
interesting, because in dimension 3 it corresponds to Coulomb interactions .... However,
now it is homogeneity which will save the game. A quick glance at formula (116) may give
the impression that if B is homogeneous of degree -N in the relative velocity variable,
then S = 0!! Of course, this is a trap: S should be defined as a principal value operator, and
after a little bit of algebra, one finds the following corollary to Proposition 6:
COROLLARY 6.2. If B(v - v,, or)- Iv- v,l-N flo(cosO), then
S=~60,
where 3o is the Dirac measure at the origin, and
f
ro~2
fl0(cos0) logcos(0/2) sinN-2 0 dO.
Z=--IsN-2I[sN-1IjO
Note that ~ is finite as soon as the cross-section for momentum transfer (63) with b = rio,
is.
The compensation lemma of Proposition 6 has been a crucial tool, (1) to obtain
sharp entropy dissipation estimates, see next section; (2) to derive a sharp renormalized
formulation for the Boltzmann operator without cut-off, see Section 4.6.
4.3. Entropy dissipation estimates
As we have explained in Section 4.1, under certain assumptions the entropy dissipation
estimate yields a partial regularity bound on ff, when the collision kernel is singular. But
does it imply a true regularity estimate on f itself?
The first result in this direction is due to Lions [316]. He proved that if B(v - v,, or) >~
q~(lv - v,l)b(cosO), where q~ is smooth and positive, and sinN-2 0 b(cos0) ~> KO -l-v,
then for all R > 0 there is a constant CR such that](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-172-320.jpg)
![A review of mathematical topics in collisional kinetic theory 171
[Iv/f(, .)112
, ns(Ivl<e)
<~CR[D(f) 1/2 + IlfllL~],
v( 1 )
s<s0=~ l+NV___~_
1 9
(119)
The exponent so here is not optimal. The idea of the proof was clever, and a very
unexpected application of the Q+ regularity in L2 form (formula (99), with explicit
bounds needed). Starting from the entropy dissipation and formula (113), one finds some
smoothness-type estimate on v/f'f~ - ~ in ~;~U • ~U )< sN-1. Then one integrates
this estimate with respect to the variables v, and a after multiplication by an artificial,
well-chosen collision kernel B~(v - v,, a). An estimate on Qe (Vc-f, v/f) follows, where
A A
Qe is the Boltzmann operator associated with Be. Then one writes
,/7(L 9 ,/7) = 0 + (,/7, ,/7) - (,/7, ,/7),
where AAe-- fsU_~ Be do. The regularity of 0 + and the estimates on Qe (~/-f, ~/c-f) yield
the conclusion in the limit e ~ 0, after quite a bit of intricate computations.
A completely different method [449], based on the compensation lemma of last
paragraph and on Carleman's representation (91), led the author to a better estimate under
more stringent assumption on f: assume that
8(v - v,, ~)/> ~0(Iv - v,l)bo(cosO), (120)
where
~o(Izl) is continuous, ~0(Izl) > 0 if Izl > 0, (121)
and (as usual)
b0(cos0) sinN-2 0 >/KO -(l+v), K > O, v >/O. (122)
Further assume that f is positive, then
II f(t, )112 [D ],
nv/Z(lvl<R ) ~ Cg (f) + Ilfll2{ (123)
where CR depends on f only via inf{f(v); Ivl ~ R}.
Exponent v/2 is optimal, as shown by a variant of the proof. But the assumption of
local bound below for f is too strong. In a joint work with Alexandre, Desvillettes and
Wennberg [10], we were able to extend the scope of (123), and eventually obtain sharp
entropy dissipation estimates. The main difference with the previous argument was the
replacement of Carleman's representation for the use of Fourier transform. In the next
paragraph we shall make this a little bit more precise, for the moment we precisely state our
main result in [10]. To this date this may be considered as the most general manifestation
of the regularizing effects of grazing collisions.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-173-320.jpg)
![172 C. Villani
PROPOSITION 7. Let f ~ L~ A L log L(~ N) be a nonnegative distribution function, and
let B be a collision kernel satisfying assumptions (120)-(122). Then, for all R > 0 there is
a constant C -- CR such that
11,/711H./2(IvI<R) <~CR[D(Z) + 11/112~], (124)
where IlfllL~ = f f(v)(1 + Ivl2) dv and CR (which is explicit) depends only on N, ~o, K,
v, R, a lower boundfor f f dr, and an upper boundfor f f(1 + Ivl2 + [log f[).
REMARKS. (1) In the limit case v = 0, one recovers local estimates in log H. As a very
general fact, as soon as the function b0(cos0) sinN-2 0 is not integrable, i.e., when
s
Z'Oo w-~ b0(cos0) sinN-2 0 dO
diverges as O0 -+ O, then one can estimate x/'-f in Xloc, where X is the functional space
defined in Fourier representation by
X={F6L2(~N); fill>>.1
(1)}
(2) This entropy dissipation estimate is asymmetric, and this is quite in contrast with the
estimates which we shall discuss in the study of trend to equilibrium. In fact, the estimate
holds just the same for
f Q(g, f) log f
(which is not always a nonnegative expression!), if one imposes B = ~0b0 instead of
B ~>~0b0, and replaces [IfllL~ in (124) by IIfIIL~ + IlgllL~; then the constant CR would
not depend on f but only on g.
By (124), one can guess a precise heuristic point of view for the regularity properties
associated with the non-cutoff Boltzmann operator: if the angular collision kernel is
singular of order v (assumption (122)), and g is a fixed distribution function with finite
mass, energy and entropy, then the linear operator f w-~ Q(g, f) "behaves" in the
same way as thefractional diffusion operator --(--A) v/2. For Maxwellian molecules, the
intuition of this result goes back to Cercignani [138], who had noticed, thirty years ago,
that the eigenvalues of the linearized Boltzmann operator behaved like those of the power
1/4 of the Fokker-Planck operator.
4.4. Boltzmann-Plancherel formula
A key step in the proof of Proposition 7 is the use of Fourier transform. As we said earlier,
in the context of Boltzmann equation, it is only for Maxwellian collision kernels that the](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-174-320.jpg)
![A review of mathematical topics in collisional kinetic theory 173
Fourier transform leads to simple expressions. So one step of the proof (based on a lot
of fine surgery) is the reduction to the purely Maxwellian case: 45o = 1 in the previous
notations. Then, it all reduces to a sharp estimate from below of expressions of the form
dvdv, fs do" g(v,)[F(v') - F(v)] 2
2N N-1
where g is an approximation of f (say, f multiplied by a smooth cut-off function), and
F is an approximation of ~/-f. The following Plancherel-type formula, established in [10]
after the ideas of Bobylev, is the appropriate ingredient.
PROPOSITION 8. With the notations ~+ = (~ 4-I~lrr)/2,
f• fs b(k.cr)g,(F'- F)2dvdv, dcr (125)
2N N-1
(2rr) N N N-1
) 2 2
m
- 29i (~(~-)F'(~+)F'(~)))d~ &r,
with 92standingfor realpart.
This general formula can be used in many other regularity problems, in particular to
establish Sobolev-regularity estimates for the spatially homogeneous Boltzmann equation
without cut-off [185].
4.5. Regularization effects
As a consequence of Proposition 7, one can derive some (rather weak) regularization
theorems. This is immediately seen in the spatially homogeneous situation. Combining
the entropy dissipation estimate
fO
T
D(f (t, .))dt + H(f (T, .)) <<.H(fo)
with Proposition 7, the a priori bound
v/2 N
,/7 L2([o,r]; Hloc(< 11 (126)
follows at once.
Weak as it is, this regularization estimate is already useful to the existence theory for
singular collision kernels. Indeed, assume that B(v - v,, or) = Iv - v,l• where](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-175-320.jpg)
<. cllfoll , 11r L2(Hv/2)
for some well-chosen q > 1 (everything being understood in local sense).
It is worth pointing out that inequality (127) always holds for collision kernels coming
out from inverse-power forces in dimension 3. We also note that the case which appears the
most delicate to treat now, is the one of a collision kernel which is singular in the relative
velocity variable but not in the angular variable; soon we shall encounter a similar problem
in the spatially inhomogeneous setting.
Other variants of these entropy dissipation estimates lead to (strong) compactness
results. For instance, let (fn (t, l)))n6N be a sequence of probability distributions, satisfying
sup{f0
T n Dn(fn)dt + sup (llfnllL~ogL + IlfnllL~)} < -+-~,
t6[0,T]
where Dn is the entropy dissipation associated to a collision kernel Bn, approximating (in
almost everywhere sense for instance) a singular collision kernel B. Then, (fn) is strongly
compact in L1. This holds true even if there is not necessarily a uniform smoothness
estimate.
The smoothing effects which we just discussed are rather weak, but using the same kind
of techniques one can bootstrap on the regularity again and again, at least if the collision
kernel is smooth with respect to the relative velocity variable. The key inequality can be
formally written as
d
dt Ilfll2H~~ -KIIfll2 2
- - HOt+v~
2 + C IIf IIH~.
This easily leads, after integration on [0, T], to the immediate appearance of the H a+v/2
norm of f if the H a norm of f is initially finite- and, by induction, to immediate C ~
regularization. 9
9In fact this methodis but an adaptationof the "energymethod"in the studyof parabolic regularity,where
similarestimateswouldholdwiththe constantv replacedby 2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-176-320.jpg)
![A reviewof mathematical topics in collisionalkinetic theory 175
Such a study is currently worked out by Desvillettes and Wennberg, who have
announced a proof of C ~ instantaneous regularization for solutions to the spatially
homogeneous Boltzmann equation, starting from an initial datum which has finite entropy.
This result had already been proven in certain particular cases by Desvillettes [172,171,
173], and his student Prouti~re [390], with the use of Fourier-transform techniques.
4.6. Renormalized formulation, or F formula
The treatment of regularizing effects for the full, spatially inhomogeneous Boltzmann
equation without cut-off requires an additional tool because of the difficulty of defining
the collision operator. As we explained in Section 1.3, a renormalized formulation of
the collision operator, together with entropy dissipation estimates (in the sharp form of
Proposition 7), is enough to prove appearance of strong compactness.
For a long time this problem stood open, until Alexandre came up with a very clever
idea [6]. The implementation of Alexandre's ideas, based on pseudo-differential theory,
suffered from intricate computations and the impossibility to cover physically realistic
collision kernels. In a joint work [12] with Alexandre, we have given a very general
definition, based on the use of the cancellation lemma of Section 4.2, and the idea of using
the asymmetric Boltzmann operator. Here is the renormalized formulation of [12], given
in asymmetric formulation:
fl'(f) Q(g, f) -- [ffl'(f)- fl(f)] ~NxSN_ ,
+ Q(g' fl(f)) -- s215 1
dr, do- B(v - v,, o-) (g', - g,)
dr, do- Bg~,F(f, f'), (128)
where
r (f, f') = fl (f') -- fl (f) -- fl' (f) (f' -- f). (129)
If/3 is a strictly concave function or strictly convex function, then F has a fixed sign. In
the context of the study of renormalized solutions, it will be convenient to choose/3 to be
concave (sublinear), satisfying fl'(f) ~<C/(1 + f). Let us explain why each of the three
terms in (128) is then well-defined.
For the first one, we may assume f~'(f) - fl(f) 6 L ~, and then this term will satisfy
an L1oc bound as a result of the cancellation lemma of Section 4.2.
As for the second term in (128), it can be given a distributional sense, by means of
formula (112). The estimate works in a spatially inhomogeneous context because the
arguments of the collision operator are g (6 L 1, say) and fl(f) (6 L ~, say). This is the
point where it is very important to have an asymmetric weak formulation!
In the end, the third term is nonnegative as soon as/3 is concave, and since all other terms
are well-defined, it satisfies an a priori estimate in L~oc for free -just as in the argument
for the gain term in the DiPerna-Lions renormalization. 1~
l~ Section3.6.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-177-320.jpg)
![176 C. Villani
Notice that this renormalization procedure can be understood as a "commutator"
problem: find a nice expression for
fl'(f)Q(g, f)- Q(g, fl(f)). (130)
Such commutators are widely used in the study of linear diffusion operators. When L is
a diffusion operator, then fl'(f) Lf - Lfl (f) = -fl" (f) I"(f), where F is the "Dirichlet
form" associated with L. This justifies our terminology of"F formula". As a matter of fact,
the renormalization procedure above presents some similarities with the renormalization
of parabolic equations by Blanchard and Murat [76,77], and is also very close to the
renormalization of the Landau operator given in Lions [311 ].
As an illustration of the drawbacks of "soft" theories, we note that the construction of
renormalized solutions with the preceding definition is still an open problem. Instead, one
is led to introduce the following, slightly weaker, definition:
DEFINITION 2. Let f e C(~ +, 79t(]1~
xNX ]1~
vN))fq L~ (II~+ LI(~ u x N
, R v )). It is said to
be a renormalized solution of the Boltzmann equation with a defect measure, if for any
nonlinearity fl e C2(~ +, R +) such that fl(0) = 0, 0 ~<fl'(f) ~<C/(1 + f), fl"(f) < O,
one has
0
--fl(f) + v . Vxfl(f) ) fl'(f)Q(f, f), (131)
Ot
in distributional sense, and moreover f satisfies the law of mass-conservation:
Yt~>0, fR f(t'x'v)dxdv= f~ 2N (132)
We insist that this is really a notion of weak solution, not just sub-solution. Indeed, if f
were smooth, then the combination of (131) and (132) shows that there is equality in (131).
See [12], and also DiPerna and Lions [190] for similar situations.
Finally, we note that formula (128) is a general tool which finds applications outside
the theory of renormalized solutions, for instance in the study of regularity for the
spatially homogeneous equation (or even for the spatially inhomogeneous one, if suitable
integrability bounds are assumed). In this context, it is convenient to choose fl to be
convex when studying regularization for initial data which belong to L log L or LP spaces,
and concave when studying regularization for initial data which are only assumed to
be probability measures. In fact, in some sense the F formula plays for the Boltzmann
equation the same role as integration by parts plays in the energy method for diffusion
operators; therefore one should not be surprised of its great utility.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-178-320.jpg)
![A review of mathematical topics in collisional kinetic theory 177
4.7. Summary
The following two theorems summarize our current knowledge of regularizing effects,
respectively in the spatially homogeneous setting and in the framework of renormalized
solutions. We restrict to the model cases
B(v - v,, or) -- r - v,l)b(cosO), (133)
where q0(Izl) > 0 for Izl ~ 0, and b satisfies the usual singularity condition,
sinN-2 0 b(cos0) ~ KO -(l+v) as 0 --+ 0. (134)
THEOREM 9. Let B satisfy Equations (133)-(134), and let fo be a probability density on
R N, with bounded energy; fo may have a singular part, but should be distinctfrom a Dirac
mass. 11 Then,
(i) if cp is smooth and bounded from above and below, then there exists a solution f (t, v)
to the Boltzmann equation with initial datum fo, which lies in C~((0, +cx~) x RN);
(ii) if q~(lv -- v.[) --Iv -- v.[ • where Y + v > -2, and fo has finite entropy, then there
exists a weak solution f (t, v) to the Boltzmann equation with initial datum fo, such that
MY~2
&
(iii) if r - v,I) = Iv - v,I • where Y + v > O, then, without further assumptions on
fo there exists a weak solution f (t, v) to the Boltzmann equation with initial datum fo,
such that
Yt > O, f (t, .) ~ L logL(I~u).
Point (i) of this theorem is work in progress by Desvillettes and Wennberg if one assumes
that f0 has finite entropy. Then, in the case where one only assumes that f0 has finite mass
and energy, work in progress by the author [440] shows that the entropy becomes finite for
any positive time (actually, one proves estimates in L~oc(dr; L p (RN)), for arbitray large p).
Key tools in these works are the Plancherel-like formula of Section 4.4 and the cancellation
lemma of Section 4.2.
Related to point (i) are probabilistic works by Fournier [217], Fournier and Mrlrard [219,
220] who prove immediate appearance of an L 1 density if the initial datum is not a Dirac
mass, and C ~ smoothness for Maxwellian collision kernel in two dimensions [217]. The
results by Fournier and Mrlrard are considerably more restricted because of strong de-
cay assumptions on the initial datum, stringent assumptions on the smoothness of the ki-
netic collision kernel and restrictions on the strength of the singularity. However, they have
the merit to develop on Tanaka's approach [415] and to build a stochastic theory of the
Boltzmann equation, whose solution is constructed via a complicated nonlinear stochastic
l lBecause a Dirac mass is a stationary solution of the spatially homogeneous Boltzmann equation! so, starting
from a Dirac mass does not lead to any regularization.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-179-320.jpg)
![178 c. Villani
jump process. These works constitute a bridge between regularization tools stated here, and
Malliavin calculus. They also have applications to the study of stochastic particle systems
which are used in many numerical simulations [177,221,223,222,256,257]. In particular,
they are able to study the numerical error introduced in Monte Carlo simulations when
replacing a non-cutoff Boltzmann equation by a Boltzmann equation with small cut-off. 12
As for point (ii), it follows from the entropy dissipation estimates in Alexandre,
Desvillettes, Villani and Wennberg [10] and by now standard computations which can be
found, for instance, in Villani [446]. One would expect that when y > 0, C ~ smoothness
still holds; current techniques should suffice to prove this, but it remains to be done. Point
(iii) is from [440].
Uniqueness is still an open problem in this setting, on which the author is currently work-
ing. This question is related to smoothing: if one wants to use a classical Gronwall strategy,
like in the proof of uniqueness for the spatially homogeneous Landau equation [182], then
one sees that the key property to prove is that (essentially) the non-cutoff bilinear Boltz-
mann operator is not only "at least" as singular as the fractional Laplace operator of order v,
but also "at most" as singular as this one, in the sense that it maps L 2 into H -v/2 (locally).
We do hope for rapid progress in this direction!
In the case of the spatially homogeneous Landau equation, then the same regularization
results hold true, and are easier to get because the Landau equation already looks like a
nonlinear parabolic equation. Hence the smoothing effect can be recovered by standard
estimates (only complicated), bootstrap and interpolation lemmas between weighted
Sobolev and Lebesgue spaces. It is possible to go all the way to C ~ smoothness even
in cases where q/is not so smooth: for instance, qJ(Iv - v.I) = KIv - v.I z+2, y > 0. This
study was performed in Desvillettes and Villani [182]. For this case the authors proved
immediate regularization in Schwarz space, and uniqueness of the weak solution, in the
class of solutions whose energy is nonincreasing, as soon as the initial datum satisfies
f f2(v)(1 + Ivl2s) dv < +c~, 2s > 5y + 12 + s (N = 3). By the way, this uniqueness
theorem of a weak solution, building on ideas by Arsen' ev and Buryak [41], required some
precise Schauder-type estimates for a linear parabolic equation whose diffusion matrix is
not uniformly elliptic in the usual sense, and our work has motivated further research in
this area [11].
We emphasize that the picture is much less complete in the case y < 0. In particular,
for the Landau equation with Coulomb potential (y = -3 in dimension N = 3), nothing is
known beyond existence of weak solutions (see Villani [446] or the remarks in [10]).
We now turn to the spatially inhomogeneous setting. It is a striking fact that no theorem
of existence of classical small solutions of the Boltzmann equation without cut-off has ever
been proven to this day, except maybe for the isolated results in [9] which still need further
clarification. So we only discuss renormalized solutions.
12Apparently, Monte Carlo methods cannot be directly applied to the study of the non-cutoffBoltzmann
equation. The onlymethodwhichseemsableto directlydealwithnon-cutoffcollisionkernels,withoutmaking
somea prioritruncation,is the Fourier-baseddeterministicschemedescribedin Section4.9.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-180-320.jpg)
![A reviewofmathematicaltopicsincollisionalkinetictheory 179
THEOREM 10. Assume that the collision kernel B is given by (133)-(134), and ~([v -
v,I) = Iv - v,I • with
0 ~<v < 2, g ~> -N, y + v < 2. (135)
Let (fn) be a sequence of solutions 13 of the Boltzmann equation, satisfying uniform
estimates of mass, energy, entropy and entropy dissipation:
sup sup fR
n~N tc[0,T] N •
fn(t,x, v)[1 + Ixl2 + Ivl2 + logfn(t,x, v)] dx dv < -+-~.
(136)
fo T
sup D( fn (t, x, .)) dx dt < +{x). (137)
n6N
Without loss of generality, assume that fn __+f weakly in L p ([0, T]; L I(RN • INN)).
Then,
(i) f is a renormalized solution of the Boltzmann equation with a defect measure;
(ii) automatically, fn __+f strongly in L 1.
COROLLARY 10.1. Let fo be an initial datum with finite mass, energy and entropy:
f• Co(x,v)[1 -t-Ivl 2
N •
+ Ix 12+ log f0 (x, v)] dx dv < +o~.
Then there exists a renormalized solution with a defect measure, f (t,x, v), of the
Boltzmann equation, with f (0,., .) -- Co.
This theorem is proven in Alexandre and Villani [12], answering positively a conjecture
by Lions [308]. The result holds in much more generality, for instance, it suffices that the
angular collision kernel be nonintegrable (no need for a power-law singularity), and the
kinetic collision kernel need not either take the particular form of a power-law, if it satisfies
some very weak regularity assumption with respect to the relative velocity variable. And
also, it is not necessary that the collision kernel split into the product of a kinetic and an
angular collision kernel. We mention all these extensions because they are compulsory
when one wants to include realistic approximations of the Debye collision kernel, which is
not cut-off, but not in product form ....
The strategy of proof is the following. First, by Dunford-Pettis criterion, the sequence
(fn)ncr~ is weakly (relatively) compact in L 1. Then, by the renormalized formulation,
and the averaging lemmas, one shows that velocity-averages of the fn's are strongly
compact. Then the entropy dissipation regularity estimates yield bounds of regularity in
the v variable, outside of a small set and outside of a set where the fn's are very small. As
13Eitherrenormalizedsolutions,or renormalizedsolutionswitha defectmeasure,or approximatesolutions,as
in Theorem5.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-181-320.jpg)
![180 C. Villani
a consequence, the sequence (fn)n6N can be very well approximated by velocity-averages,
and therefore it lies in a strongly compact set (as in [311 ]).
Let us comment on the range of parameters in (135). The assumption y + v < 2 is
just a growth condition on the kinetic collision kernel, and is a natural generalization of
the assumption y < 2 in the DiPerna-Lions theorem; by the way, for inverse s-powers in
three dimensions, the inequality y + v < 1 always holds true. But now, we see that there
are two extensions: first, the possibility to choose v 6 [0, 2) (which is the optimal range),
secondly, the possibility to have a nonintegrable kinetic collision kernel, provided that the
singularity be homogeneous of degree -N. This feature allows to deal with Coulomb-like
cross-sections in dimension 3.
By the way, a problem which is left open is whether the theorem applies when
the collision kernel presents a nonintegrable kinetic singularity of order -N but no
angular singularity. Such collision kernels are unrealistic, but sometimes suggested as
approximations of Debye collision kernels [162]. The renormalized formulation above
is able to handle this case (contrary to the DiPerna-Lions renormalization), but without
angular singularity the regularizing effect may be lost- or is it implied by the nonintegrable
singularity, as some heuristic considerations [12] may suggest?
A result quite similar to Theorem 10 (actually simpler) holds for the Landau equation,
see Lions [311 ], and also Alexandre and Villani [13].
To this day, no clean implementation of a regularization effect has been done in the
framework of spatially inhomogeneous small solutions. Desvillettes and Golse [176] have
worked on an oversimplified model of the Boltzmann equation without cut-off, for which
L ~ solutions can be constructed for free. For this model equation they prove immediate
H a regularization for some ot which is about 1/30.
In fact, regularization for the spatially inhomogeneous Boltzmann equation without cut-
off may be understood as a hypoellipticity problem- with the main problem that the
diffusive operator is of nonlocal, nonlinear nature. F. Bouchut has recently communicated
to us some very general methods to tackle hypoelliptic transport equations in a Sobolev
space setting, via energy-type methods; certainly that kind of tools will be important in the
future.
5. The Landau approximation
In this section, we address the questions formulated in Section 2.7. In short, how to justify
the replacement of Boltzmann's operator by Landau's operator in the case of Debye (=
screened Coulomb) potential when the Debye length is very large compared to the Landau
length?
5.1. Structureof the Landauequation
We recall here the structure of the Landau operator, in asymmetric form:
QL(g' f)= Vv "(f RN dv, a(v- v,)[g,(V f) - f (Vg),]), (138)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-182-320.jpg)
![A reviewof mathematical topics in collisional kinetic theory 181
ZiZj]
aij(Z) = q/(Izl) ~ij iz[2 . (139)
The Landau operator can also be rewritten as a nonlinear diffusion operator,
Q L(g, f)= V . ({tV f -- [~f ) - Z {tijOijf -- ~"
f ,
ij
(140)
where b = V .a, c = V. b, or more explicitly
bj = Z Oiaij, c- ~-~ Ojbj,
i j
and
~=a.g, /~=b.g, ?=c.g.
There is a weak formulation, very similar to Boltzmann's, for instance,
fR QL(g, f)~o=fR g, f TLqgdvdv,,
N 2N
where
[7-s v,) = -2b(v- v,) . Vgo(v) -+-a(v- v,) : D2qg(v). (141)
Compare this with the following rewriting of Maxwell's weak formulation of the
Boltzmann equation:
f~ QB(g, f)go= fR g*f 7"q9dv dr,,
N 2N
where
[7"~0](V, V#) -- fsN-1 B(v - v,, a)(qg' - qg)da. (142)
5.2. Reformulation of the asymptotics of grazing collisions
As we explained in Section 3.5, one expects that the Boltzmann operator reduce to the
Landau operator when the angular collision kernel concentrates on grazing collisions, the
total cross-section for momentum transfer being kept finite.
The first rigorous proofs concerned the spatially homogeneous situation: Arsen'ev and
Buryak [41] for a smooth kinetic collision kernel, Goudon [248] for a kinetic singularity](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-183-320.jpg)
![182 C. Villani
of order less than 2, Villani [446] for a kinetic singularity of order less than 4. All proofs
were based on variants of the weak formulations above, and used the symmetry v +-~ v..
In order to extend these results to the spatially inhomogeneous setting, there was need for
a renormalized formulation which would encompass at the same time the Boltzmann and
Landau collision operators. This was accomplished with the results about the Boltzmann
equation without cut-off in [12]. Here is the renormalized formulation of the Landau
equation:
fl'(f)QL(g, f) = -~[ffl'(f) - fl(f)] + V. [V. (~fl(f)) - 2/~fl(f)]
fl"(f) aVfl(f) Vfl(f) (143)
fl,(f)2
Again, fl stands for a concave nonlinearity, typically fl (f) = f~ (1 + 8f). If one notes that
the second term in the fight-hand side of (143) can be rewritten as Qc (g, fl(f)), there is an
excellent analogy between this renormalization and the renormalization of the Boltzmann
operator which was presented in Section 4.6. This is what makes it possible to pass to the
limit.
The convergence of the first and second terms in the renormalized formulation can be
expressed in terms of the kernels S (appearing in the cancellation lemma) and T. This
allows one to cover very general conditions for the asymptotics of grazing collisions, and
this generality is welcome to treat such cases as the Debye approximation. Here we only
consider a nonrealistic model case.
Let (Bn)n~N be a sequence of collision kernels
Bn(v - v., or) -- clg(Iv - v.I) bn(cosO), (144)
where the kinetic collision kernel ~/, satisfies
9 (Izl) >0, (145)
Izl~c~
[ ~(~lzl) - ~(Izl) ]
q'(lzl), sup 6 L~oc(RN) (146)
1<z~<J~L )~-- 1
and the family of angular collision kernels (bn)n6Nconcentrates on grazing collisions, in
the sense
V00 > 0, sup bn (cos 0) ~ 0,
0/>00 n--+c~
L bn(k"
or)(1 - k. or) dtr ~/z > 0, Ikl = 1.
N-1 n---+ Oo
(147)
Let Sn be the kernel associated to Bn as in Section 4.2, and Tn be the linear operator
associated to Bn as in formula (142). Moreover, let
Izl2~(Izl)
(Izl) = 4(N - 1)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-184-320.jpg)
![A reviewofmathematicaltopicsin collisionalkinetictheory 183
and let QL, TL be the associated quantities entering the Landau operator. Then,
(z )
Sn(IZl) n__,~(N-1)V" ~~(IZl)
in weak-measure sense, and
% >7c
I"/----->oo
in distributional sense. In this sense one can say that the sequence of Boltzmann kernels Qn
approaches QL.
These lemmas are not enough to pass to the limit. It still remains (1) to gain strong
compactness in the sequence of solutions to the Boltzmann equation, (2) to pass to the
limit in the last term of the renormalized solution. Task (2) is a very technical job, based
on auxiliary entropy dissipation estimates and quite intricate computations, from which the
reader is unlikely to learn anything interesting. On the other hand, we explain a little bit
about the strong compactness.
5.3. Damping of oscillations in the Landau approximation
As we have seen earlier, entropy dissipation bounds for singular Boltzmann kernels entail
the appearance of strong compactness, or immediate damping of oscillations. In the case
of the Landau equation, this is the same. It turns out that it is also the same if one considers
a sequence of solutions of Boltzmann equations in which the collision kernel concentrates
on grazing collisions, in the sense of (147). This is a consequence of the following variant
of our joint results in [10]:
PROPOSITION 11. Assume that Bn(v - v,, a) ~ 450(Iv- v,I)bo,n(cosO), where 49o is
continuous, q~(lzl) > 0 for Izl > 0, and bo,n concentrates on grazing collisions, in the
sense of (147). Then there exists #: > 0 and a sequence or(n) --+ 0 such that
f0 a(n) sinN-2 0 bn(cosO) (1 - cos0) dO >/z' > 0,
n---+ oo
f Jr bn(cosO) = ~(n)
sinN-2 0 dO
(n)
> _t_oo,
n---+ oo
(148)
and there exists K > 0 such that
f0 rr sinN-2 0 bn(cosO) (021~12A 1)dO /> K min[ap(n), 1~12]. (149)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-185-320.jpg)
![184 C Villani
In particular, for any distribution function f, let F = X~ be obtained by multiplication
of q/-f with a smooth cut-offfunction X, then
I~>R
1 1 )[Dn(f) -+-IlfllL2~
]
d~: ~<C max ap(n)' R2
where Dn is the entropy dissipation functional associated with Bn, and C depends on f
only via a lower boundfor f f dv and an upper bound for f f(1 + Ivl2 + Ilog f[) dr.
As a consequence of this proposition, strong compactness is automatically gained in the
asymptotics of grazing collisions. By the way, this simplifies already existing proofs [446]
even in the spatially homogeneous setting.
5.4. Summary
Here we give a precise statement from [13].
THEOREM 12. Let Bn be a sequence of collision kernels concentrating on grazing
collisions, in the sense of (144)-(147). Further assume that ~(Izl) > 0 as Izl > 0. Let
(fn)nrN be a sequence of renormalized solutions of the Boltzmann equation (with a defect
measure)
ofn
Ot
-~-1). 7xf n = Qn(f n, fn),
satisfying uniform bounds of mass, energy, entropy, entropy dissipation. Without loss of
generality, assume that fn __+f in weak L 1. Then, the convergence is automatically strong,
and f is a renormalized solution (with a defect measure) of the Landau equation with
!P(lzl) = 4(N- 1)Izl2q~(Izl)"
REMARK. Theorem 12 allows for kinetic collision kernels with a strong singularity at the
origin, but does not allow collision kernels which are unbounded at large relative velocities.
This theorem includes all preceding results in the field, however in a spatially
homogeneous situation one could reasonably hope that present-day techniques would yield
an explicit rate of convergence (as n --+ o0) when q~is not too singular. On the other hand,
when ~(Izl) = 1/Izl 3, an improvement of this theorem even in the spatially homogeneous
setting would require a much deeper understanding of the Cauchy problem for the Landau
equation for Coulomb interaction. 14
14Seethe discussionin Section1.3ofChapter2E.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-186-320.jpg)
![A review of mathematical topics in collisional kinetic theory 185
6. Lower bounds
We conclude this chapter with estimates on the strict positivity of the solution to the
Boltzmann equation. Such results are as old as the mathematical theory of the Boltzmann
equation, since Carleman himself proved one of them. At the present time, these estimates
are limited to the spatially homogeneous setting, and it is a major open problem to get
similar bounds in the full, x-dependent framework in satisfactory generality. Therefore, we
restrict the ongoing discussion to spatially homogeneous solutions. Even in this situation,
more work remains to be done in the non-cutoff case.
6.1. Mixing effects
First consider the case when Grad's angular cut-off is satisfied, and Duhamel's for-
mula (100) applies. Then one is allowed to write
f0t
f (t, v) >~ e-f;' Lf(r'v)dr Q+(f, f)(s, v)ds, (150)
f(t, v) ~ e-fo Lf(r,v)dr fo(U), (151)
where Lf = A 9 f, A(z) -- f B(Z, a)da.
As a trivial consequence of (151), if f0 is strictly positive (resp. bounded below by a
Maxwellian), then the same property will be true for f (t, .).
But a much stronger effect holds true: whatever the initial datum, the solution will be
strictly positive at later times. Just to get an idea of this effect, assume that A is bounded
from above and below, so that
u t 6 [0, T], e-f;' Lf(r,v)dr ~ KT > 0
for some constant KT depending on T. Then, as a consequence,
f0t
f (t, v) ~ KT Q+ (f, f)(s, v) ds, 0 ~<t ~<T. (152)
Further assume that
f0 >~otls, c~>0, (153)
where 18 is the characteristic function of some ball B in velocity space, without loss of
generality B is centered on 0. From (151) it follows that
f (t, .) >/OtKT1B, O~ t <<.T.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-187-320.jpg)
![186 C. Villani
Now, plug this inside (152), to find that
fO
t
f(t, v) >~ot2K3T Q+(1B, 1B)(S, v)ds, O<~t<.T.
But Q+(1 B, 1B) is positive and bounded below in all the interior of the ball (1 + 6)B for
6 small enough. In particular, there is a positive constant 13 such that
f (t, v) >~ot2K3Tfl l(l+a)B.
By an immediate induction,
Yt > 0, Yv ~ ]1~
N, f (t, v) > O.
Precise estimates of this type have enabled A. Pulvirenti and Wennberg [392,393] to
prove optimal (Gaussian-type) bounds from below on f, for the spatially homogeneous
Boltzmann equation with Maxwellian or hard potentials. In this respect they have improved
on the old results by Carleman [119], who obtained a lower bound like e-Ivl2+e (e > 0) in
the case of hard spheres. Assumption (153) can also be dispended with, by use of the Q+
regularity.
Also the proofs in [393] are sharp enough to prove existence of a uniform (in time)
Maxwellian lower bound.
6.2. Maximumprinciple
The author suggests another explanation for the immediate appearance of strict positivity,
which is the maximumprinciple for the Boltzmann equation. The study of this principle
is still under progress, so we cannot yet display explicit lower bounds obtained with this
method; the most important feature is that it applies in the non-cutoff case. Let us just give
an idea of it.
Rewrite the spatially homogeneous Boltzmann equation as
-- B f, (f - f)
Ot N•
+ f(fRN• dr, dcr B(f~- f,)). (154)
We assume that we deal with a C ~ solution, which is reasonable when the kinetic collision
kernel is nice and when there is a nonintegrable angular singularity. The good point about
the decomposition (154) is that it is well-defined 15 even in the non-cutoff case.
Assume now, by contradiction, that there is some point (to, v0) (to > 0) such that
f(to, v0) = 0. Obviously, 8f/St = 0 at (to, v0). Thus the left-hand side of (154), and also
15By cancellation lemma, for instance, see Section 4.2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-188-320.jpg)
![A reviewof mathematical topics in collisionalkinetic theory 187
the second term on the right-hand side vanish at (to, v0). But, when v = v0, f' - f ~>0, for
all vf. Thus the integrand in the first term on the left-hand side of (154) is nonnegative, but
the integral vanishes, so f' = f = 0, for all vt. This entails that f is identically 0, which
is impossible. In other words, we have recovered the weak result that f(t, .) is strictly
positive on the whole of ~U for t > 0.
6.3. Summary
THEOREM 13. Let B be a collision kernel of the form B(v - v., or) = Iv - v.l•
where y ~ O. Let fo be an initial datum with finite mass and energy, and f (t,-) be a
solution of the spatially homogeneous Boltzmann equation. Then,
(i) if Grad's angular cut-off condition holds, then for any to > O, there exists a
Maxwellian distribution M(v) such that for all t ~ to, f (t, v) >1M(v);
(ii) if Grad's angular cut-off condition does not hold, and f (t, v) is a C ~ function on
(0, +oo) x IRy, then for any t > O, v ~ It~N, f (t, v) > O.
Point (i) is due to A. Pulvirenti and Wennberg [392,393]. Point (ii) was first proven
by Fournier, using delicate probabilistic methods, in the special case of the Kac equation
without cut-off [218], then also for the two-dimensional Boltzmann equation under
technical restrictions [218]. Then it was proven in a much simpler way by the author, with
the analytical method sketched above. Current work is aiming at transforming this estimate
into a quantitative one.
We note that in the case of the Landau equation with Maxwellian or hard potential [182],
one can prove a theorem similar to that of A. Pulvirenti and Wennberg by means of the
standard maximum principle for parabolic equations. 16
16Actually,in [182] the statedresult is not uniform in time, but, as suggested to us by E. Carlen,a uniform bound
is easily obtainedby tracing back all the constants: sincethey are uniform for t ~ (e, 2e) and do not depend on
the initial datum, it follows that they are uniform in t > e.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-189-320.jpg)
![Preface
The motion of fluids has intrigued scientists since antiquity and we may say that the
field of mathematical fluid dynamics originated more than two centuries ago. In 1755
Euler [2] gave a mathematical formulation of the principle of conservation of mass in
terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear
and angular momentum by PDEs. Material symmetry and frame invariance were used by
Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of
internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint-
Venant [11] and Stokes [12].
In the 19th century no sharp distinction was drawn between mathematicians and
physicists as we sometime see in more recent times. The formulation of the equations
of fluid motion could be considered as either mathematics or physics. The first work
in fluid dynamics that has a "modern" mathematical flavor may have been done by
Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous
problem. Riemann recognized the mathematical nature of the entropy. This notion led
him to his duality method for solving the non-characteristic Cauchy problem for linear
hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his
contemporaries. What we now call the Cauchy problem for a PDE and the search for its
solution did not have the significance that it is accorded nowadays. Even Poincar6 did not
raise that kind of question in his Th~orie des tourbillons.
For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly
accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the
Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28.
Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of
the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much
as Riemann, Leray developed new mathematical tools which proved to have independent
interest: e.g., weak solutions (that we now call Leray's solutions in this context) and
topological degree (a joint work with Schauder [4]).
Since the 1930s, the interest that mathematicians devote to fluid dynamics has
unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von
Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a
Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on
the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic
displays such a variety of models and questions that thousands of scientists, among them
many mathematicians, focus their research on fluid dynamics.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-191-320.jpg)
![A review of mathematical topics in collisional kinetic theory 191
In Chapter 2A we have discussed Boltzmann's H theorem, and the natural conjecture that
the solution of Boltzmann's equation converges towards statistical equilibrium, which is
a global Maxwellian distribution. In this chapter we shall study this problem of trend to
equilibrium, and also enlarge a little bit the discussion to models of collisional kinetic
theory which are variants of the Boltzmann equation: for instance, Fokker-Planck-type
equations, or simple models for granular media. The Cauchy problem for these equations
is usually not so challenging as for the Boltzmann equation, but the study of trend to
equilibrium for these models may be very interesting (both in itself, and to enlighten the
Boltzmann case).
As a general fact, one of the main features of many collisional kinetic systems is their
tendency to converge to an equilibrium distribution as time becomes large, and very often
a thermodynamical principle underlies this property: there is a distinguished Lyapunov
functional, or entropy, and the equilibrium distribution achieves the minimum of this
functional under constraints imposed by the conservation laws. In Section 1 we shall review
some of these models. For each example, we shall be interested in the functional of entropy
dissipation, defined by the equation
d] e[:<t)],
D(fo) =-~--~ t=0
where E is the Lyapunov functional, and (f(t))t>~o the solution to the equation under
study, f (0) = f0. We shall use the denomination "entropy dissipation" even when E is not
the usual Boltzmann entropy.
Traditional approaches for the study of trend to equilibrium rely on soft methods,
like compactness arguments, or linearization techniques, which ideally yield rates of
convergence. In Section 2 we briefly review both methods and explain why they cannot
yield definitive answers, and should be complemented with other, more constructive
methods. This will lead us to discuss entropy dissipation methods, starting from Section 3.
In Section 4, we expose quantitative versions of the H theorem for the Boltzmann and
Landau operators, in the form of some functional inequalities. Then in Section 5 we show
how these inequalities can be used for the study of the trend to equilibrium for the spatially
homogeneous Boltzmann and Landau equations.
Section 6 is devoted to a class of collision models which exhibit a particular gradient
structure. Specific tools have been devised to establish variants of the H theorem in this
case.
Finally, Section 7 deals with the subtle role of the position variable for spatially
inhomogeneous models. The construction of this area is only beginning.
1. A gallery of entropy-dissipating kinetic models
Let us first review some of the basic models and the associated entropy functionals,
equilibria, entropy dissipation functionals. We shall not hesitate to copy-cut some of the
formulas already written in our introductory chapter.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-192-320.jpg)
![192 C. Villani
1.1. Spatially homogeneous models
These models read
of
= Q(f), t ~ O, 1) e ]I~N ,
Ot
where the collision operator Q, linear or not, may be
(1) the Boltzmann operator,
"
Q(f) = QB(f, f)= dv, da B(v - v,, a)(f f, - ff,);
N N-1
(155)
then there are three conservation laws: mass, momentum and energy. Moreover, the natural
Lyapunov functional is the H-functional,
H (f) = fRN f log f,
and its dissipation is given by the by now familiar functional
if ,,
D(f) -- -~ dv dr, dcr B(v - v,, a)(f'f~, - ff,) log f f* ~>O. (156)
ff,
Define p, u, T by the usual formulas (1), then the equilibrium is the Maxwellian
M(v)--Mf(v)=
Iv-ul2
e 2T
(2Jr T)N/2 "
IMPORTANT REMARK. We shall only consider here the case of the Boltzmann equation
with finite temperature. In the case of infinite temperature, almost nothing is known, except
for the very interesting recent contribution by Bobylev and Cercignani [81].
It should be noted that, since M has the same moments as f up to order 2,
H (f) - H (M) = fR f log f
N M'
which is nothing but the Kullback relative entropy of f with respect to M, and that we
shall denote by H (f[ M). Generally speaking, the Kullback relative entropy between two
probability densities (or more generally two nonnegative distributions) f and g is given by
the formula
H(flg) = f f log f.
g
(157)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-193-320.jpg)
![A reviewof mathematical topics in collisional kinetictheory 193
It is well-known 1that H(flg) ~ 0 as soon as f and g have the same mass;
(2) the Landau operator,
Q(f)=QL(f,f)=Vv.(s dv, a(v-v,)[f,(Vf)- f(Vf),]),
aij (z) -- O(Izl) aij izi2 ,
(158)
(159)
in this case there are also three conservation laws, and the natural Lyapunov functional is
also the H-functional. Now the entropy dissipation is
1s
DL(f ) -- -~ NxRUff, lP(Iv- v,l)lFI(v- v,)(V(log f)- [V(log f)],) 12,
(160)
where H (z) stands for the orthogonal projector onto z• As for the equilibrium state, it is
still the same as for the Boltzmann equation;
(3) the linear Fokker-Planck operator,
Q (f) = Q FP (f) -- Vv" (gv f + f v). (161)
In this case there is only one conservation law, the mass (p = f f dr), and the natural
Lyapunov functional is the free energy:this is the sum of the H-functional and the kinetic
energy,
E(f) fir f log f + fR fly]2
= dr. (162)
N N 2
Moreover, the entropy dissipation is
DFP(f) = fRu f -+v
2
dr,
which can be rewritten as the so-called relative Fisher information of f with respect to M,
thereafter denoted by I (flM). More generally,
I (fig) -- ~N f
f
V log g (163)
1The classical proof is to rewrite (157) as f f[- log(g/f) + g/f - 1] (or as f g[(f/g) log(f/g) - (f/g) + 1])
and to use the inequality log X ~<X - 1 (or X log X ~>X - 1). Compare with the Cercignani-Lampis trick of
Equation (50).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-194-320.jpg)
![194 C. Villani
Compare with the definition of the relative Kullback entropy (157);
(4) a coupled Fokker-Planck operator, like
p"v~. [rv~f + f(v - u)],
where 0 ~<a ~< 1 and p, u, T are coupled to f by the usual formulas (1). In this case there
are three conservation laws, the natural Lyapunov functional is the H-functional, and the
entropy dissipation is
P~fR~u f 12__pOt f)
Vv log ~--f I(flM .
The equilibrium is the same as for the Boltzmann operator.
Other couplings are possible: one may decide to couple only T, or only u ... ;
(5) some entropy-dissipating model for granular flow, like the one-dimensional model
proposed in [70],
Q(f) = Vv . (f Vv(f 9 u)), (164)
where U (z) = Izl3/3. Then there are two conservation laws, mass and momentum; and the
natural Lyapunov functional is
1s f(v)f(w)U(v-w) dvdw,
E(f) = -~ 2N
(165)
while its dissipation is
D(f) = s flVU . fl 2.
Moreover the equilibrium is p6u, i.e., a multiple of the Dirac mass located at the mean
velocity.
A particular feature of this model is its gradient flow structure. Generally speaking,
models of the form
Ofat-- V " (f V6-~)' (166)
where E is some energy functional and 6E/6f stands for its gradient with respect to
the usual L 2 structure, can be considered as gradient flows [364,365], via geometric and
analytical considerations which are strongly linked with the Wasserstein distance. 2 An
2TheWassersteindistanceis definedby Equation(244).The gradientstructureis explainedin Section6.1.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-195-320.jpg)
![A review of mathematical topics in collisional kinetic theory 195
integration by parts shows that solutions of (166) admit E as a Lyapunov functional, and
the dissipation is given by
D(f) = fIRNf
6E
Falling into this category is in particular the model for granular flow discussed in [68],
in which one adds up the collision operators (164) and (161).
1.2. Spatially inhomogeneous models
These models can be written in the general form
0f
+ v. Vxf + F(x). Vvf -- Q(f), t ~ O, X E •N, V E R N, (167)
Ot
where F is the sum of all macroscopic forces acting on the system, and Q is one of
the collision operators described in the previous paragraph (acting only on the velocity
variable!).
In the sequel we shall only consider the situation when the total mass of the gas is finite;
without loss of generality it will be normalized to 1. We mention however that the case of
infinite mass deserves interest and may be studied in the spirit of [310].
If the total mass is finite, then among the forces must be a confinement which prevents
the system from escaping at infinity, and ensures the existence of a relevant equilibrium
state. There are several possibilities:
Potential confinement. Assume that the particles interact with the background environ-
ment via some fixed potential, V(x). Then the force is just
F(x) = -VV(x).
The minimum requirement for V to be confining is e-v 6 L 1. Since V is defined up to an
additive constant, one can assume without loss of generality that
fR e-V(x) dx - 1.
N
The presence of the confining potential does not harm the conservation of mass, of
course; on the other hand, when Q is a Boltzmann-type collision operator (with three
conservation laws), it usually destroys the conservations of momentum and energy. Instead,
there is conservation of the total mechanical energy,
E v2]
f (x, v) V (x) + dx dv
N • --2 "](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-196-320.jpg)
![196 C. Villani
And as far as the entropy is concerned, it is not changed for Boltzmann-type models:
this is still the usual H-functional, the only difference being that now the phase space is
f
H (f) -- / f log f.
JRN•
This similarity is a consequence of the physical assumption that collisions are localized in
space.
For the linear Fokker-Planck equation, the situation is different: to the free energy one
has to add the potential energy, so the natural Lyapunov functional is
E(f) = LN xRN f log f + f~N >
<
~
N
f (x v)[V(x) + I~ 2]
, ~ dxdv.
Box confinement. Another possible confinement is when the system is enclosed in a box
X C NN, with suitable boundary conditions. The most standard case, namely specular
reflection, is a limit case of the preceding one: choose V = +cx~ outside of X, V -- const.
within X. When specular reflection is imposed, then the energy conservation is restored
(not momentum conservation), and the Lyapunov functional is the same as in the spatially
homogeneous case, only integrated with respect to the x variable. For other boundary
conditions such as diffusive, the entropy functional should be modified [141,143].
Torus confinement. This is the most convenient case from the mathematical point of view:
set the system in the toms TN, so that there are no boundaries. Physicists also use such
models for discussing theoretical questions, and numerical analysts sometimes find them
convenient.
Additional force terms. Many models include other force terms, in particular self-
consistent effects described by mean-field interactions: typically,
F(x) = -74~(x),
I~ =~,p, P = f]t{N f d v ,
where r is an interaction potential between particles. As we already mentioned, from the
physical point of view it is not always clear whether interactions should be modelled via
collisions, or mean-field forces, or both .... A very popular model is the Vlasov-Fokker-
Planck equation, in which the collision operator is the Fokker-Planck operator and the
forces include both confinement and self-consistent interaction. Let us rewrite the model
explicitly:
Of
-57 + v . Vxf + F(x) . Vvf = Vv . (Vvf + fv),
F -- -V(V + r 9 p), p = fRN f dr.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-197-320.jpg)
![A review of mathematical topics in collisional kinetic theory 197
If the interaction is Coulomb, then one speaks of Vlasov-Poisson-Fokker-Planck model;
this case is very singular, but it has a lot of additional structure because (by definition) the
potential 4~is the fundamental solution of the Laplace operator.
In the self-consistent case, one has to add a term of interaction energy to the free energy:
2 N•
p(x)p(y)ck(x - y) dx dy.
Then the entropy dissipation is unchanged.
Let us now turn to equilibrium states. Their classification in a spatially inhomogeneous
context is quite a tedious task. Many subcases have to be considered, the dimension of the
space comes into play, and also the symmetries of the problem. We are not aware of any
systematic treatment; we shall only consider the most typical situations.
9 The Boltzmann (or Landau) equation in a box. Then, in dimension N = 2, 3 there is
a unique steady state which takes the form of a global Maxwellian: f(x, v) = M(v).
The mass and temperature of the Maxwellian are determined by the conservation
laws, while the mean velocity is 0. This result holds true on the condition that the box
be not circular in dimension N = 2, or cylindric in dimension N = 3 (i.e., with an
axis of symmetry). For this one can consult [254,167,143].
9 The Boltzmann (or Landau) equation in a confining potential. Then, the unique steady
state has the form f(x, v) = e-V(X)M(v). Again, the mass and temperature of M are
determined by the conservation laws, and the mean velocity is 0. This result holds
true if the potential V is not quadratic; if it is, then there exist periodic (in time)
solutions, which can be considered as stationary even if they are time-dependent. This
was already noticed by Boltzmann (see, for instance, [143]).
9 The Boltzmann (or Landau) equation in a toms. Then, the unique steady state has the
form f (x, v) = M (v) where M is an absolute Maxwellian. The mass and temperature,
but also the mean velocity of M are determined by the conservation laws.
9 The Fokker-Planck equation in a confining potential. Then, the unique steady state is
f(x, v) -- e-V(x)M(v), where M is the Maxwellian with unit temperature and zero
mean. The mass is of course determined by the conservation law.
9 The Vlasov-Fokker-Planck equation in a confining potential. In general there is a
unique steady state in this situation, and it takes the form f(x, v) = p~(x)M(v),
where M is the Maxwellian with unit temperature and zero mean. The density p~ is
nonexplicit, but solves a nonlinear equation of the form
e-(V+4~,p~)
fRN e-(V+g'*P~ dx
There are also variational formulations of this problem. In the case of the Vlasov-
Poisson-Fokker-Planck equation, a detailed survey of the situation is given by
Dolbeault [197].
In all the preceding discussion, we have avoided the models for granular collisions,
Equation (164). A naive guess would be that the natural Lyapunov functional, in the
spatially inhomogeneous case, is obtained by integrating its spatially homogeneous](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-198-320.jpg)
![198 C. Villani
counterpart, Equation (165), in the x variable. This is false! Because the transport operator
-v. Vx may have an influence on the evolution of this functional.
1.3. Related models
The following models are not kinetic models, but have come to be studied by members of
the kinetic community because of the unity of methods and problematics.
- The spatial Fokker-Planck equation, or Smoluchowski 3 equation [399]
Op
-- Vx " (Txp -Jr-RVV(x)), t ~ 0, x E ~N. (168)
0t
One always assume e-v E L 1, and without loss of generality e-v should be a
probability measure, just as p. For this equation the natural Lyapunov functional is the
free energy, or relative entropy, H (p le- v), and the entropy dissipation coincides with
the relative Fisher information, I (pie -v). For a summary of recent studies concerning
the trend to equilibrium for (168), the reader may consult Arnold et al. [39], or
Markowich and Villani [330].
- Equations modelling porous medium with confinement:
Op
= 7x. (Vx P(p) + pVV(x)), t ~ O, X E ]I~N, (169)
Ot
where P is a nonlinearity, P(p) standing for a pressure term, for instance P(p) = p•
In this last case (power law), equations like (169), with a quadratic confinement po-
tential, naturally arise as rescaled versions of their counterparts without confinement.
The natural Lyapunov functional for (169) is
f A(p)dx + fpvr
where P (p) = p A' (p) - A (p).
The trend to equilibrium for (169) has been studied independently by Carrillo and
Toscani [131], Dolbeault and Del Pino [163], Otto [364] for the power law case, then
more generally by Carrillo et al. [129].
One of the most remarkable features of Equations (168) and (169) is that they have the
form of a gradient flow,
Ot
For a general discussion of the implications, see, for instance, Otto and Villani [365],
Markowich and Villani [330].
3There are several types of equations which are called after Smoluchowski!](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-199-320.jpg)
![A review of mathematical topics in collisional kinetic theory 199
1.4. General comments
In this section we shall informally discuss the features which may help the trend to
equilibrium, or on the contrary make it more difficult- both from the physical and from
the mathematical point of view.
9 First of all, the distribution tails are usually at the origin of the worst difficulties.
By distribution tails, we mean how fast the distribution function decreases as Ivl ~ ~,
or Ix l~ c~. This is not only a technical point; Bobylev has shown that large tails could
be a true obstacle to a good trend to equilibrium for the Boltzmann equation, even in the
spatially homogeneous case. More precisely, he proved the following result [79]. Consider
the spatially homogeneous Boltzmann equation with Maxwell collision kernel (with or
without cut-off), and fix the mass, momentum, energy of the initial datum f0. Let M(v) be
the corresponding equilibrium state. Then, for any e > 0 one can construct an initial datum
f0 = f~ such that the associated solution f~ (t, v) of the Cauchy problem satisfies
vt Ilf (t,.)-Mll Kee-et, Ke >0.
At this point we should make a remark to be honest: an eye observation of a plot of
these particular solutions will show hardly any departure from equilibrium, because most
of the discrepancy between fe and M is located at very high velocities - and because the
constant Ke is rather small. This illustrates the general fact that precise "experimental"
information about rates of convergence to equilibrium is very difficult to have, if one wants
to take into account distribution tails.
9 Moreover, recent studies have shown that the Boltzmann equation, due to its nonlocal
nature, is more sensitive to this tail problem than diffusive models like Landau or Fokker-
Planck equations. For the latter equations, it is not possible to construct "pathological"
solutions as Bobylev; the trend to equilibrium is typically exponential, with a rate which is
bounded below. We shall come back to this point, which by the way is also folklore in the
study of Markov processes: it is known that jump processes have more difficulties in going
to equilibrium than diffusion processes.
9 Next, it is clear that the more collisions there are, the more likely convergence is bound
to be fast. This is why the size of the collision kernel does matter, in particular difficulties
arise in the study of hard potentials because of the vanishing of the collision kernel at zero
relative velocities; and also in the study of soft potentials because of the vanishing of the
collision kernel for large relative velocities. A common belief is that the problem is worse
for soft potentials than for hard. Also note that hard potentials are associated with a good
control of the distribution tails, while soft potentials are not.
Studies of the linearized operator show that in principle, one could expect an exponential
decay to equilibrium for the spatially homogeneous Boltzmann equation with hard or
Maxwellian potentials (under strong control of the distribution tails), while for soft
potentials the best that one could hope is decay like O(e -t~) for some ot E (0, 1) (see
Caflisch [111 ]). This is of course related to the fact that there is a spectral gap in the first
case, not in the second one.
9 In the case of Boltzmann or Landau models (or some versions of coupled Fokker-
Planck), the collision frequency also depends on the density of particles. This of course can](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-200-320.jpg)
![200 C. Villani
be seen via the fact that Boltzmann and Landau operators are quadratic, while the linear
Fokker-Planck is not. As a consequence, the trend to equilibrium should be extremely slow
at places where the density stays low: typically, very large positions. Therefore, the trend
to equilibrium is expected to hold on extremely long scales of times when one considers
the Boltzmann equation in a confinement potential, as opposed to the Boltzmann equation
in a finite box.
1~ In the x-dependent case, a strong mathematical difficulty arises: the existence of
local equilibria. These are states which make the entropy dissipation vanish, but are not
stationary states. In fact they are in equilibrium with respect to the velocity variable, but
not with respect to the position variable; for instance they are local Maxwellians Mx(v),
with parameters p, u, T depending on x. Of course the trend to equilibrium is expected to
be slowed down whenever the system comes close to such a state. We shall discuss this
problem in more detail in Section 7.
t~ Finally, a gradient flow structure often brings more tools to study the trend to
equilibrium. We shall see this in the study of such models as (161) or (164). As we
mentioned in Section 2.4 of Chapter 2A, in the case of the spatially homogeneous
Boltzmann equation no gradient flow structure has been identified. Moreover, for all the
spatially inhomogeneous equations which are considered here, the existence of the local
equilibria rules out the possibility of such a structure.
2. Nonconstructive methods
In this section, we briefly review traditional methods for studying the convergence to
equilibrium.
2.1. Classical strategy
A preliminary step of (almost) all methods is to identify stationary states by searching
for solutions of the functional equation D(f) = 0, or more generally ff D(f(t)) dt = O.
Once uniqueness of the stationary solution has been shown, then weak convergence of
the solution towards equilibrium is often an easy matter by the use of compactness tools.
Uniqueness may hold within some subclass of functions which is left invariant by the flow.
For instance, in the case of the spatially homogeneous Boltzmann equation, it is easy
to prove weak convergence as n --+ cx~ of f(n + t, U)nEN towards the fight Maxwellian
distribution in weak-LP ([0, T] x ~U), as soon as
/,
lim limsup [ f(t, v)lvl 2dv = 0.
R--+ cx~ t--~ cx~ ,] lv l>/ R
(170)
Condition (170), thereafter referred to as "tightness of the energy", ensures that there
is no leak of energy at large velocities, and that f (t, .) does converge towards the fight
Maxwellian distribution- and not towards a Maxwellian with too low temperature. In all](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-201-320.jpg)
![A review of mathematical topics in collisional kinetic theory 201
the sequel, we will assume that the moments of f are normalized, so that the equilibrium
distribution is the standard Maxwellian M with zero mean and unit temperature.
As a typical result, under general conditions Arkeryd [17] proved that the solution to
the spatially homogeneous Boltzmann equation with hard potentials does converge to M,
weakly in L 1, as t ~ cx~.This result is facilitated by the fact that Equation (170) is very
easy to prove for hard potentials, while it is a (seemingly very difficult) open problem for
soft potentials.
In the framework of the spatially homogeneous Boltzmann equation with Max-
wellian collision kernel, other approaches are possible, which do not rely on the
entropy dissipation. Truesdell [274] was the first one to use such a method: he proved
that all spherical moments satisfy closed differential equations, and converge towards
corresponding moments of M. This implies weak convergence of f (t, .) towards M.
Also contracting metrics4 can be used for such a purpose along the ideas of Tanaka [414,
415].
A refinement is to prove strong convergence of f(t, .) towards M as t --~ cxz, for
instance, as a consequence of some uniform (in time) smoothness estimates. The first result
of this kind is due to Carleman [118]: he proved uniform equicontinuity of the family
(f(t, "))t>~o when f is the isotropic solution of the spatially homogeneous Boltzmann
equation with hard spheres, assuming that the initial datum decays in O(1/Ivl6). As
a consequence, he recovered uniform convergence to equilibrium. This method was
improved by Gustafsson [270] who proved strong L p convergence for the solution of
the spatially homogeneous Boltzmann equation with hard potentials, under an ad hoc LP
assumption on the initial datum.
In the much more general framework of the spatially inhomogeneous Boltzmann
equation, by use of the Q+ regularity, Lions [308] proved strong L 1 compactness as
t --+ cx~,say when the system is confined in a torus. This however is not sufficient to prove
convergence, because there is no clue of how to prove the spatially-inhomogeneous variant
of (170),
lim limsupf~, dxfv
R--+ oe t--+ o~ N i/>R
f(t, v)lvl 2 dv = 0.
At this point we have to recognize that there is, to this date, no result of trend to equilibrium
in the spatially inhomogeneous context, except in the perturbative framework of close-
to-equilibrium5 solutions: see, for instance, [286] (perturbation setting in whole space)
or [404] (in a bounded convex domain) - with just one exception: the case of a box
with uniform Maxwellian diffuse boundary conditions, which was solved by Arkeryd
and Nouri [35] in a non-perturbative setting. On the contrary, it is rather easy to prove
convergence to equilibrium for, e.g., the spatially inhomogeneous linear Fokker-Planck
equation.
Once strong convergence to equilibrium has been established (for instance, in the
case of the spatially homogeneous Boltzmann equation with hard potentials), a natural
4SeeSection2 in Chapter2D.
5Of courseit is not a verysatisfactorysituationif one is ableto proveconvergenceto equilibriumonlywhen
one startsextremelycloseto equilibrium....](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-202-320.jpg)
![202 C.Villani
refinement is to ask for a rate of convergence. In the "good" cases, a hard work
leads to exponential rates of decay thanks to linearization techniques and the study of
the spectral gap of the linearized operator. This strategy was successfully applied by
Arkeryd [23], Wennberg [456] to the spatially homogeneous Boltzmann equation with
hard, or Maxwellian potentials. In the spatially inhomogeneous context, it was developed
by the Japanese school under the assumption that the initial datum is already extremely
close to equilibrium.
In the case of soft potentials, though there is no spectral gap for the linearized operator,
Caflisch [111] was able to prove convergence to equilibrium like e-t~ for some exponent
fl 6 (0, 1) - also under the assumption that the initial datum belong to a very small
neighborhood of the equilibrium.
2.2. Why askfor more ?
The preceding results, as important as they may be, cannot be considered as a definitive
answer to the problem of convergence to equilibrium. There are at least two reasons for
that:
(1) Non-constructiveness. The spectral gap (when it exists, which is not always the case !)
is usually nonexplicit: for the Boltzmann equation with hard or soft potentials there is only
one exception, the spatially homogeneous operator with Maxwellian collision kernel. What
is more problematic, nobody knows how to get estimates on its size: usual arguments for
proving its existence rely on Weyl's theorem, which asserts that the essential spectrum is
invariant under compact perturbation. But this theorem, which is based on a compactness
argument, is nonexplicit ....
Another problem arises because the natural space for the linearized operator (the space
in which it is self-adjoint) is typically LZ(M-1), endowed with the norm []f]122(M_l) =
f fZ/M, which is of course much narrower than the natural spaces for the Cauchy problem
(say, Lebesgue or Sobolev spaces with polynomial weights). A new compactness argument
is needed [457] to prove the existence of a spectral gap in these much larger spaces.
REMARK. This problem of functional space arises even for linear equations! For instance,
if one considers the Fokker-Planck equation, then the spectral gap exists in the functional
space L2(M -1), but one would like to prove exponential convergence under the sole
assumption that the initial datum possess finite entropy and energy.
(2) Nature of the linearization procedure. In fact, even if linearization may predict an
asymptotic rate of convergence, it is by nature unable to yield explicit results. Indeed,
it only shows exponential convergence in a very small neighborhood of the equilibrium: a
neighborhood in which nonlinearities are negligible in front of the linear terms. It cannot
say anything on the time the solution needs to enter such a neighborhood ....
This of course does not mean that linearization is in essence a bad method, but that it is
a valuable method onlyfor perturbations of equilibria.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-203-320.jpg)
![204 C. Villani
is the H theorem. This approach has the merit to stand upon a clear physical basis, and
experience has shown its robustness and flexibility.
RULE 1. The "discrepancy" between a distribution function f and the equilibrium f~
should not be measured by the L 1 norm, but rather by E[flf~] = E(f) - E(f~),
thereafter called relative entropy by abuse of language. Thus, one should not try to prove
that f(t) converges to f~ in L 1, but rather show that E(f(t)) --+ E(fe~) as t --+ ~, which
will be called "convergence in relative entropy". A separate issue is to understand whether
convergence in relative entropy implies convergence in some more traditional sense.
RULE 2. One considers as a main object of study the entropy dissipation functional D. Of
course, the definition of the entropy dissipation relies on the evolution equation; but it is
important to consider D as a functional that can be applied to any function, solution or not
of the equation.
RULE 3. One tries to quantify the following idea: if at some given time t, f (t) is far
from f~, then E (f) will decrease notably at later times.
Before turning to less abstract considerations, we comment on the idea to measure the
distance in terms of the entropy, rather than, say, in terms of the well-known L 1 distance.
A first remark is that there is no physical meaning, in the context of kinetic equations,
in L1 distance. Some rather violent words by Truesdell will illustrate this. After proving
exponential convergence of all moments in the framework of the spatially homogeneous
Boltzmann equation with Maxwell collision kernel, he adds [274, p. 116] "Very likely
it can be shown that [the solution] itself approaches Maxwellian form, but there is little
interest in this refinement." A justification of this opinion is given on p. 112: "Since apart
from the entropy it is only the moments of the distribution function that have physical
significance, the result sought is unnecessarily strong". Thus, at the same time that he
attacks the relevance of L 1results, Truesdell implicitly supports entropy results ....
A second remark is that, very often, convergence of the entropy implies convergence in
L 1 sense. In the case of the H-functional, or more generally when E (f) - E (f~) takes
the form of a relative Kullback entropy, this is a well-known result. Indeed, the famous
(and elementary) Csisz~ir-Kullback-Pinsker inequality states that whenever f and g are
two probability distributions,
1 f i
2llf-gll 2
- L' ~< f log -- = H (fig).
g
In many other instances, especially when a gradient flow structure is present, the quantity
E (f) - E (f~) can also be shown to control some power of the Wasserstein distance. 7 For
this see in particular Otto and Villani [365]. A basic example is the Talagrand inequality,
1W(f, M) 2 <~H(fIM),
2
7Equation(244)below.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-205-320.jpg)
![A review of mathematical topics in collisional kinetic theory 205
where M is the zero-mean, unit-temperature Maxwellian distribution. Usually, one can
then obtain control of the L 1 norm via some ad hoc interpolation procedure [130].
As a final remark, we comment on the entropy dissipation equality itself- say in the case
of the Boltzmann equation. As we saw, formally, solutions of the spatially homogeneous
Boltzmann equation satisfy the identity
d
--H(f(t .)) -- -D(f)
dt ' '
but when is this rigorous? It was actually proven by Lu [326] that this equality always holds
for hard potentials with cut-off, under the sole assumptions that the initial datum has finite
mass, energy and entropy. In fact, we shall always work under much stronger conditions.
Thus, in all the sequel, we shall always consider situations in which the estimates for
the Cauchy problem are strong enough, that the entropy dissipation identity can be made
rigorous. Such is not the case, for instance, in the framework of the DiPerna-Lions theory
of renormalized solutions, s
3.2. Entropy-entropy dissipation inequalities
When trying to implement the preceding general principles, one can be lucky enough to
prove an entropy-entropy dissipation inequality" this is a functional inequality of the type
D(f) >~O(E[flfoc]), (171)
where H ~ O (H) is some continuous function, strictly positive when H > 0. The main
idea is that "entropy dissipation controls relative entropy".
Such an inequality implies an immediate solution to the problem of trend to equi-
librium. Indeed, let f(t) be a solution of the evolution equation. Since D(f(t))=
-(d/dt)E[f(t)lf~], it follows that the relative entropy H(t) -- E[f(t, ")lfc~] satisfies
the differential inequality
dt
---H(t) ~ O(H(t)). (172)
This implies that H(t) --+ 0 as t --+ +oo, and if the function 69 is known with enough
details, one can compute an explicit rate of convergence. For instance, a linear bound like
D(f) >i 2~.E[fJfcc]
will entail exponential convergence to equilibrium, relative entropy converging to 0 like
e-2xt. On the other hand, an exponent bigger than 1,
D(f) >1KE[flfool l+a (K > O, ot > O)
8In any case, this theory should be hopelessly excluded from any study of trend to equilibrium until energy
conservation, and even local energy conservation, has been proven.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-206-320.jpg)
![206 C. Villani
will entail "polynomial" rate of convergence to equilibrium, the entropy going down like
O(t-1/a).
Situations in which the exponent is lower than 1 are very rare; in such cases the system
converges to equilibrium in finite time. This occurs in certain simple model equations for
granular media [426].
Very often, one cannot hope for such a strong inequality as (171), but one can prove such
an inequality in a restricted class of functions:
D(f) ~ Of(E[flf~]), (173)
where the explicit form of Of may depend on some features of f such as its size in some
(weighted) Lebesgue spaces, its strict positivity, its smoothness, etc.: all kinds of a priori
estimates which should be established independently.
In collisional kinetic theory, there are many situations in which entropy-entropy
dissipation inequalities cannot hold true, in particular for spatially inhomogeneous models
when the collisions only involve the velocity variable. As we shall see, in such cases
it is sometimes possible to use entropy-entropy dissipation inequalities from spatially
homogeneous models.
As a final remark, the interest of entropy-entropy dissipation inequalities is not restricted
to proving theorems of trend to equilibrium. Entropy-entropy dissipation inequalities may
also in principle be applied in problems of hydrodynamic (as opposed to long-time) limits,
yielding rather explicit estimates. For this one may consult the work by Carlen et al. [124]
on a baby model, the recent paper by Saint-Raymond [400] on the hydrodynamic limit for
the BGK model, or the study by Berthelin and Bouchut [74] on a complicated variant of
the BGK model. However, to apply this strategy to more realistic hydrodynamic limits, say
starting from the Boltzmann equation, we certainly have to wait for very, very important
progress in the field.
3.3. Logarithmic Sobolev inequalities and entropy dissipation
We illustrate the preceding discussion on the simple case of the spatially homogeneous
Fokker-Planck equation,
~f
----Vv. (Vvf + f v).
Ot
Recall that the entropy functional is the Kullback relative entropy of f with respect to the
standard Gaussian M,
H (flM) = LN f log fM](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-207-320.jpg)
![A reviewof mathematical topics in collisional kinetic theory 207
(equivalently, the additive constant in the free energy has been chosen in such a way that
the equilibrium state has zero energy). And the entropy dissipation functional is the relative
Fisher information,
I(flM)--~Nf
f
Vv log (174)
The archetype of (171) is the Stam-Gross logarithmicSobolevinequality[411,261 ]. In
an information-theoretical language, this inequality can be written most simply as
I(flM) >~2H(fIM). (175)
Inequality (175) was first proven, in an equivalent formulation, in a classical paper
by Stam9 [411]. The links between the theory of logarithmic Sobolev inequalities and
information theory have been pointed out for some time [45,120,165,16].
Of course, inequality (175) immediately implies that the solution to the Fokker-Planck
equation with initial datum f0 satisfies
H(f (t)lM) <<.
e-2tH(folM).
This is a complete l~ and satisfactory solution to the problem of trend to equilibrium for the
Fokker-Planck equation.
Actually, the interplay between functional inequalities and diffusion equations goes in
both directions [330]. As was noticed in a famous work by Bakry and Emery [45], some
properties of trend to equilibrium for the Fokker-Planck equation can be used to prove
inequalities such as (175). We shall discuss their approach in Section 6, together with
recent developments.
By the way, as a general rule, logarithmic Sobolev inequalities are stronger than spectral
gap inequalities [261]. As a typical illustration: if one lets f = M(1 + eh) in (175), where
f Mh = 0, and then lets e go to 0, one finds the inequality
f Mh--O >f MIVhl2) f Mh2, (176)
which is the spectral gap inequality for the Fokker-Planck operator. Inequality (176)
implies the following estimate for solutions of the Fokker-Planck equation:
fo GL2(M-1) ~ [If(t, .)- MIIL~(M_,) ~<e-t lifo - MIIL2(M-,).
9Stam proved the inequality N'(f)I (f) >~N, which is equivalent to (175) by simple changes of variables,
in dimension 1. Here N" is the entropy power functional of Shannon, formula (52). The proof of Stam was not
completely rigorous, but has been fixed.
10The assumption that the initial datum possess finite entropy can even be relaxed by parabolic regularization.
For instance, one can prove [366] that H(f(t)lM) = O(1/t) as soon as f fo(v)lvl 2dv is finite.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-208-320.jpg)
![208 C. Villani
4. Entropy dissipation functionals of Boltzmann and Landau
In this section, we discuss entropy-entropy dissipation inequalities for functionals (156)
and (160).
A common feature of both functionals is monotonicity: the Boltzmann entropy
dissipation is a nondecreasing function of the collision kernel B, while the Landau entropy
dissipation is a nondecreasing function of qJ. This property makes it possible to only treat
algebraically simplified cases where B (resp. q/) is "small". As a typical application, if we
find a lower bound for D when the collision kernel is Maxwellian, then we shall have a
lower bound for all collision kernels whose kinetic part is bounded below. This reduction is
interesting because Maxwellian collision kernels do have many additional properties. We
shall see some of these properties in a moment, and shall dig more deeply into them in
Chapter 2D. All known lower bounds for the entropy dissipation functionals of Boltzmann
or Landau have been obtained from a preliminary study of the Maxwellian case.
As a consequence, it will be natural to define an "over-Maxwellian" collision kernel as
a collision kernel B which is bounded below by a Maxwellian collision kernel.
In the sequel, we shall assume without loss of generality that the first moments of the
distribution function f are normalized:
(1)v =(1)
fI~ f (v) dv 0
N [1312 N
(177)
and denote by M the associated Maxwellian.
4.1. Landau's entropy dissipation
We start with the case of the Landau equation, because its diffusive nature entails better
properties of the entropy dissipation functional. Let us first state the main results, then we
shall comment on them.
THEOREM 14. Let f be a probability distribution satisfying (177).
(i) "Over-Maxwellian case": Let qJ(lzl) ~> ]el2, and let DL be the associated entropy
dissipation functional, formula (160). Then there exists a constant ~.(f) > O, explicit and
depending on f only via an upper boundfor H (f), such that
DL(f) >.X(f)I(flM) >~2~,(f)H(flM). (178)
More precisely, one can choose
f
,k(f)=(N-1) inf I f(v)(v.e) 2dr
eES N-1 JRN
(179)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-209-320.jpg)
![A reviewofmathematicaltopicsincollisionalkinetictheory 209
(ii) "Soft potentials"" Let ~(Izl) ~ Izl2(1 -Jr-Izl) -~, ~ > 0. Then, for all s > O, there
exists a constant Cs(f), explicit and depending on f only via an upper bound for H(f),
such that
DL(f) >~Cs(f)H(flM)l+~7 Fs ~, (180)
where Fs = Ms+2(f) Jr-Js+2(f), and
Ms+2(f) fR f(v)(1 -+-Iv12)s+2
= N dr,
Iv [ (l+lol )
N du.
(iii) "Hard potentials"" Let q/(Izl) ~ Izly§ 9/> 0. Then, there exists constants K1 (f),
K2 (f), explicit and depending on f only via an upper boundfor H (f), such that
DL(f) ~ Kl(f) min[I(flM), l(flM)l+-~]
~>K2(f) min[H(flM), H(fIM)I+-~].
(181)
(182)
REMARKS. (1) Note that the constant )~ given by (179) has the dimensions of a
temperature, and can vanish only if f is concentrated on a line. This is the typical
degeneracy of the Landau equation; in particular, the operator in (30) is always strictly
elliptic unless f is concentrated on a line. But the finiteness of the entropy prevents such
a concentration, and allows one to get a bound from below on ~(f). Of course, other
estimates are possible: for instance, by use of some L p, or L c~, or smoothness bound
on f. Or, if f is radially symmetric, then automatically )~(f) = 1.
(2) Also, as we shall see in the next section, it may sometimes be wiser to estimate from
below ~(f) in terms of the entropy dissipation of f!
(3) Further note that the inequalities on the fight in (178) and in (182) are nothing but
the logarithmic Sobolev inequality (175).
In the preceding theorem, point (i) is the starting point for the remaining cases. It
was established in Desvillettes and Villani [183] by two different methods. The first one
relies on some explicit computations performed in Villani [443], whichare recalled in
formula (30). The second strategy is a variant of Desvillettes' techniques, inspired by a
method due to Boltzmann himself [93]. It consists in "killing", with a well-chosen operator,
the symmetries of the functional DL which correspond to the equilibrium state. 11To be just
a little bit more precise, one writes
D(f) = f dvdv, ff, lR(v, v,)[ 2
11 See Boltzmann'sargumentin Section4.3.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-210-320.jpg)
![210 C. Villani
where R" ]1~2N ---+ ]I~N, and one finds a linear operator T = T(v, v,)" ]l~N ---> 11~N such that
T R is identically 0 if and only if D = 0; then
DL (f) >~
l/
iiTll2 dvdv, ff, ITR(v, V,)I 2
A careful choice of the operator T enables a very simple computation of the fight-hand
side in this inequality.
Point (ii) is proven in [429]. The idea is that the vanishing of qJ(lv - v,l)/lv - v,[ 2 as
Iv - v,[ --+ cr can be compensated by some good estimates of decay at infinity, in the form
of the constant Fs+2 (which involves both moments and smoothness).
As for point (iii), it is rather easy to get by "perturbation" from point (i), see Desvillettes
and Villani [183]. The idea is that the contribution of small Iv - v,[ is negligible. One
writes
Iv - v,I • ~> e y Iv - v,[ 2 - 6g+2,
then one estimates from below the contribution of e• - V,[2 to the entropy dissipation,
and from above the contribution of the small constant function ey+2. A few algebraic
tricks [183] lead to the estimate (181) without further bounds on the concentration of f:
the constant K in this estimate is essentially )~(f)l+g/2.
Theorem 14 gives explicit and satisfactory answers to the quest of entropy-entropy
dissipation estimates for the Landau equation; in the next section we shall see that they
can be used efficiently for the study of the trend to equilibrium, at least in the spatially
homogeneous situation. However, we should avoid triumphalism: it is abnormal that the
exponent in the case of hard potentials (which is 1 + g/2) be worse than the exponent in
the case of soft potentials (1 + e, with e as small as desired, if f has a very good decay
and smoothness at infinity). One would expect that for hard potentials, the inequality
DL(f) >~K(f)H(fIM)
hold true.
4.2. Boltzmann's entropy dissipation: Cercignani's conjecture
Now we turn to the more complicated case of the functional (156). Some parts of the
following discussion are copied from [442].
An old conjecture by Cercignani, formulated at the beginning of the eighties, was that
the Boltzmann equation would satisfy a linear entropy-entropy dissipation inequality. We
state this conjecture here in a slightly more precise form than the original. There are two
forms of it, a weak and a strong.
CERCIGNANI'S CONJECTURE. Let B ~ 1 be a collision kernel and (156) be the
associated entropy dissipation functional. Let f (v) be a probability distribution on R N,](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-211-320.jpg)
![A reviewofmathematicaltopicsin collisionalkinetictheory 211
with unit temperature, and let M be the associated Maxwellian equilibrium. Then,
(strong version) there exists )~ > O, independent of f, such that
D(f) >~2~H(fIM); (183)
(weak version) there exists )~(f) > O, depending on f only via some estimates of moments,
Sobolev regularity, lower bound, such that
D(f) >~2~(f)H(f]M). (184)
It soon appeared that the strong version of this conjecture had to be false. Indeed,
it would have implied a universal exponential rate of convergence for solutions of the
spatially homogeneous Boltzmann equation with a collision kernel B ~> 1. But, as we
mentioned in Section 1.4, Bobylev [79, p. 224] was able to produce a family of initial
data (f~)~>0 with unit mass and temperature, such that the associated solutions of the
Cauchy problem (with Maxwellian collision kernel, say B = 1) converge to equilibrium
slowly, in the sense
Yt ~>0, [[fe(t, .)- M[I ~>Kee -e', Ke > 0.
These initial data are constructed more or less explicitly with the help of the Fourier
transform apparatus, and hypergeometric functions.
Later, Wennberg [461] produced direct counterexamples to (183), covering the case of
hard potentials as well.
Finally, Bobylev and Cercignani [87] disproved even the weak version of the conjecture.
They exhibited a family of distribution functions for which (184) does not hold for a
uniform )~, while these distribution functions do have uniformly bounded LP or H k norms
(whatever p, k), uniformly bounded moments of order k (whatever k), and are bounded
below by a fixed Maxwellian. These counterexamples are obtained by adding a very tiny
(but very spread) bump, at very high velocities, to the equilibrium distribution. They again
illustrate the principle that distribution tails are the most serious obstacle to a good trend
to equilibrium for the Boltzmann equation.
Thus, Cercignani's conjecture is false.* It may however be that (184) hold true under
more stringent assumptions:
- under very strong decay conditions, for instance, f 6 L2(M-1), as in the linearized
theory; 12
-or under an assumption of nonintegrable angular singularity, which may help.
This conjecture would be supported by the good behavior of the Landau entropy
dissipation.
*Note added in proof: To my own surprise, after completion of this review, I discovered that Cercignani's
conjecture does hold true when B(v- v,, a) ~>1-4-Iv- v,]2. This is not in contradiction with the Bobylev-
Cercignani counterexamples,becausetheyassumef Bdo"~<C(1 + iv - v, lY),y < 2!
12Averyrecent,deepresultby Balland Bartheaboutthe centrallimittheoremsuggeststhatthereis somehope
if f satisfiesa Poincar6inequality.Thismay be the first step towardsidentifyingsome"reasonable"conditions
for Cercignani'sconjectureto be true.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-212-320.jpg)
![212 C. Villani
4.3. Desvillettes' lower bound
The first interesting lower bound for Boltzmann's entropy dissipation functional was
obtained by Desvillettes [166]. His idea was to go back to Boltzmann's original argument
for the identification of cases where the entropy dissipation vanishes. As many proofs of
similar results, Boltzmann's proof relies on some well-chosen linear operators which "kill
symmetries". Let us sketch this argument (slightly modified) in a nutshell, since it may
enlighten a little bit the discussion of the most recent results in the field.
BOLTZMANN'S THEOREM. Let N >~2, and let f (v) be a smooth positive solution of the
functional equation
V ( U, U, , o") E ]1~N >( I[~N X S N-l, f' f~, = f f,. (185)
Then f is a Maxwellian distribution; in other words there exist constants )~ ~ R, lZ ~ I[~N
such that
VV E ~N, V log f (v) = Zv +/z. (186)
BOLTZMANN'S ARGUMENT. Average (185) over the parameter o" 6 SN-1 , to find
1 fs (f' f*~)do".
ff* = IsN-11 u-, (187)
It is easy to convince oneself that the function
is ! !
f f, do" = G(v, v,)
N-1
depends only on the sphere S(v, v,) with diameter [v, v,]. Actually, up to a Jacobian
factor (Iv- v,[/2) N-l, G is just the mean value of the function f(w)f(Co) on this sphere,
where t~ stands for the velocity on S which is diametrically symmetric to w. The spheres
S(v, v,) are in turn parametrized by only N + 1 parameters, say (v + v,)/2 and Iv - v,I;
or, equivalently, by the physical variables
m = v + v, [total momentum of colliding particles];
Ivl2 Iv, I2
e = -~ + ~ [total kinetic energy of colliding particles].
(188)
Thus we shall abuse notations by writing G(v, v,) = G(m, e).
Now, introduce the linear differential operator T = (v - v,)/x (V - V,) (or, which
amounts to the same, H(v - v,)(V - V,), where 17(v - v,) is the orthogonal projection](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-213-320.jpg)
![A review of mathematical topics in collisional kinetic theory 213
on (v - v,)• Its kernel consists precisely of those functions that depend only on m and e.
If we apply this operator to the equation
log ff, = log G (m, e),
we find
(v - v.) A [V log f - (g log f),] ----O.
In words, for all v, v, there exists a real number •v,v, such that
V log f (v) - V log f (v,) = ~,v,v,(v - v,).
This functional equation, set in ]~N,N ~>2, implies the conclusion at once.
(189)
REMARK. The very last part of the proof, starting from (189), is exactly what one needs
to identify cases of equality for Landau's entropy dissipation functional. This can make us
suspect a deep connection between the entropy dissipations of Boltzmann and Landau. We
shall soon see that there is indeed a hidden connection.
With the help of the open mapping theorem, Desvillettes was able to produce a
"quantitative" version of Boltzmann's argument, leading to the
THEOREM 15. Let B >~1, and let D be the associated entropy dissipation functional (47).
Let f be a nonnegative density on IRN, with finite mass and energy. Without loss of
generality, assume that the first moments of f are normalized by (177). Then, for all R > 0
there is a constant KR > O, depending only on R, such that
f
D(f) >~Kit inf 1
m~.Ad Jlv [<~R
llog f - logmldv,
where M is the space of all Maxwellian distributions.
Note that the quantity on the right is always positive for some R > 0 if f is not
Maxwellian. Several variants were obtained, with better estimates and simpler proofs, and
recently Desvillettes [175] found a way to avoid the use of the open mapping theorem, and
get explicit constants. Also Wennberg [455] extended the result to hard potentials.
Although Desvillettes' result is rather weak, it was important as the very first of its kind.
Subsequent developments were partly motivated by the search for stronger estimates.
4.4. The Carlen-Carvalho theorem
At the beginning of the nineties, Carlen and Carvalho [121,122] made a crucial contribution
to the subject by using the tools of information theory and logarithmic Sobolev inequalities.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-214-320.jpg)
![214 C. Villani
They proved that there always exists an entropy-entropy dissipation inequality for
Boltzmann's collision operator as soon as one has some (very weak) control on the decay at
infinity and smoothness of the distribution function. In their general result, decay at infinity
of a distribution function f is measured by the decay of
x f "R ~ fv f (v)lvl 2 dv
as R 1"c~, while the smoothness is measured by the decay of
7rf "X w-~ H(f) - H(Sz f)
as )~$ 0. Here (St)t>>ois as usual the semigroup generated by the Fokker-Planck operator;
sometimes it is called the adjoint Ornstein-Uhlenbeck semigroup.
Carlen and Carvalho's general theorem [121] can be stated as follows:
THEOREM 16. Let B(v - v,, or)/> 1 be a collision kernel. Let Xo, #/o be two continuous
functions, decreasing to 0 as R t +cx~ and X $ 0 respectively. Let then f be a probability
distribution function with unit mass and temperature, and let M be the associated
Maxwellian distribution. Assume that
Xf ~ Xo, ~f <<.~PO. (190)
Then, there exists a continuous function 0 = Ox,~/, strictly increasing from O, depending
on f only via Xo and r such that
D(f) >~69(H(flM)).
REMARKS. (1) This result crucially uses the special properties of Maxwellian collision
kernels, explained in Chapter 2D.
(2) The result in [121] is stated for a collision kernel which is bounded below in co-
representation. 13 Recent works have shown that this assumption can be relaxed (see the
references in Chapter 2D).
The main ideas behind the proof of Theorem 16 are (1) the reduction to Maxwellian
collision kernel by monotonicity, (2) the inequality 14
D(f) >~H(f) - H(Q+(f, f)) ~ O,
which holds true for a Maxwellian collision kernel b(cos 0) such that f b(cos 0) sinN-2 0 dO
= 1, and (3) show that when f satisfies (190) and H(f) - H(M) >, s, then f lies in a com-
pact set of probability measures on which H - H (Q+) attains its minimum value.
13See Section4.6 in Chapter2A.
14See Section3.2 in Chapter2D.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-215-320.jpg)
![A review of mathematical topics in collisional kinetic theory 215
One of the key ingredients is a study of the Fisher information functional I (f) --
f IVfl2/f, and the representation formula
f0 ec
H(Q+(f, f)) - H(M)= [I(Q+(Sxf, Szf)) - I(Sxf)]d)~. (191)
This formula and related estimates are explained in Chapter 2D. A crucial point is to bound
below the integrand in (191), for )~positive enough, by the method of Carlen [120].
We note that there is no assumption of lower bound on f in the Carlen-Carvalho theo-
rem, though they actually use lower bounds in their estimates. There is no contradiction,
because Maxwellian lower bounds are automatically produced by the semigroup (Sz).
However, these lower bounds are rather bad, and so are the resulting estimates. Better
bounds can be obtained if the probability density f is bounded below by some Maxwellian
distribution.
In a companion paper [122], Carlen and Carvalho showed how to extend their method
to physically realistic cases like the hard-spheres kernel, B(v - v,, or) = Iv - v,I, and gave
a recipe for computing the function 0.
These results were the first entropy dissipation estimates which would find interesting
and explicit applications to the Boltzmann equation, see Section 5. More importantly, they
set new standards of quality, and introduced new tools in the field. However, the Carlen-
Carvalho entropy-entropy dissipation inequalities are not very satisfactory because the
function 69 is quite intricate, and usually very flat near the origin.
4.5. Cercignani's conjecture is almost true
As we mentioned earlier, the "linear" entropy-entropy dissipation inequality conjectured
by Cercignani (O(H) = const.H) is in general false. Nevertheless, it was proven a few
years ago by Toscani and Villani [428] that one can choose O(H) = const.H l+e, with e
as small as desired. Here is a precise statement from [428]. We use the notation
IlfllL~ -- fRN f(v)(1 + IV12)s/2 dv
and its natural extension
IlfllLl~ogL-- fR~vf(V)log( 1 + f(v)) (1 -+-Ivl2)~/2dv.
THEOREM 17.* (i) "Over-Maxwellian case": Let B >~1 be a collision kernel and D be the
associated entropy dissipation functional Equation (156). Let f be a probability density
on IRN with unit temperature, and let M be the associated Maxwellian equilibrium. Let
e > 0 be arbitrary, and assume thatfor some ~ > O, A, K > O,
IlfllL~4+2/~+~ Ilfll 1 < -31-oo
' L2+2/E+~log L '
*Noteaddedinproof:All the resultsin thistheoremhavebeenimprovedin recentworkby the author.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-216-320.jpg)
![216 C. Villani
f (v) >/Ke -Alvl2. (192)
Then, there exists a positive constant Cs(f), depending only on N, s, ~, [JfllL~+2/~+a,
IIf II , A and K, such that
L2+2/s+alog L'
D(f) > Cs(f)H(fIM) l+e. (193)
As an example (choosing 8 = 1), the following more explicit constant works"
D(f) >~KTf FZSH(flM) l+~, (194)
where K is an absolute constant (not depending on f), Tf is the "temperature" given
by (179), and
( 1 )
Fe= log~+A Ilfll 2 Ilfll ,
L~+2/~ L3+2/~log L"
(ii) "Soft potentials": Assume now that
B(v - v,, ~) > (1 + Iv - v,I) -e, /~ > O.
Then, for all s > O, Equation (194) still holds with
(1)
Fe = log ~ + A IIf II2
tl5+(2+fl)/sIIf IIt~+(2+fl)/elogL"
(iii) "Hard potentials"" Assume now that
B(v-v,,o')>~lv-v,I • g>O.
Assume, moreover, that f ~ L p, for some p > 1, and tc large enough. Then, there exists
> 1, C > O, depending on N, y, p, x, IIf IILff, and on A, K in (192), such that
D(f) >~CH(flM) ~.
Thus Cercignani's conjecture is "almost" true, in the sense that any power of the relative
entropy, arbitrarily close to 1, works for point (i), provided that f decays fast enough and
satisfies a Gaussian lower bound estimate.
This theorem is remindful of some results in probability theory, about modified
logarithmic Sobolev inequalities for jump processes, see Miclo [345]. Even if the situation
considered in this reference is quite different, and if the methods of proof have nothing in
common, the results present a good analogy. From the physical point of view, this is not
surprising, because the Boltzmann equation really models a (nonlinear) jump process.
Let us briefly comment on the assumptions and conclusions.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-217-320.jpg)
![A review of mathematical topics in collisional kinetic theory 217
(1) The main improvement lies in the form of the entropy-entropy dissipation inequality,
which is both much simpler and much stronger.
(2) The lower bound assumption can be relaxed into f(v) >~ge -Alvlp for some p > 2,
provided that more moments are included in the estimate.
(3) Strictly speaking, this theorem is not stronger than the Carlen-Carvalho theorem,
because the assumptions of decay at infinity are more stringent. On the other hand, it does
not require any smoothness condition.
As regards the proof, it is completely different from that of the Carlen-Carvalho
theorem, and relies strongly on Theorem 14, point (i). Since this is quite unexpected, we
shall give a brief explanation in the next paragraph.
Once again, the result for hard potentials is not so good as it should be, because the power
in point (iii) cannot be chosen arbitrarily close to 1. We have hope to fix this problem by
improving the error estimates for small relative velocities which were sketched in [428]. 15
4.6. A sloppy sketch of proof
In this survey, we have chosen to skip all proofs, or even sketches of proof. We make
an exception for Theorem 17 because of its slightly unconventional character, and also
because of its links with Boltzmann's original argument 16 about cases of equality in the
entropy dissipation - with ideas of information theory coming into play. Of course, we
shall only try to give a flavor of the proof, and not go into technical subtleties, which by the
way are extremely cumbersome. Also we only consider point (i), and set B -- 1, or rather
B = ISN-11-1, so that f Bdo- = f da = 1. Thus the functional to estimate from below is
1s dodo,
f
D(f) = ~ 2N sN-1
da (f' f~, - f f,) log
f'f',
ff,
The three main ingredients in Theorem 17 are
- a precise study of symmetries for the Boltzmann collision operator, and in particular
the fact that the entropy dissipation can be written as a functional of the tensor product
f|
- a regularization argument ~ la Stam;
- our preliminary estimate for the Landau entropy dissipation, Theorem 14.
Stam's argument. At the end of the fifties, Stam [411] had the clever idea to prove the
so-called Shannon-Stare inequality, conjectured by Shannon:
H(v x + ,/1 + (1 (195)
15As this review goes to print, we just managed to prove the desired result, at the expense of very strong
smoothness estimates (in all Sobolev spaces).
16See Section 4.3.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-218-320.jpg)
![218 C. Villani
actually equivalent to (52), as a consequence of the Blachman-Stam inequality, which he
introduced on that occasion:
I(x/~X + ~/1 -otY) <~o~I(X) + (1 -a)I(Y). (196)
In inequalities (195) and (196), X and Y are arbitrary independent random variables on ~N,
and one writes H(X) = H(f) = f f logf, I(X) = I(f) = f [v fle/f whenever f is the
law of X.
Stam found out that (196) is essentially an infinitesimal version of (195) under heat
regularization. Think that I is nothing but the entropy dissipation associated to the
heat equation .... A modern presentation of Stam's argument is found in Carlen and
Soffer [125]: these authors replace H and I by their relative counterparts with respect to
the standard Gaussian M, and obtain (195) by integrating (196) along the adjoint Ornstein-
Uhlenbeck semigroup (St)t>o. More explicitly, since I (f IM) is the derivative of H (flM)
along regularization by St, and since also Stf -+ M as t --+ oo, one can write
f0 ~176
H(fIM) = I(StflM)dt.
The strategy in [428] is inspired from this point of view: we would like to start from
+~ d f)] dt.
D(f)-- fo [--~D(St
This identity is formally justified because Stf ~ M as t ~ oo, and D(M)= 0. Then
one can hope that for some reason, the derivative -dD/dt will be easier to handle that
the entropy dissipation functional D. This is the case in the proof of the Shannon-Stam
inequality, and also here in the framework of the Boltzmann entropy dissipation.
It actually turns out, rather surprisingly, that
Kf +~
D(f) ~ -R~ DL(Stf) dt, (197)
with K > 0, and R a typical size for the velocity. In other words, the entropy dissipation
for the Landau equation is a kind of differential version of the entropy dissipation for
the Boltzmann equation! Admit for a while (197), and combine it with the result of
Theorem 14, in the form
DL(Stf) >~(N- 1)TstfI(StflM)
>~(N - 1)Tf I(St flM).
It follows that
D(f) >~C(f) fo +~176
l (Stf lM) dt = C(f)H(flM);](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-219-320.jpg)
![Areviewofmathematicaltopicsincollisionalkinetictheory 219
which is the statement in Cercignani's conjecture.
Of course, we know that Cercignani's conjecture is false, which means that (197) cannot
rigorously hold true. A precise variant is established in [428]. The technical problem which
prevents (197) is the presence of large velocities, as one could expect. Controlling the
contribution of large velocities to the entropy dissipation is the most technical point in the
proof presented in [428]. It means for instance establishing quantitative bounds on the tails
of the entropy dissipation, like
fo +~ fix Cs
dt ]xIZ(StF - StG) log StF dX <~
I>~R StG Rs
for arbitrary probability densities F(X) and G(X) in ~2N, where Cs is a constant
depending on s and on suitable estimates on F and G (moments, lower bound...).
In the next two pages, we shall skip all these technicalities and present a sketch of proof
of (197) under the absurd assumption that all velocities are bounded, just to give the reader
an idea of the kernel of the proof.
SLOPPY SKETCH OF PROOF FOR (197). First we introduce the adjoint Ornstein-
Uhlenbeck semigroup (St), and we try to compute (-d/dt)D(Stf). At first sight this
seems an impossible task to perform in practice, due to the number of occurrences of f
in the entropy dissipation functional, and the complicated arguments v', v~,. But a first
observation will help: D(f) is actually a functional of the tensor product f | f = ff,.
And it is easily checked that the following diagram is commutative, with T standing for
tensorization,
7
f > F=ff,
I St 7" I St
St f > St F.
(198)
(Here we use the same symbol for the semigroups St in L I(R N) and in L1 (R2N).)
This enables to replace in computations (St f)(St f), by St(ff,). One could hope that,
similarly,
F=ff, > f'f',
I St I St
St F > St (f' ff,)
(199)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-220-320.jpg)
![220 C. Villani
is commutative. This is false! The point is that the angular variable a is not intrinsic to the
problem. To remove this flaw, we integrate with respect to the parameter a. Since
x
(x, y) w-~ (x - y) log -
Y
is a jointly convex function of its arguments, by Jensen's inequality
s
-- 1 dv dr, If, - da f' f2 log f da f' f~,
D(f) >i D(f) = -~ 2N (200)
Now it is true, even if not immediate at all, 17 that
7- ,,4
f >F f f, > G f ''
= = daff,
7- A
&f ~ &f > &G
(201)
with A standing for the averaging operation over the sphere, is an entirely commutative
diagram. This actually is a consequence of the fact that (St) is a Gaussian regularization
semigroup.
This suggests to work with D instead of D, and to write D(Stf) in the form
D(St F, St G), with the abuse of notations
-- lf• (F-G) logF
D(F, G) = -~ 2N -~ dX, X m. (1), 13,) E ~2N.
After these preliminaries, it is not hard to compute
dD(StF, StG)= lfR (StF+StG)
-dt 4 2N
V(&F) V(StG)
StF StG
2
dX. (202)
Here, of course,
V = [V~, V~,]
is the gradient in ]~N x Rv
N .
Under suitable assumptions one can also prove that t ~ D(St f) is a continuous function
as t ~ 0, and goes to 0 as t ~ +cx~. Then
D(f) = -~ dt 2N(St F + St G)
V(StF) 7(StG)
StF StG
2
dX
17A weaker property, sufficient for the argument, is that St preserves the class of functions which only depend
on v + v, and Ivl2 + Iv, I2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-221-320.jpg)
![A review of mathematical topics in collisional kinetic theory 221
lfo+ f
) -~ dt 2N StF
V(StF)
StF
V(ScG)
(203)
Since StG is a very complicated object, we would like to get rid of it. Recall from
Boltzmann's original argument that StG, being an average on spheres with diameter
[v, v, ], does not depend on all of the variables v, v,, but only upon the reduced variables
m = v + v,, e = Ivl2/2 + Iv,12/2. Accordingly, we shall abuse notations and write
StG(v, v,)-- StG(m,e).
Now comes the key point: there is a conflict of symmetries between StG, which only
depends on a low-dimensional set of variables, and StF, which is a tensor product.
In Boltzmann's argument, the Maxwellian distribution pops out because it is the only
probability distribution which is compatible with both symmetries.
Here these different structures of StF and StG reflect at the level of their respective
gradients:
V(StF)
StF I
V(St f) (VSt f), ]
= gy ; (204)
V(StG) 1 VmStG -k- 1) , VmStG d- 1), 9 (205)
StG StG Oe Oe
In particular, V(StG) always lies (pointwise) in the kernel of the linear operator
P "[A, B] ~ ]~2N~ 17(1) -- v,)[A - B] ~ ]~N,
where H(z) is the orthogonal projection upon z• Of course IIPII = 4~ as a linear
operator,18 and so
V(StF) V(StG)
StF &G
2 2
1
IIPII2
p(VStF
_1
2 17(v - v,) Stf
(gstf)_*]12"
(Stf), (206)
By combining (200), (203) and (206),
if +~
D(f) >~-~ dt fR2u(St f)(St f),
VStf (VStf), ]
2
dv dr,.
The reader may have recognized a familiar object in the integrand of the right-hand side.
Actually, apart from a factor Iv - v, 12, it is precisely the integrand in the Landau entropy
dissipation, computed for Stf! If we now use our absurd assumption of boundedness of
18In contrast with the linear operator appearing in Boltzmann's proof, which was unbounded.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-222-320.jpg)
![222 C. Villani
all relative velocities, in the form Iv - v, I ~<R, we get
D(f) >~-~ dt 2u(St f)(St f)*lv -- v,I 2
VStf
x H(v-v,) Stf
1 fo+~
8R2 dt DE (St f).
(VSt f), ]
(St f),
2
dvdv,
I-1
4.7. Remarks
We shall point out a few remarks about the preceding argument. First of all, in the
course of the rigorous implementation, it is quite technical to take into account error
terms due to large velocities. One has to study the time-evolution of expressions like
f~o(X)(StF- StG)log(StF/StG). But the calculations are considerably simplified by
a striking "algebraic" property: a local (not integrated) version of (202) holds true. Let
F
h (F, G) -- (F - G) log G' j(F,G)=(F+G) VFF VG]
2"G
Then, one can check that
d
dt t=O
[St, h]-j,
in the sense that for all (smooth) probability distributions F and G,
d
dt t=O
(Sth(F, G) - h(St F, StG)) - j (F, G).
This property is somewhat reminiscent of the F calculus used for instance in Bakry
and Emery [45] and Ledoux [294]. It yields another bridge between entropy dissipation
inequalities and the theory of logarithmic Sobolev inequalities.
Our second remark concerns the use of the Fokker-Planck semigroup regularization. As
we have seen, the main point above was to estimate from below the negative of the time-
derivative of D(f) along the semigroup (St)t 90. As was already understood by Carlen and
Carvalho, and even a long time ago by McKean [341 ] in the framework of the Kac model,
this estimate has to do with the behavior of the Fisher information I (f) -- f IvflZ/f along
the Boltzmann semigroup. Note that I (f) is the dissipation of the H-functional along the
semigroup (St)t~o. As we shall explain in Chapter 2D, the semigroup (Bt), generated by](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-223-320.jpg)
![A reviewofmathematicaltopicsin collisionalkinetictheory 223
the spatially homogeneous Boltzmann equation with Maxwell collision kernel, commutes
with (St), and it follows that
dt t=o
d
D(& f) -- --~
t=0
I (Bt f). (207)
We shall see in Chapter 2D that the right-hand side of (207) is always nonnegative; this
could be considered as an a priori indication that the functional D behaves well under
Fokker-Planck regularization.
Actually, in the simpler case of the Kac model, 19 McKean [341, Section 7, Lemma d)]
used relation (207) the other way round! He proved directly, with a very simple argument
based on Jensen's inequality, that the left-hand side of (207) is nonnegative for the Kac
model. His argument can be transposed to the Boltzmann equation with Maxwell collision
kernel in dimension 2, and also to the case where the collision kernel is constant in co-
representation: see [428, Section 8].
As a third remark, we insist that the above argument, besides being rather intricate, is
certainly not a final answer to the problem. The use of the average over a seems crucial
to its implementation, while for some applications it would be desirable to have a method
which works directly for arbitrary Maxwellian collision kernels b(cos0). There is no clue
of how to modify the argument in order to tackle the problem of Cercignani conjecture
(with exponent 1) for very strongly decaying distribution functions. It also does not manage
to recover spectral gap inequalities for Maxwellian collision kernels, which are known to
be true. Applied to simpler models than Boltzmann's equation, it yield results which are
somewhat worse than what one can prove by other, elementary means! However, in terms
of lower bounds for Boltzmann's entropy dissipation, at the moment this is by far the best
that we have.
Our final remark concerns the problem of solving (53). As mentioned in Section 2.5,
many authors have worked to prove, under increasing generality, that these solutions are
Maxwellian distributions. The problem with Boltzmann's proof was that it needed C 1
smoothness. However, as suggested by Desvillettes, the use of the Gaussian semigroup
(St) (or just the simple heat regularization) allows one to save Boltzmann's argument: let
f be a L 1 solution of (53) with finite energy; without loss of generality f has unit mass,
zero mean and unit temperature. Average (53) over a to get
f f. -" G(m, e)
as in formula (187). Then apply the semigroup (St) to find
(St f)(St f), = StG(m, e).
Since St f is C ~ for t > 0, Boltzmann's proof applies and St f is a Gaussian, which has
to be M by identification of first moments. Since this holds true for any t > 0, by weak
continuity f = M.
19Equation(21).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-224-320.jpg)
![224 C. Villani
5. Trend to equilibrium, spatially homogeneous Boltzmann and Landau
As we already explained, in principle the trend to equilibrium is an immediate consequence
of an entropy-entropy dissipation inequality and of suitable a priori estimates. However,
there are some interesting remarks to make about the implementation.
5.1. The Landau equation
By Theorem 14, one obtains at once convergence to equilibrium for the spatially
homogeneous Landau equation
- with explicit exponential rate if q'(lz[)/> K[zl2;
- with explicit polynomial rate if ~(Izl)/> Klzl • Y > 0.
These results hold in the sense of relative entropy, but also in any Sobolev space, thanks
to the regularization results which we discussed in Chapter 2B and standard interpolation
inequalities.
An interesting feature is that the rate of convergence given by the entropy-entropy
dissipation inequality is likely to improve as time becomes large, by a "feedback" effect.
Indeed, when f approaches equilibrium, then the constant Tf in (179) will approach
the equilibrium value TM = 1. In the case q'(Izl) = Izl2, this enables one to recover an
asymptotically optimal rate of convergence [183].
The case of soft potentials (y < 0) is more problematic, because the moment estimates
are not uniform in time - and neither are the smoothness estimates which enter the constant
Fs+2 in Theorem 14.
The fact that we do not have any uniform moment estimate for some moment of order
s > 2 may seem very serious. It is not clear that condition (170) should be satisfied.
Compactness-based methods spectacularly fail in such a situation.
However, and this is one of the greatest strengths of the entropy method, it is not
necessary that the constant Fs+2 be uniformly bounded. Instead, it is sufficient to have
some estimate showing that it does not grow too fast, say in O(t ~) for c~ small enough.
With this idea in mind, Toscani and Villani [429] prove the following theorem:
THEOREM 18. Let tP(lzl) --Izl2t~D([z[), where r is smooth, positive and decays like
[z[-~ at infinity, 0 < fl < 3. Let fo be an initial datum with unit mass and temperature, and
let M be the associated Maxwellian distribution. Assume that fo is rapidly decreasing, in
the sense that
Vs > 0, IlfollL~ < -+-~.
Then, for all e > 0 there exists so > 0 and a constant Ce(fo), depending only on e, N,
and [[f0[lL2o, such that the unique smooth solution of the spatially homogeneous Landau
equation with initial datum f o satisfies
H(f (t, ")IM) ~<CE(fo) t -1/e.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-225-320.jpg)
![A review of mathematical topics in collisional kinetic theory 225
We note that this theorem does not cover the interesting case fl = 3 (Coulomb potential
in dimension 3): the proof in [429] just fails for this limit exponent. Including this case
would be a significant improvement. We also note that this theorem deals with a smooth qJ,
while realistic q/'s would present a singularity at the origin. This singularity cannot harm
the entropy-entropy dissipation inequality, but may entail serious additional difficulties in
getting the right a priori estimates, z~
5.2. A remark on the multiple roles of the entropy dissipation
Numerical applications for the constant TU appearing in (179) are very disappointing (say,
10-2~ .... ) This is because the entropy is quite bad at preventing concentration. Much
better estimates are obtained via L ~ bounds for instance (which can be derived from
regularization).
Another possibility is to use the entropy dissipation as a control of concentration for f.
The idea is the following: if the entropy dissipation is low (which is the bad situation
for trend to equilibrium), then the distribution function cannot be concentrated too much
close to a hyperplane, because the entropy dissipation measures some smoothness. As a
consequence, Tf cannot be too small. More explicitly, say if q/(lzl) ~> Izl2, then [183,
Section 5]
TS>
(N - 1)2
N + DL (f) "
N
By re-injecting this inequality in the proof of Theorem 14, one finds the following
improvement (still under the assumption ~(Izl)/> Izl2)
DL(f) ~ ~ZN(N- 1)2H(fIM)-~
N 4 N 2
4 2
This in turns implies exponential convergence to equilibrium with realistic bounds, which
we give explicitly as an illustration.
THEOREM 19. Let !It(Izl) ~ Izl2, and let fo be a probability distribution on ]1~N, with zero
mean velocity and unit temperature. Let M be the associated Maxwellian distribution. Let
f (t, .) be a classical solution of the Landau equation with initial datum fo. Then, for all
time t >~O,
IITCt,')-MII ,
~/N C~e-~ e N
<
<
" N-1
20Seethe discussionin Section 1.3of Chapter2E.
CO 2 t 2(N-1)2t
+ ~/-N(N- 1) e~c~ e----w--, (208)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-226-320.jpg)
![226 C. Villani
where
Co = ~2N (N - 1)2H (folM) -k
N 4 N 2
4 2
This estimate shows that satisfactory bounds can sometimes be obtained by cleverly
combining all elements at our disposal!
5.3. The Boltzmann equation
Once again, one has to separate between over-Maxwellian collision kernel, hard potentials
or soft potentials. In order to apply Theorem 17, we need
Moment estimates. They hold true for hard potentials without any assumption on the
initial datum, and for Maxwellian collision kernels if a sufficient number of moments
are finite at the initial time; we have discussed all this in Chapter 2B. In the case of soft
potentials, these estimates are only established locally in time, but in some situations one
can control the growth well enough.
Lower bound estimates. Uniform such bounds were proven by A. Pulvirenti and
Wennberg for Maxwellian collision kernels or hard potentials with cut-off In the case
of soft potentials, uniform bounds are an open problem. But, still under the cut-off
assumption, local (in time) bounds are very easy to obtain as a consequence of Duhamel's
formula (100), if the initial datum satisfies a lower bound assumption. Such a crude bound
as
f (t, v) ~ Ke -Atlvl2 At- (1 + t)
is sufficient in many situations [428, Section 4]. The case of non-cutoff collision kernels is
still open.
LP estimates. In the model case of hard potentials with Grad's cut-off assumption, such
estimates are a consequence of the studies of Arkeryd and Gustafsson, as discussed in
Section 3 of Chapter 2B. For instance, if the initial datum lies in L ~ with suitable
polynomial decay, then the solution will be bounded, uniformly in time. Also the case
of Maxwell collision kernel can be treated in the same way. However, when the collision
kernel decays at infinity, things become more intricate. The search for robust estimates
led the authors in [428] to a new way of controlling L p norms by the Q+ smoothness,21
moment estimates, and a lot of interpolation.
21For simplicity,kinetic collisionkernels q~(Iv- v,I) consideredin [429] were smooth and bounded from
above and below.The authors had forgottenthat in such a case the Q+ smoothnesscould not apply directly,
because q~(0) > 0. The proofis howevereasyto fixby treatingseparatelyrelativevelocitieswhichare closeto 0;
this has beendonerecentlyby Mouhot.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-227-320.jpg)
![A review of mathematical topics in collisional kinetic theory 227
On the other hand, in the case of non-cutoff collision kernels, LP estimates are obtained
via Sobolev estimates and regularizing effect.
Here we see that many of the estimates which we discussed in Chapter 2B can be
combined to yield a qualitative theorem for solutions of the Boltzmann equation: trend
to equilibrium with some explicit rate. Since the general panorama of a priori estimates
for the spatially homogeneous Boltzmann equation is not completely settled yet, we do
not have a general theorem. Let us give one which encompasses the few cases that can
be treated completely. We put very strong conditions on the initial datum so that a unified
result can be given for different kinds of collision kernels.
THEOREM 20. Let B(v - v,, o') --~(Iv - v,l)b(cosO) be a collision kernel satisfying
Grad's angular cut-off let fo be an initial datum with unit mass and temperature, and let
f (t, .) be a strong solution of the Boltzmann equation with initial datum fo. Assume that
fo lies in L ~ and decays at infinity like O(]v[ -k) for any k >10. Assume moreover that
fo(v) ~ Ke -alvl2 for some A, K > O. Then
(i) if ~ =_ 1, then H(fIM)- O(t-~);
(ii) if~(lv - v,I) --Iv - v,[ • V > 0, then H(fIM) -- O(t-K) for some tc > 0;
(iii) if q~(lv - v,[) is bounded, strictly positive and decays at infinity like Iv - v,I -r
with 0 < fl < 2, then H(fIM) --O(t-~).
Moreover, all the constants in these estimates are explicitly computable.
Of course, O(t -~) means O(t -K) for any tc > 0. For parts (i) and (ii), see [428]; for
part (iii), see [429]. Also we insist that result (i) also holds when q> is bounded from above
and below. In Chapter 2D, we shall see that the structure of the particular case q~ = 1
allows a better result, in the form of an explicit exponential rate of convergence.
5.4. Infinite entropy
We conclude this section with an interesting remark due to Abrahamsson [1]. Of course, it
seems intuitive that entropy dissipation methods require an assumption of finiteness of the
entropy. This is not true! In some situations one can decompose the solution of the spatially
homogeneous Boltzmann equation into a part with infinite entropy, but going to zero in
L 1 sense, and a part with finite entropy, on which the entropy dissipation methods can be
applied. If the estimates are done with enough care, this results in a theorem of convergence
with explicit rate, even when the initial datum has infinite entropy. It is important here to
have a good control of the entropy-entropy dissipation inequality which is used, in terms
of the initial datum.
With this technique, Abrahamsson [1] was able to prove convergence to equilibrium in
L 1 for the spatially homogeneous Boltzmann equation with hard spheres, assuming only
that the initial datum has finite mass and energy. On this occasion he used the Carlen-
Carvalho theorem, and also some iterated Duhamel formulas in the a priori estimates.
Note that this problem mainly arises for cut-off collision kernels, because for most
kernels with an angular singularity, entropy becomes finite for any positive time [440]
by regularization effects.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-228-320.jpg)
![228 C.Villani
This ends our review of applications of entropy methods in the spatially homogeneous
Boltzmann and Landau equations. Before discussing spatially inhomogeneous models, we
shall briefly consider another class of spatially homogeneous systems, characterized by
their gradient flow structure.
6. Gradient flows
This section is a little bit outside the main stream of our review, but reflects active trends
of research in kinetic theory, and may enlighten some of the considerations appearing here
and there in this chapter. The main application to Boltzmann-like equations is Theorem 21
below, for simple models of granular flows.
6.1. Metric tensors
As we explained before, several equations in kinetic theory have a gradient flow structure:
they can be written
Of=vv.(fVv6E )
at -~- (209)
for some energy functional f ~ E(f), which we shall always call the entropy for
consistency. Typical examples are the Fokker-Planck equation, for which
E(f) = flog f + flvl 2dr;
N 2 N
or the model from [68] for granular flow,
af = Vv. (f(f * VU)) +crAvf +OVv. (fv), (210)
Ot
with U(v) - Ivl3/3, and tr, 0 > 0; then
1 ~ f(v)f(w)U(v- w)dvdw
E(f) = -~ 2
s
+ (r f log f + -~ flvl 2dr. (211)
Among examples outside kinetic theory, we have also mentioned the heat equation, the
spatial Fokker-Planck equation, the porous medium equation ....
Generally speaking, a gradient flow is an equation of the form
dX
n
dt
- - grad E(X(t)). (212)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-229-320.jpg)
![A review of mathematical topics in collisional kinetic theory 229
Underlying the definition of the gradient operator is that of a Riemannian metric tensor
on some "manifold" in which the unknown X lives. Thus, to explain why (209) is a
gradient flow, we first have to explain how to define a meaningful metric tensor on the
"manifold" of all probability measures. Of course this is a formal point of view, because
infinite-dimensional Riemannian geometry usually does not make much sense, even if it
is sometimes enlightening, as the well-known works by Arnold [40] in fluid mechanics
illustrate.
In our context, the relevant metric tensor is defined as follows. Let f be a probability
density (assume that f is smooth and positive, since this is a formal definition). Let Of/Os
be a "tangent vector": formally, this just means some function with vanishing integral.
Then define
Of
2
= inf,
f }
flu] 2dr; -~s + Vv" (fu) = 0 . (213)
The infimum in (213) is taken over all vector fields u on ]t~N such that the linear transport
equation Of/Os + Vv. (fu) = 0 is satisfied. By polarization, formula (213) defines a
metric tensor, and then one is allowed to all the apparatus of Riemannian geometry
(gradients, Hessians, geodesics, etc.), at least from the formal point of view. Then, an easy
computation shows that (209) is the gradient flow associated with the energy E, on the
"manifold" of all (smooth, positive) probability measures endowed with this Riemannian
structure.
The metric tensor defined by (213) has been introduced and studied extensively by
Otto [364]. One of its important features is that the associated geodesic distance is nothing
but the Wasserstein distance on probability measures. 22 This is part of the whole area of
mass transportation, whose connections with partial differential equations are reviewed in
Villani [452].
6.2. Convergence to equilibrium
A general property of gradient flows is that they make the entropy decrease. From
formula (212) one sees that
dE(x(t)) =- IIgrad E(XCt)) ll
dt
And the equilibrium positions of (212) are the critical points for E: typically, minima. In
all the cases which we consider, there is a unique minimizer for E, which is therefore the
only equilibrium state.
Now, it is a general, well-known fact that the rate of convergence to equilibrium for a
gradient flow is very much connected to the (uniform, strict ...) convexity of the energy
functional. A typical result is the following: assume that the energy E is uniformly convex,
22See formula (244) below.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-230-320.jpg)
![230 C. Villani
in the sense that its Hessian is bounded below by some positive multiple of the identity
tensor,
Hess E ~>)~Id.
Then, E admits a unique minimizer X~, and the gradient flow (212) satisfies the linear
entropy-entropy dissipation inequality
IIgrad Ell ~ 2/.[E(X) - E(X~)]. (214)
A possible strategy to prove (214) is to go to the second derivative of the entropy
functional with respect to time. From (212) and the definition of the Hessian,
d 2
dt [[gradE(X(t))II - 2(Hess(E) 9VE, VE). (215)
The functional which just appeared in the right-hand side is the dissipation of entropy
dissipation. Therefore, the assumption of uniform positivity of the Hessian implies
d 2
dr IIgrad E (X (t))II -2~llgrad E(X<t>)ll 2
Integrating this inequality in time, one easily arrives at (214) if everything is well-
behaved. 23
This remark shows that the trend to equilibrium for Equation (209) can in principle be
studied via the properties of convexity of the underlying energy E. But the right notion
of convexity is no longer the usual one: it should be adapted to the definition (213). This
concept is known as displacement convexity, and was first studied by McCann [338,340],
later by Otto [364], Otto and Villani [365].
DEFINITION 3. Let f0, fl be two (smooth, positive) probability measures on I[~N. By a
classical theorem of Brenier [103,339] and others, there exists a unique gradient of convex
function, Vqg, such that
v~o#fo = f~,
meaning that the image measure24 of fo by the mapping Vq9 is the measure fl. Let us
define the interpolation (fs)o<~s<<.lbetween f0 and fl by
fs -- [(1 - s)Id + sV~o]#fo.
23There are also other, simpler derivations of (214) based on Taylor formula. The above procedure was
chosen because this is precisely a way to understand the famous Bakry-Emery method for logarithmic Sobolev
inequalities.
24By definition of the image measure: for all bounded continuous function h, f(h o V~o)f0 = f hfl. If ~ois C2,
then for all v, fo(v) = fl (V~0(v)) det(D2~0)(v).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-231-320.jpg)
![A reviewofmathematicaltopicsincollisionalkinetictheory 231
Then, the functional E is said to be displacement convex if whatever fo, fl,
s w-~E (f~) is convex on [0, 1].
It is furthermore said to be uniformly displacement convex with constant )~ > 0 if whatever
fo, f~,
d2
ds 2
~E(fs) ~ )~W(fo, fl) 2 (0~<s ~< 1),
where W stands for the Wasserstein distance25 between f0 and fl.
REMARKS. (1) To get a feeling of this interpolation procedure, note that the interpolation
between (~a and 6b is 6(1-s)a+sb,instead of (1 - S)6a + s3t,. Further note that the preceding
definition reduces to the usual definition of convexity if the interpolation (fs)o~<s~<l is
replaced by the linear interpolation.
(2) Let us give some examples. The functionals f f log f, or f fP (p >/ 1) are
displacement convex as one of the main results of McCann [340]. The functional f fV is
displacement convex if and only if the potential V is convex. Moreover, if V is uniformly
convex with constant )~, the functional f f V is uniformly displacement convex, with the
same constant.
Another interesting example is the case of functionals like f•2N f(v)f(w)U(v-
w) dv dw. Such a functional is never convex in the usual sense, except for some very
peculiar potentials (power laws ...). On the other hand, it is displacement convex as soon
as U is convex.
Among the results in Otto and Villani [365], we mention the following statement. If E
is uniformly displacement convex, with constant )~ > 0, then the associated gradient flow
system (209) satisfies a linear entropy-entropy dissipation inequality of the form
fRNf
EI2
7-~- dv >i-2s E(f~)],
where f~ is the unique minimizer of the energy. This is not a true theorem, because the
proof is formal, but this is a general principle which can be checked on each example of
interest. A standard strategy of proof goes via the second derivative of the entropy, as we
sketched above. In the context of the linear Fokker-Planck equation (168), this strategy
of taking the second derivative of the entropy is known as the Bakry-Emery strategy, and
goes back to the mid-eighties.
To check the assumption of uniform displacement convexity, it is in principle sufficient
to compute the Hessian of the entropy. When this is done, one immediately obtains the
dissipation of entropy dissipation via formula (215). This calculation is however very
intricate, as one may imagine. This is where Bakry and Emery [45] need their so-called
25 Seeformula(244)below.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-232-320.jpg)
![232 C. Villani
F2 calculus, which is a set of formal computation rules involving linear diffusion operators
and commutators. On the other hand, the formalism developed by Otto [364], Otto and
Villani [365] enables simpler formal computations, and can be adapted to nonlinear cases
such as granular flows [130].
Just to give an idea of the complexity of the computations, and why it is desirable to have
efficient formal calculus here, let us reproduce below the dissipation of entropy dissipation
which is associated to the gradient flow for Equation (210):
fo ]2
DD(f) = 2rr f (v) -~v~(V) dv
+ 20 f(v) I~(v)12dv
+ f(v)f(w)(D2U(v - w)
x - - (216)
where
a[ Iv12 1
(v)=~ (rlogf(vl+O--~+U,f , U(v) = Ivl3/3.
6.3. A survey of results
Let us now review some results of trend to equilibrium which were obtained via the
considerations above, or which can be seen as related. A survey paper on this subject is
Markowich and Villani [330].
The first partial differential equation to be treated in this way was the spatial Fokker-
Planck equation,
Op
Ot
=Vx . (Vxp -t- pVV(x)).
The classical paper by Bakry and Emery [45] shows that the solution to this equation
converges exponentially fast, in relative entropy sense, to the equilibrium e-v (assuming
f e- v _ 1), at least if v is uniformly convex with constant )~.The decrease of the entropy is
like e-2~t . Underlying entropy-entropy dissipation inequalities are known under the name
of logarithmic Sobolev inequalities, and have become very popular due to their relationship
with many other fields of mathematics (concentration of measure, hypercontractivity,
information theory, spin systems, particle systems ... see the review in [16]).
The Bakry-Emery strategy, and the corresponding proof of the Stam-Gross logarithmic
Sobolev inequality, were recently re-discovered by Toscani [422,423] in the case of the
kinetic Fokker-Planck equation. Instead of F2 calculus, Toscani generalized a lemma by
McKean [341] to compute the second derivative of the entropy functional. Though this](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-233-320.jpg)
![A review of mathematical topics in collisional kinetic theory 233
work was mainly a re-discovery of already known results, it had several merits. First,
it suggested a more physical way of understanding the Bakry-Emery proof, in terms of
entropy dissipation and dissipation of entropy dissipation. Also, Toscani directly studied
the Fokker-Planck equation Otf -- Af + V. (f v), while previous authors mainly worked
on the adjoint form, Oth -- Ah - v 9Vh. Last but not least, his paper made these methods
and techniques popular among the kinetic community, which began to work on this subject:
see in particular the recent synthesis by Arnold, Markowich, Toscani and Unterreiter [39].
Then, these results were generalized to the porous medium equation with drift,
Op
-- Axp m -k- Vx . (px).
Ot
It was found that when m ~> 1 - 1/N, solutions to this equation converge exponentially
fast (with relative entropy decreasing like e-2t) towards a probability density known
as Barenblatt-Pattle profile. These results were obtained independently by Otto [364],
Carrillo and Toscani [131], and Del Pino and Dolbeault [163]. All three papers established
nonlinear analogues of the logarithmic Sobolev inequalities. The paper by Otto made the
link with mass transportation problem. Various generalizations of all these results can be
found in [129].
Let us now come back to kinetic, Boltzmann-like systems and display recent results
about the equations for granular media suggested in [70,68]. These results were proven
by Carrillo, McCann and Villani [130] by using the ideas above, and in particular those
of gradient flows, Wasserstein distance and Bakry-Emery strategy. The theorem which we
state is slightly more general: the dimension is arbitrary, and the interaction potential is not
necessarily cubic.
THEOREM 21. Let U be a convex, symmetric potential on ~N, and or, 0 ~ O. Let
E(f)--a f fl~ +Of f(v)lv122dv
1s f (v) f (w)U(v - w) dvdw.
Let moreover
iol2 )
~-V crlogf +O-~+U. f .
We consider the equation
of = V. (f~), (217)
at
for which E is a Lyapunov functional, whose time-derivative is given by the negative of
D(f) - f fill2 dr.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-234-320.jpg)
![234 C. Villani
Let fo be an initial datum, and f (t) be the solution 26 to (217). Let moreover fc~ be the
unique minimizer of E (if 0 -- O, the unique minimizer of E which has the same mean
velocity as fo), and let
E(flf~)= E(f)- E(f~).
Then,
(i) if 0 > O, then D(f) ~ 20E(flf~) for all probability distribution f, and
E(f (t)lf~) <<.e-20tE(fOIfoo);
(ii) if O = 0 and U is uniformly convex, D2U ~ s then D(f) ~ 2s for all f
with the same mean velocity as foo, and E(f (t)lf~) <~e-2xt E(folf~);
(iii) if 0 = 0 and U is strictly convex, in the sense
D2U(z) ~ g(Izl '~ A 1), g,o~ > O, (218)
then D(f) ~ CE(flfc~) # for some positive constants C, r, and for any f with the same
mean velocity as f~, and E(f (t)lfc~) converges to 0 at least in O(t -K) for some x > 0;
(iv) if 0 = O, U is strictly convex in the sense of (218), and moreover a > O, then
for all f with the same mean velocity as f~, one has D(f) ~ s for some
s > 0 which only depends on an upper bound for E(f). Moreover, E(f (t)lf~) <.
e-X~ for some positive constant )~o which only depends on an upper bound
for E(fo).
REMARKS. (1) The assumptions on the mean velocity reflect an important physical
feature: in the cases in which they are imposed, the entropy is translation-invariant.
(2) The motivation to study strictly convex, but not uniformly convex potentials comes
from the physical model where U (z) = Izl3/3. Then, lack of uniformity may come from
small values of z. This difficulty is of the same type than in the study of hard potentials
for the Boltzmann equations; this is not surprising since the model in [70] can be seen as a
limit regime for some Boltzmann-type equation with inelastic hard spheres.
(3) In general, point (iii) cannot be improved into exponential decay: when a -- 0 --- 0
and U(z) = Izl3/3, then the decay of the energy is O(1/t) and this is optimal.
(4) We note that part (iv) of this theorem is the most surprising, because in this case the
energy functional is not uniformly displacement convex; yet there is a "linear" entropy-
entropy dissipation inequality (not uniform in f). This result raises hope that the entropy-
entropy dissipation inequalities described in Section 4 for the Landau equation with hard
potentials may be improved into inequalities of linear type.
To conclude this section, we mention that the trend to equilibrium for Equation (217)
has been studied by Malrieu [329] with the F2 calculus of Bakry and Emery. Even though
his results are much more restrictive (only a, 0 > 0) and the constants found by Malrieu
are not so good, on this occasion he introduced several interesting ideas about particle
26WeassumethatU is sufficientlywell-behavedthatexistenceofaunique"nice"solutionto(217)is guaranteed.
Thisis quitea weakassumption.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-235-320.jpg)
![A review of mathematical topics in collisional kinetic theory 235
systems and his work provided yet another connection between the kinetic and probabilistic
communities.*
7. Trend to equilibrium, spatially inhomogeneous systems
We now turn to the study of spatially inhomogeneous kinetic systems like the ones
presented in Section 1.2. We first make several remarks.
(1) Once again, we are mainly interested in explicit results, and wish to cover situations
which are not necessarily perturbations of the equilibrium. Thus we do not want to use
linearization tools, and focus on entropy dissipation methods.
(2) For many of the spatially inhomogeneous models which we have introduced, the
entropy and the entropy dissipation functionals are just the same as in the spatially
homogeneous case, up to integration in x. Also, the transport part does not contribute
to the entropy dissipation. Thus, one may think, the same entropy-entropy dissipation
inequalities which we already used for the spatially homogeneous case will apply to the
spatially inhomogeneous case. This is completely false, as we shall explain! And the
obstruction is not a technical subtlety, but stands for a good physical reason.
(3) Nevertheless, it is plainly irrelevant to ask for an x-dependent version of the entropy-
entropy dissipation inequalities presented in Section 4, since the entropy dissipation does
not make the x variable play any role.
(4) The boundary conditions, and the global geometry of the spatial domain, are
extremely important in this study. In this respect the problem of trend to equilibrium
departs notably from the problem of the hydrodynamic limit, which fundamentally is a
local problem.
(5) To work on the trend to equilibrium, one should deal with well-behaved solutions,
satisfying at least global conservation laws. In the sequel, we shall even assume that
we deal with very well-behaved solutions, for which all the natural estimates of decay,
smoothness and positivity are satisfied. Of course, for such equations as the Boltzmann
or Landau equation, nobody knows how to construct such solutions under general
assumptions .... Therefore, the results dealing about these equations will be conditional, in
the sense that they will depend on some strong, independent regularity results which are not
yet proven. It is however likely that such regularity bounds can be obtained with present-
day techniques in certain particular situations, like weakly inhomogeneous solutions [32].
We wish to insist that even if we assume extremely good a priori estimates, the problem of
convergence to equilibrium remains interesting and delicate, both from the mathematical
and from the physical point of view!
7.1. Local versus global equilibrium
When studying the trend to equilibrium in a spatially dependent context, a major obstacle to
overcome is the existence of local equilibrium states, i.e., distribution functions which are
*Note added in proof: In a more recent, quite clever work, Malrieu was able to remove the condition 0 > 0.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-236-320.jpg)
![236 C. Villani
in equilibrium with respect to the velocity variable, but not necessarily with respect to the
position variable. For instance, for the Boltzmann or Landau equation, a local equilibrium
is a local Maxwellian,
Iv-u(x)l 2
e 2T(x)
f (x, v) -- p(x) tZ" lt )) /'''7r~'x''N/2
. (219)
For the linear Fokker-Planck equation, a local equilibrium is a distribution function of the
form
Ivl2
e 2
f (x, v) -- p(x) (2:rr)N/-----------------
~ = p(x)M(v). (220)
Local equilibria are not equilibrium distributions in general, but they make the entropy
dissipation vanish. This shows that there is no hope to find an entropy-entropy dissipation
inequality for the full x-dependent system.
If the system ever happens to be in local equilibrium state at some particular time to,
then the entropy dissipation will vanish at to, and it is a priori not clear that the entropy
functional could stay (almost) constant for some time, before decreasing again. This may
result in a strong slowing-down of the process of trend to equilibrium. This difficulty has
been known for a long time (even to Boltzmann! as pointed out to us by C. Cercignani),
and is discussed with particular attention by Grad [254], Truesdell [430, pp. 166-172] and,
in the different but related context of hydrodynamic limits for particle systems, Olla and
Varadhan [362].
On the other hand, whenever the solution happens to coincide with some local
equilibrium state Mloc at some time, then the combined effect of transport and confinement
will make it go out of local equilibrium again, unless Mloc satisfies some symmetry
properties which ensure that it is a stationary state. In fact, in most situations these
symmetry properties select uniquely the stationary state among the class of all local
equilibria.
Note the fundamental difference with the problem of hydrodynamical limit: in the latter,
one wishes to prove that the solution stays as much as possible close to local equilibrium
states, while here we wish to prove that if the solution ever happens to be very close to
local equilibrium, then this property will not be preserved at later times.
Thus, one can see the trend to equilibrium for spatially inhomogeneous systems as the
result of a negociation between collisions on one hand, transport and confinement on the
other: by dissipating entropy, collisions want to push the system close to local equilibrium,
but transport and confinement together do not like local equilibria- except one. This is why
transport phenomena, even if they do not contribute in entropy dissipation, play a crucial
role in selecting the stationary state. Our problem is to understand whether these effects
can be quantified.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-237-320.jpg)
![A review of mathematical topics in collisional kinetic theory 237
An answer to this question was recently obtained by the author in a series of
collaborations with Desvillettes. In the sequel, we shall explain it on a simple case: the
linear Fokker-Planck equation, with potential confinement,
~f
+ v. Vxf - VV(x). Vvf = Vv. (Vvf + fv). (221)
at
The trend to equilibrium for this model was studied by Bouchut and Dolbeault [100]
with the help of compactness tools, so no explicit rate of convergence was given. Also,
Talay [413] proved exponential decay with a probabilistic method (based on general
theorems about recurring Markov chains) which does not seem to be entirely constructive.
Other probabilistic approaches have been proposed to study this model, but they also
strongly depend on the possibility to interpret (221) as the evolution equation for the law
of the solution of some stochastic differential equation with particular properties.*
In the sequel, we shall explain how the entropy method of Desvillettes and Villani [184]
leads to polynomial (as opposed to exponential), but fully explicit estimates. This method
is robust in the sense that it can be generalized to smooth solutions of nonlinear equations,
in particular Boltzmann or Vlasov-Poisson-Fokker-Planck equations.
With respect to the Boltzmann equation, the model (221) has several pedagogical
advantages. First, one can prove all the a priori estimates which are needed in the
implementation of the method. What is more important, the local equilibrium only depends
on one parameter (the density), instead of three (density, mean velocity and temperature).
This entails significant simplifications in the computations and intermediate steps, which
however remain somewhat intricate.
7.2. Local versus global entropy: discussion on a model case
To use entropy methods in a spatially dependent context, the main idea is to work at the
same time at the level of local and global equilibria; i.e., estimate simultaneously how far
f is from being in local equilibrium and how far it is from being in global equilibrium.
(1) One first introduces the local equilibrium associated with f, i.e., the one with the
same macroscopic parameters as f. For instance, in the case of the linear Fokker-Planck
equation, the local equilibrium is just (220), with p(t, x) = f f(t, x, v)dr. In the case of
the Boltzmann equation, the local equilibrium is the local Maxwellian (219), with p, u, T
given by (1).
How close f is from local or global equilibrium will naturally be measured by relative
entropies. Thus one defines Hglo to be the relative entropy of f with respect to the
global equilibrium, and Hloc to be the relative entropy of f with respect to the associated
local equilibrium. In the Fokker-Planck (resp. Boltzmann) case, Hloc is H(flpM) (resp.
H(fIMf)); note that this is an integral over Rff • R N now. Then, one looks for a system
of differential inequalities satisfied by Hglo and Hloc.
*Note added in proof: A muchmorecomplete,verysatisfactorystudywas recentlyperformedby Hrrau and
Nier; see alsothe referencesprovidedin theirwork.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-238-320.jpg)
![238 C. Villani
(2) The first equation is given by an entropy-entropy dissipation estimate of the same
type that the ones we discussed in Section 4. We just have to apply this inequality pointwise
in x. For instance, for the linear Fokker-Planck equation,
dHglo=s I(flpM)dx -- s p I( f
D(f ) -- - dt N N
M) dx, (222)
and by the logarithmic Sobolev inequality (175),
D(f)>2s pH( f
~x
M) dx = 2H(fIpM) - - 2Hloc (223)
(check the last-but-one equality to be convinced!). Note that the symbol H is used above in
two different meanings: relative entropy of two probability distributions of the v variable,
relative entropy of two probability distributions of the x and v variables.
Similarly, if we have nice uniform a priori bounds for the solution of the Boltzmann
equation, it will follow from our discussion in Section 4 that
dt
-- -- Hglo ) K Hl~c, (224)
for some constants K > 0, ot > 1.
In a spatially homogeneous context, this inequality would be essentially sufficient to
conclude by Gronwall's lemma. Here, we need to keep much more information from the
dynamics in order to recover a control on how the positivity of Hglo forces Hloc to go up
again if it ever vanishes.
(3) To achieve this, we now look for a differential inequality involving the time-behavior
of Hloc. We start with a heuristic discussion. At a time when the entropy dissipation would
vanish, then both the local relative entropy and its time derivative would vanish, since
the relative entropy is always nonnegative. Therefore, one can only hope to control from
below the second time derivative of the local relative entropy! Taking into account the first
differential inequality about Hglo and Hloc, this more or less resembles to considering the
third derivative of the entropy at an inflexion point.
It is easy to compute (d2/dr 2)Hloc at a time to when f happens to be in local equilibrium.
For instance, in the case of the linear Fokker-Planck equation, we have the remarkably
simple formula
d21 s
H(fIpM) - p
dt2 t--to N
vp 1: v).
§ VV dx =_Ix (pie- (225)
P
Here Ix is the Fisher information, applied to functions of the x variable. We do not describe
here the corresponding results for the Boltzmann equation, which are of the same nature but
much, much more complicated [181]. Here we shall continue the discussion only for the
Fokker-Planck equation, and postpone the Boltzmann case to the end of the next section.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-239-320.jpg)
![A review of mathematical topics in collisional kinetic theory 239
If V is well-behaved, the logarithmic Sobolev inequality, applied in the x variable, yields
d 2
dt 2 t=to
H(flpM) >~KH(ple -V) (226)
for some positive constant K depending only on V. This is the piece of information that
was lacking! Indeed, for the linear Fokker-Planck equation,
Hglo-- Hloc4- H(ple-V);
thus Equation (226) turns into
d 2
dt 2 t=to
H (f IpM) >~KHglo- K Hloc. (227)
Note that the use of the logarithmic Sobolev inequality in the x variable is the precise
point where the geometry of the boundary conditions (here replaced by a confinement
potential) comes into play. The fact that this effect can be quantified by a functional
inequality is very important for the method; see the remarks in the end of the chapter
for the analogous properties in the Boltzmann case.
Of course, the preceding calculations only apply at a time to where f happens to be in
local equilibrium- which is a very rare event. Therefore, one establishes a quantitative
variant of (227), in the form
d2 K
dt
----TH(flpM) ~> z---x-(H~I~ Hloc)- J(flpM), (228)
where J (flpM) is a complicated functional which vanishes only if f -- pM:
1 fR (pu)2 l fR IVx.(pu)12
- J (flpM) -- ~ dx 4-
4 U p -4 N [9
dx
UR IVx" (puQu)] 2 fR
4- dx4-
U /9 U
IVx[p(T- 1)]12
P
fR [Vx . S]2
+ ~ dx + Iv(f[pM)
U p
dx
1
4- -~Iv(flp M) 1/2 Ix(flp M) 1/2. (229)
Here p, u, T are the usual macroscopic fields, and S is the matrix defined by the equation
p(x)u(x) | u(x) + p(x)T(x)IN 4- S(x) -- f•N f (x, V)V | vdv. (230)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-240-320.jpg)
![240 C. Villani
(4) The next step of the program is to control J in terms of Hloc, in order to have a closed
system of differential inequalities on Hloc and Hglo. This is done by some ad hoc nonlinear
interpolation procedure, which yields
d 2
dt 2
K
--H(fIpM) >~-~ H(flfc~)- Cs(f) H(flpM) 1-E
Here e is an arbitrary positive number in (0, 1) and Ce(f) is a constant depending on f
via moment bounds, smoothness bounds, and positivity estimates on f.
All these bounds have to be established explicitly and uniformly in time, which turns out
to be quite technical but feasible [184] (see also Talay [413]); then the constant Ce = Ce (f)
can be taken to be independent of t.
In the case of the Boltzmann equation, it is possible to perform a similar interpolation
procedure; the only missing step at the moment is establishing the a priori bounds.
(5) Summing up, for solutions of the Fokker-Planck equation we have obtained the
system of differential inequalities
d
-- ~-THglo > 2Hloc,
d2 K
d-~ Hloc/> -~- Hglo- Ce Hlloc e 9
(231)
The last, yet not the easiest step, consists in proving that the differential system (231)
alone implies that Hglo converges to 0 like O(t-K). Since there is apparently no comparison
principle hidden behind this system, one has to work by hand .... The bound established
in Desvillettes and Villani is
Hglo = O(tl-1/e),
which is presumably optimal. Thus, the global entropy converges to 0 with some explicit
rate, which was our final goal.
7.3. Remarks on the nature of convergence
Solutions to (231) do have a tendency to oscillate, at least for a certain range of parameters.
In fact, were it not for the positivity of relative entropies, system (231) would not imply
convergence to 0 at all! We expect "typical" solutions of (231) to decrease a lot for small
times, and then converge to 0 more slowly as t --+ +c~, with some mild oscillations in the
slope. This kind of behavior is completely different from what one can prove in the context
of spatially homogeneous kinetic equations. 27 We think that it may reflect the physical
nature of approach to equilibrium for spatially inhomogeneous systems. As time becomes
large and the system approaches global equilibrium, it is more and more likely to "waste
27The rate of convergence typically improves as t --+ +c~, see Section 5.1.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-241-320.jpg)
![A reviewof mathematical topics in collisionalkinetic theory 241
time" fighting against local equilibria .... And this may result in oscillations in the entropy
dissipation.
But examination of a particular, "integrable" case, suggests that (1) these oscillations
may be present only when the confinement potential is strong enough, (2) the decay should
be exponential. This case corresponds to the quadratic confinement potential, V(x) =
0921xl2/2 -+-C. For this particular shape of the potential, the Fokker-Planck equation can
be solved in semi-explicit form [399, Chapter 10], and the rate of decay is governed by the
quantities exp(-Xt), where
1
1 - ~/1 - 4092 092 ~< 4
X= 2 ' -'
- 1
X-- 1 4- i~/4092 1 092 >
2 ' 4
(232)
(in [399], these equations are established only in dimension 1). Thus the decay is always
exponential, the rate being given by the real part of X. When the confinement is very tiny,
then the convergence is very slow (think that there is no trend to equilibrium when there
is no confinement); when the confinement becomes stronger then the rate increases up
to a limit value 1/2. For stronger confinements, the rate does not improve, but complex
exponentials appear in the asymptotics of the solution. Note that in the same situation, the
rate of convergence for the spatially homogeneous equation would be equal to 1.
Another integrable case is when there is no confinement potential, but x 6 TN, the N-
dimensional toms. Then the decay is always exponential and the rate depends on the size
of the periodic box. In dimension 1 of space, it is maximal (equal to 1) when the side of
the box has length ~<2zr [152].
It is yet an open problem to generalize the above considerations to nonintegrable cases,
and to translate them at the level of entropy dissipation methods. In our opinion, these
examples show that a lot of work remains to be done to get an accurate picture of the
convergence, even in very simplified situations.*
7.4. Summary and informal discussion of the Boltzmann case
We now sum up the state of the art concerning the application of entropy dissipation
methods to spatially inhomogeneous systems. The following theorem is the main result
of Desvillettes and Villani [184].
THEOREM 22. Let M(v) denote the standard Maxwellian probability distribution on ]KN
with zero mean velocity and unit temperature. Let V be a smooth confining potential
on ~N, behaving quadratically at infinity:
V(x) = ~o2' '
'x'2 + ~(x),
2
*Note addedinproof: Forprogressonthesequestions,the recentworkby HrrauandNieris recommended.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-242-320.jpg)
![242 C. Villani
where co > 0 and 9 9 ~k>~OHk(RN) 9Assume without loss of generality that f e-v(x) dx --
1, and let
foe(X, v) = e-V(X)M(v)
denote the unique global equilibrium of the Fokker-Planck equation
af
Ot
- - + v . Vx f - VV(x). Vv f = Vv . (Vv f + f v).
Let fo = fo(x, v) be a probability density such that fo/f~ is bounded from above and
below, and let f (t) = f (t, x, v) be the unique solution of the Fokker-Planck equation with
initial datum fo. Then, for all e > 0 there exists a constant CE(fo), explicitly computable
and depending only on V, fo and e, such that
H(f (t)lf~) <<.C~(fo) t-1/e.
We already pointed out several shorthands of this result: in particular, the convergence
should be exponential. We consider it as a major open problem in the field to compute the
optimal rate of decay, in relative entropy, as a function of the confinement potential V.
Let us now turn to nonlinear situations. The following result was recently obtained by
the author in collaboration with Desvillettes [181].
THEOREM 23. Let B be a smooth collision kernel bounded from above and below. Let
fo = fo(x, v) be a smooth probability density on I-2x x N3v,where [2 is a smooth bounded,
connected open subset of ]R3 with no axis of symmetry, and let f (t) - f (t,x, v) be a
smooth solution of the Boltzmann equation
af
Ot
--+ v. Vx f = Q(f, f), t>~O, xeX2, yen 3,
with specular reflection boundary condition. Let moreover f~ (x, v) be the unique global
equilibrium compatible with the total mass and kinetic energy of fo. Assume that all the
moments of f are uniformly bounded in time, that f is bounded in all Sobolev spaces,
uniformly in time, and that it satisfies a lower bound estimate,
f (t, x, v) >/poe-A~
for some p >~2, Po > O, Ao > O, uniformly in time. Then, for all e > 0 there is a constant
Ce, depending only on (finitely many of) the requested a priori bounds, such that
H(f (t)lf~) <. Cet -1/e.
We do not display here the system of differential inequalities - much, much more
complicated than (231) - which underlies this result, and we refer to [181] for more
information. An unexpected feature was revealed by this study: not only is the entropy](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-243-320.jpg)
![A review of mathematical topics in collisional kinetic theory 243
dissipation process indeed slowed down when the distribution function becomes a
local Maxwellian state, but also, it is much more slowed down for some particular
local Maxwellian states, in particular those of the form p(x)M(v), i.e., with constant
temperature and zero velocity. More precisely, the entropy dissipation vanishes in time
up to order 4 (instead of 2) when going through such a Maxwellian state.
From the mathematical point of view, this entails a spectacular complication of the
arguments, and the need for at least three differential inequalities: apart from the behavior
of the global entropy, one studies at the same time the departure of f with respect to Mf
and the departure of f with respect to pM.
From the physical point of view, this additional degeneracy could be interpreted as an
indication that the relaxation of the density typically holds on a longer time-scale, than the
relaxation of the temperature and the local velocity- although we should be cautious about
this.
The proof of Theorem 23 combines the general method of Desvillettes and Villani [184]
with the entropy dissipation results of Toscani and Villani [428]. In the computations,
the natural functionals H(flMU), H(flpM) were traded for the simpler substitutes
Ilf - mf 112L2' Ilf - PMIIZL
2 which enable one to weaken significantly the assumptions of
Theorem 23. Of course, these assumptions are still very strong, even though rather natural
after our discussions in Chapter 2B.
The influence of the shape of the box is quantified by the values of several "geometric"
constants related to it. One of these constants is the Poincar6 constant P(s defined in a
scalar setting by
L2(n)
whenever F is a real-valued function on f2, and in a vector setting by
IlVxull2
L2(n)/> P(Y2)llull 2
L2 (f2) '
whenever u is a vector field in X2, tangent to 0f2. Another constant which appears in the
proof is what we call Grad'snumber,defined in [254]:
G(f2)-inf {fs2~ij ( Ovi OvJ)2"
coo6SN-1 .. ~ + OXi ~] '
V. v--O, V/x v-coo, v.n--Oon Of2},
n standing for the unit normal on OX2. The number G(f2) is strictly positive if and only
if s has no axis of symmetry. This number contributes to a lower bound for the constant
K (s in a variant of the Korn inequality which reads
VxU+rVxU
L2(n)
>>.
K(~)llVxul122(n). (233)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-244-320.jpg)
![244 C. Villani
The Korn inequality, of paramount importance in elasticity theory [201], is naturally
needed to establish the system of differential inequalities which we use to quantify the
trend to equilibrium. Our proof of (233) is partly inspired by Grad [254].
The whole thing adapts to the case of the toms, or to the bounce-back boundary
condition, with significant simplifications. On the other hand, in the case of domains with
an axis of symmetry, additional global conservation laws (angular momentum) have to be
taken into account, and the case of a spherical domain also has to be separated from the
rest. These extensions are discussed by Grad [254], but have not yet been transformed in a
quantitative variant along the lines above.
REMARK. As we have seen in Chapter 2B, if the initial datum is not very smooth and if
the Boltzmann collision kernel satisfies Grad's cut-off assumption, then the solution of the
Boltzmann equation is not expected to be very smooth. But in this case, as we discussed
in Section 3.5 of Chapter 2B, one can hope for a theorem of propagation of singularities in
which a vanishingly small (as t ~ c~) singular part could be isolated from a very smooth
remainder, and, as in [1], the entropy dissipation strategy would still apply.
Theorem 23 certainly calls for lots of improvement and better understanding. Yet, it
already shows that, in theory, entropy dissipation methods are able to reduce the problem
of trend to equilibrium for the full Boltzmann equation, to a problem of uniform a priori
estimates on the moments, smoothness and strict positivity of its solutions. Moreover, it
shows that there is no need for stronger a priori estimates than the ones which are natural
in a nonlinear setting" in particular, no estimates in L2(M -1) are needed. We hope that
these results will also provide a further motivation for the improvement of known a priori
bounds.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-245-320.jpg)
![Preface
The motion of fluids has intrigued scientists since antiquity and we may say that the
field of mathematical fluid dynamics originated more than two centuries ago. In 1755
Euler [2] gave a mathematical formulation of the principle of conservation of mass in
terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear
and angular momentum by PDEs. Material symmetry and frame invariance were used by
Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of
internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint-
Venant [11] and Stokes [12].
In the 19th century no sharp distinction was drawn between mathematicians and
physicists as we sometime see in more recent times. The formulation of the equations
of fluid motion could be considered as either mathematics or physics. The first work
in fluid dynamics that has a "modern" mathematical flavor may have been done by
Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous
problem. Riemann recognized the mathematical nature of the entropy. This notion led
him to his duality method for solving the non-characteristic Cauchy problem for linear
hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his
contemporaries. What we now call the Cauchy problem for a PDE and the search for its
solution did not have the significance that it is accorded nowadays. Even Poincar6 did not
raise that kind of question in his Th~orie des tourbillons.
For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly
accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the
Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28.
Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of
the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much
as Riemann, Leray developed new mathematical tools which proved to have independent
interest: e.g., weak solutions (that we now call Leray's solutions in this context) and
topological degree (a joint work with Schauder [4]).
Since the 1930s, the interest that mathematicians devote to fluid dynamics has
unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von
Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a
Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on
the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic
displays such a variety of models and questions that thousands of scientists, among them
many mathematicians, focus their research on fluid dynamics.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-247-320.jpg)
![A reviewof mathematicaltopicsin collisionalkinetictheory 247
In this chapter we focus on the Boltzmann collision operator when the collision kernel only
depends on the deviation angle:
B(v - v,, or) = b(cosO). (234)
As recalled in Chapter 2A, the modelling of Maxwell molecules, i.e., more or less
fictitious particles interacting via repulsive forces in 1/r 5, in three dimensions of space,
leads to a collision kernel B which satisfies (234). By extension, we shall call Maxwellian
collision kernel any collision kernel of the form (234).
Assumption (234) entails a number of particular properties. The gain part of the
Boltzmann collision operator
Q+(g,f)=s ao,f
N N-I
cos0= Iv-v,l'a'
dcr b(cosO)g(v',) f (v'),
(235)
shares many features with the (more symmetric) rescaled convolution operator,
g. f ----gl 9 f!, (236)
2 2
where the rescaling operation is defined by
1 (v t
fk (1)) -- )~N/2f --~ " (237)
Note that if X and Y are independent random variables with respective law f and g,
then f * g is the law of (X + Y)/~/2. Therefore, with the analogy between the Q+
and * operations in mind, the theory of the spatially homogeneous Boltzmann equation
with Maxwellian collision kernel resembles that of rescaled sums of independent random
variables.
In the sequel, we shall insist on some peculiar topics which illustrate the originality
of Maxwellian collision kernels: in Section 1 the Wild sum representation, which is
an appealing semi-explicit representation formula for solutions in terms of iterated Q+
operators; in Section 2, the existence and applications of several contracting probability
metrics compatible with the Boltzmann equation. Finally, in Section 3, we describe some
interesting connections with information theory.
For more standard issues concerning the Cauchy problem or the qualitative behavior of
solutions, the best reference is the long synthesis paper by Bobylev [79], entirely based on
the use of Fourier transform, which also reviews many contributions by various authors.
Before embarking on this study, we recall that besides its specific interest, the study
of Maxwellian collision kernels is often an important step in the study of more general
properties of the Boltzmann operator. 1
1See,for instance,Section4.4 in Chapter2Bor Sections4.3 to4.6 in Chapter2C.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-248-320.jpg)
![248 C. Villani
1. Wild sums
If the collision kernel is Maxwellian and Grad's cut-off assumption is satisfied, then one
can assume without loss of generality that for some (and thus any) unit vector k,
is fo
N-1 b(k. a) da = IsN-2I b(cos0) sinN-2 0 dO = 1. (238)
Then the spatially homogeneous Boltzmann equation can be rewritten as
~f
= Q+ (f, f)- f. (239)
Ot
However, when the collision kernel is nonintegrable, then one can only write the general
form
Ofot= f~tN dr, fSN-1 da b(cosO)[fl f~, - ff,]. (240)
As was noticed by Wild [464], given any initial datum f0, Equation (239) can be solved
recursively in terms of iterated Q+ operators (this is nothing but a particularly simple
iterated Duhamel formula, if one considers (239) as a perturbation of Otf = --f). One
finds
oo
f(t,.)=e-tZ(1-e-t)n-lQ+(fo) '
n=l
(241)
where the n-linear operator Q+ is defined recursively by
o+r io,
n-1
1
Qn-k(fO))"
- 1 ~ (o; Io), +
k=l
The sum (241) can also be rewritten
f(t, v) -- Ze-t(1 e-t) n-1 +
- ot(ylQ• ,
n=l y )
(242)
where F(n) stands for the set of all binary graphs with n leaves, each node having zero
or two "children", and Q+ (f0) is naturally defined as follows: if y has two subtrees Yl
and Y2 (starting from the root), then Q+ (f0) = Q+ (Q• (f0), Q+ (f0)). Moreover, ct(y)
are combinatorial coefficients. Wild sums and their combinatorial contents are discussed
with particular attention by McKean [341], and more recently by Carlen, Carvalho and
Gabetta [126].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-249-320.jpg)
![A review of mathematical topics in collisional kinetic theory 249
It follows from the Wild representation that a solution of the Boltzmann equation (239)
can be represented as a convex combination, with time-dependent weights, of terms of the
form
fo, Q+(fo, fo), Q+(Q+(fo, fo), fo),
Q+(Q+(fo, fo), Q+(Q+(fo, fo), Q+(fo, fo))),
etc.
This representation is rather intuitive because it more or less amounts to count collisions:
the f0 term takes into account particles which have undergone no collision since the initial
time, the term in Q+ (f0, f0) corresponds to particles which have undergone only one
collision with a particle which had never collided before, Q+ (f0, Q+ (f0, f0)) to particles
which have twice undergone a collision with some particle having undergone no collision
before ....
This point of view is also interesting in numerical simulations: in a seemingly crude
truncation procedure, one can replace (241) by
No
f (t, V)- e-t E(1 --e-t) n-1Q+(fo)+ (1- e-t)NOM,
n--1
where M is the Maxwellian distribution with same first moments as f0. Later in this
chapter, we shall explain why such a truncation is rather natural, how it is related to the
problem of trend to equilibrium and how it can be theoretically justified. 2
2. Contracting probability metrics
In this section, probability metrics are just metrics defined on a subset of the space of
probability measures on IRN.
We call a probability metric d nonexpansive along solutions of Equation (240) if,
whenever f (t, .) and g(t, .) are two solutions of this equation, then
d(f (t), g(t)) <~d(fo, go). (243)
We also say that d is contracting if equality in (243) only holds for stationary solutions,
i.e. (in the case of finite kinetic energy), when f0, go are Maxwellian distributions.
2.1. The Wassersteindistance
In his study of the Boltzmann equation for Maxwellian molecules, Tanaka [414,353,415]
had the idea to use the Wasserstein (or Monge-Kantorovich) distance of order 2,
W(f, g) = inf{y/ElZ - YI2; law(X) - f, law(Y) - g}. (244)
2Thisis the purposeofTheorem24(v).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-250-320.jpg)
![250 C.Villani
Here the infimum is taken over all random variables X, Y with respective law f and g.
It is always assumed that f and g have finite moments of order 2, which ensures that
W(f, g) < +oo.
In analytical terms, W can be rewritten as
W (f, g) : inf,
~f]~Nx~NIv - wl2dzr(v, w); Jr E/7(f, g) },
where/7 (f, g) stands for the set of probability distributions Jr on ]1~
N X ]1{N which admit
f and g as marginals, More explicitly, zr 6/7 (f, g) if and only if
c0(R • c0(R ),
I +f
The Wasserstein distance and its variants are also known under the names of Frrchet,
HOffding, Gini, Hutchinson, Tanaka, Monge or Kantorovich distances.
The infimum in (244) is finite as soon as f and g have finite second moments. Moreover,
it is well-known that convergence in Wasserstein sense is equivalent to the conjunction of
weak convergence in measure sense, and convergence of the second moment:
V(tg E C o (]~N ) , f f "~odVn__+
~>f f ~od. ,
>0 ~---~
w(fn' f)n----~cx~ " " f fn[ul2dun--+cxz>
f flul2du"
(245)
Tanaka'stheorem [415] states that whenever f, g are two probability measures with the
same mean, and b is normalized by (238), then
W(Q+(f, f), Q+(g, g)) <~w(f, g). (246)
Tanaka's representation (74) entered the proof of this inequality, which is formally similar
to a well-known inequality for rescaled convolution, W(f, f, g, g) <<.W(f, g). Thanks to
the Wild formula, Tanaka's theorem implies that W is a nonexpansive (in fact contractive)
metric, say when restricted to the set of zero-mean probability measures. On this subject,
besides Tanaka's papers one may consult [391].
As a main application, Tanaka proved theorems of convergence to equilibrium for
Equation (240) without resorting to the H theorem. In fact, convergence to equilibrium
follows almost for free from the contractivity property, since the distance to equilibrium,
W(f(t), M), has to be decreasing (unless f is stationary). So one can prove that
W(f(t),M) >0
l----~O0
as soon as f0 has just finite energy, not necessarily finite entropy.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-251-320.jpg)
![A reviewofmathematicaltopicsincollisionalkinetictheory 251
Since that time, Tanaka's theorem has been largely superseded: more convenient metrics
have been found, and entropy methods have become so elaborate as to be able to cover
cases where the entropy is infinite.3 Yet, Tanaka's theorem reminds us that Boltzmann's
H theorem is not the only possible explanation for convergence to equilibrium. The next
sections will confirm this.
2.2. Toscani's distance
Since Bobylev's work, it was known that the Fourier transform provides a powerful tool
for the study of the spatially homogeneous Boltzmann equation with Maxwellian collision
kernel. To measure discrepancies in Fourier space, Toscani introduced the distance
d2(f, g) -- sup . (247)
~C]~N I~12
The supremum in (247) is finite as soon as f and g have finite second moments, and the
same mean velocity:
Nf (V)Vdv - - fi[{Ng(V)V dr, fR (f + g)lvl 2dr < +ec.
N
Also, convergence in d2 sense is equivalent to convergence in Wasserstein sense (245). It
turns out [427] that, under the normalization (238),
d2(Q+(f, f), Q+(g, g)) ~ d2(f, g), (248)
with equality only if f, g are Maxwellian distributions. As a consequence, d2 is a
contracting probability metric along solutions of (239) (when one restricts to probability
measures with some given mean).
As shown by Toscani and the author [427], this contracting property remains true for
Equation (240) with a singular collision kernel. As a main consequence, the Cauchy
problem associated with (240) admits at most one solution. This uniqueness theorem holds
under optimal assumptions: it only requires finiteness of the energy and of the cross-section
for momentum transfer (63).
As other applications of the d2 distance, we mention
- a simple proof of weak convergence to equilibrium under an assumption of finite
energy only;
- some partial results for the non-existence of nontrivial eternal solutions [450,
Annex II, Appendix];
- some explicit estimates of rate of convergence in the central limit theorem [427], by
refinement of the inequality d2(f, f, M) ~<d2 (f, M).
3RecallSection5.4 in Chapter2C.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-252-320.jpg)
![_5_ C. Villani
2.3. Other Foltrier-based metrics
Other useful Fourier metrics are defined by
d~(g, f)- sup , s > 2. (249)
They are well-defined only when f, g have the same moments up to high enough order.
For instance, one cannot directly compare f to the associated Maxwellian distribution Mf
in distance d4 unless f fl)iVjVk dv -- f MfvivjVk dl) for all i, j, k. But this drawback is
easily fixed by subtracting from f a well-chosen Taylor polynomial.
The interest of using exponents s greater than 2 comes from the fact that the distances
ds become "more and more contracting" as s is increased, and this entails better properties
of decay to equilibrium. As soon as s > 2, one can prove [225] exponential decay to
equilibrium in distance ds, if the initial datum has a finite moment of order s. If one only
assumes that the initial datum has a finite moment of order 2, then the method also yields
exponential decay in some distance of the form
d~(L g) - sup
~RN ISe124~(~
e)
for some well-chosen function 4) with 4)(0) = 0.
By taking larger values of s, one improves the rate of convergence in ds metric; in
particular, the choice s = 4 yields the optimal rate of convergence [128], which is the
spectral gap of the linearized Boltzmann operator.4 This exponent 4 is related to the fact
that the linearized Boltzmann operator admits Hermite polynomials as eigenfunctions, and
the lowest eigenvalues are obtained for 4th-degree spherical Hermite polynomials.
Of course, this result of optimal convergence is obtained in quite a weak sense; but, by
interpolation, it also yields strong convergence in, say, L 1 sense if one has very strong
(uniform in time) smoothness and decay bounds at one's disposal. Such bounds were
established in [128], thanks to an inequality which can be seen as reminiscent of the
Povzner inequalities, but from the point of view of smoothness, i.e., with moments in
Fourier space, instead of velocity spaceS:
IIa+<f, f 112 1
H m ~ ~llfll2m -+-Cm, m E N. (250)
This inequality holds true at least when f is close enough to M f in relative entropy sense.
After establishing the optimal decay to equilibrium in d4 distance on one hand, and the
uniform smoothness bound in H m on the other hand, Carlen, Gabetta and Toscani [128]
4Recall that when the collision kernel is Maxwellian, then the spectrum of the linearized operator can be
computedexplicitly [79,p. 135].
5It is not rare in the theory of the spatiallyhomogeneousBoltzmannequation that smoothnessestimates and
decay estimates bear a formal resemblance, and this may be explainedby the fact that the Fourier transform of
the spatiallyhomogeneousBoltzmannequationis a kindof Boltzmannequation,see Equation (77) ....](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-253-320.jpg)
![A review of mathematical topics in collisional kinetic theory 253
had no difficulty in interpolating between both partial results to prove convergence to
equilibrium in L 1 at exponential rate. The interpolation can be made at the price of an
arbitarily small deterioration in the rate of convergence if m is very large. A precise
theorem will be given in Section 4.1.
REMARK. This theorem of exponential trend to equilibrium with explicit rate is at the
moment restricted to Maxwellian collision kernels. This is because only in this case are
nice contracting probability metrics known to exist. Also note that the spectral gap of the
linearized collision operator is known only for Maxwellian collision kernel. Accordingly, a
lower bound on the spectral gap is known only when the collision kernel is bounded below
by a Maxwellian collision kernel.
The preceding problem of trend to equilibrium takes its roots on the very influen-
cial 1965 work by McKean [341]. In this paper, he studied Kac's equation (21), and at
the same time proved exponential convergence to equilibrium, with rate about 0.016, sug-
gested the central limit theorem for Maxwellian molecules6 and established the decrease
of the Fisher information.7 The value 0.016 should be compared to the optimal rate 0.25,
which is obtained, up to an arbitrarily small error, in [128]. McKean's results have inspired
research in the area until very recently, as the rest of this chapter demonstrates. Another
related early work was Grtinbaum [262].
2.4. The central limit theoremfor Maxwell molecules
Once again, let us consider the Boltzmann equation with Maxwellian collision kernel, fix
an initial datum f0 with unit mass, zero mean and unit temperature, and denote by M
the corresponding Maxwellian. Recall from Section 1 that the solution to the Boltzmann
equation with initial datum f0 can be written as the sum of a Wild series, which is a convex
combination of iterated Q+ operators acting on f0.
Let us be interested in the behavior of the terms of the Wild series as t --+ +c~. It is
obvious that the terms of low order have less and less importance as t becomes large, and
in the limit the only terms which matter are those which take into account a large number
of collisions.
But the action of Q+ is to decrease the distance to equilibrium; for instance, one has the
inequality
d2(Q+(f, g), M) ~<max[d2(f, M),d2(g, M)],
as a variant of the inequalities discussed in Section 2.2. Therefore we can expect that terms
of high order in the Wild series will be very close to M, and this may be quantified into a
statement that f approaches equilibrium as time becomes large.
This however is not true for all terms of high order in the Wild series, but only for
those terms Q+ (f0) such that the corresponding tree y is deep enough (no leaves of small
6See next section.
7See Section 3.1.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-254-320.jpg)
![254 C.Villani
height). Intuitively, small depth means that all particles involved have collided sufficiently
many times. For instance, one would expect8
Ion, o+ io, ion), o+(o+ io, ZoO,o+ io, Ion))
to be rather close to M, but not
Ion, Zo), Io), Io), Io), Yo), Io)
(think that even Q+ (f0, M) is not very close to M ...). Thus, to make the argument work,
McKean [341] had to perform an exercise in combinatorics of trees, and show that the
combined weight of "deep enough" trees approaches 1 as time becomes large. These ideas
were implemented in a very clean, and more or less optimal way, by Carlen, Carvalho and
Gabetta [126], thanks to Fourier-defined probability metrics. A precise result will be given
in Section 4.1.
3. Information theory
3.1. The Fisherinformation
Among the most important objects in information theory are the Shannon entropy and
the Fisher information. Up to a change of sign, Shannon's entropy9 is nothing but the
Boltzmann H-functional. As for the Fisher information, it is defined as
I(f)-- fR IVfl2 fR
=4 ]V~[ 2 (251)
U f U
(compare with the relative Fisher information (174)). The Fisher information is always
well-defined in [0, +cx~], be it via the L2 square norm of the distribution V~/-f or by the
convexity of the function (x, y) w-~Ixl2/y. It is a convex, isotropic functional, lower semi-
continuous for weak and strong topologies in distribution sense.
Fisher [216] introduced this object as part of his theory of "efficient statistics". The
Fisher information measures the localization of a probability distribution function, in
the following sense. Let f(v) be a probability density on R N, and (Xn) a family of
independent, identically distributed random variables, with law f(.- 0), where 0 is
unknown and should be determined by observation. A statistic is a random variable
- 0 (X1 ..... Xn), which is intended to give a "best guess" of 0. In particular, 0 should
converge towards 0 with probability 1 as n --+ cx~;and also one often imposes (especially
when n is not so large) that 0 be unbiased, which means E0 = 0, independently of n. Now,
8Drawthe correspondingtrees!
9Seethe referencesin Section2.4 of Chapter2A.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-255-320.jpg)
![A review of mathematical topics in collisional kinetic theory 255
the Fisher information measures the best possible rate of convergence of 0 towards 0 in the
sense of mean quadratic error, as n --+ ~. More explicitly, if 0 is unbiased, then
N 2
Var(tg) ~> ~ . (252)
nI(f)
Inequality (252) is called the Cram6r-Rao inequality, but for an analyst it is essentially a
variant of the Heisenberg inequality, which is not surprising since a high Fisher information
denotes a function which is very much "localized" .... In fact, the standard Heisenberg
inequality in R N can be written
VI)0 E ~N,
(fo)
I (f) f(v)lv - vol2dv ~>N 2 N f dv
For given mass and energy, the Fisher information takes its minimum value for
Maxwellian distributions -just as the entropy. And for given covariance matrix, it takes
its minimum value for Gaussian distributions. This makes it plausible that the Fisher
information may be used in problems such as the long-time behavior of solutions to the
Boltzmann equation, or the central limit theorem.
The idea of an information-theoretical proof of the central limit theorem was first
implemented by Linnik [305] in a very confuse, but inspiring paper. His ideas were later
put in a clean perspective by Barron [59] and others, see the references in [156]. The
same paper by Linnik also inspired McKean [341] and led to the introduction of the Fisher
information in kinetic theory. 10
3.2. Staminequalitiesfor the Boltzmann operator
As one of the key remarks made by McKean [341 ], the Fisher information is a Lyapunov
functional for the Kac model (21). We already mentioned in Section 4.7 of Chapter 2C
that his argument can be adapted to the two-dimensional Boltzmann equation. Also
Toscani [421 ] gave a direct, different proof of this two-dimensional result.
A more general result goes via Stam-type inequalities. The famous Blachman-Stam and
Shannon-Stam inequalities 11 [411,75,125] admit as particular cases
I (f * f) ~<I (f), H (f * f) ~<H (f).
These inequalities are central in information theory [156,165]. Their counterparts for the
Boltzmann equation are
l(Q+(f, f)) <~I(f), H(Q+(f, f)) <~H(f), (253)
where Q+ is defined by (235), and the collision kernel has been normalized by (238).
10McKean used the denomination "Linnik functional" for the Fisher information, which is why this terminology
was in use in the kinetic community for some time.
11Inequalities (196) and (195), respectively.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-256-320.jpg)
![256 C. Villani
Inequalities (253) immediately entail (by Wild sum representation when (238) holds,
by approximation in the general case - or by an ad hoc application of the definition of
convexity) that the Fisher information and the H-functional are Lyapunov functionals
along the semigroup generated by the Boltzmann equation with Maxwellian collision
kernel. In particular, this gives a new proof of the H theorem in this very particular
situation. This remark is not so stupid as it may seem, because this proof of the H theorem
is robust under time-discretization, and also applies for an explicit Euler scheme le -
apparently this is the only situation in which the entropy can be shown to be nonincreasing
for an explicit Euler scheme.
Inequalities (253) were proven in dimension 2 by Bobylev and Toscani [83], and in
arbitrary dimension, but for constant collision kernel (in w-representation), by Carlen and
Carvalho [121]. Finally, the general case was proven by the author in [445]. Apart from
rather classical ingredients, the proof relied on a new representation formula for V Q+ in
the Maxwellian case:
V Q+ (f, f)
-- -~ dv, dab(k.a)(f~,(I + P~k)(VI)' + f'(I - P~k)(VI)~,), (254)
where k = (v - v,)/lv - v,I, IN" ]1~N -'-> ]I~N stands for the identity map and P~k" ]~N
R u is the linear mapping defined by
P~k(x) -- (k . a)x + (a . x)k - (k . x)a.
Formula (254) was obtained by an integration by parts on S u-1 , which crucially used
the assumption of Maxwellian collision kernel. In the non-Maxwellian case, we could
only obtain an inequality weaker by a factor 2: I(Q+(f, f)) <~ 211AIIL~I(f), where
A (z) "- fSN-1 B(Z, a) do'.
Just as in the well-known Stam proof, the first inequality in (253) implies the second
one via adjoint Ornstein-Uhlenbeck regularizafion. To be explicit, if (St)t>1ostands for the
semigroup associated to the Fokker-Planck equation, and if f has unit mass, zero mean,
unit temperature, then
H(Q+(f, f)) - H(M)= fo +~ [I ( Q+ (St f, St f)) - I (M)] dt.
Underlying this formula is of course the commutation between St and Q+, St Q+ (f, f) -
Q+ (St f, St f), which follows from Bobylev's lemma, 13 for instance.
REMARKS. (1) The decreasing property of the Fisher information also holds for solutions
of the Landau equation with Maxwell molecules. This can be seen by asymptotics
12Also one may dream a little bit and imagine that this remark could end up with new lower bounds for the
entropy dissipation!
13See Section 4.8 in Chapter 2A.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-257-320.jpg)
![A review of mathematical topics in collisional kinetic theory 257
of grazing collisions, but a direct proof is also possible, and gives much better
quantitative results [451 ]. In particular, one can prove that the Fisher information converges
exponentially fast to its equilibrium value.
(2) Still for the Landau equation, this decreasing property was compared with results
from numerical simulations by Buet and Cordier [105]. In many situations, they observed
a decreasing behavior even for non-Maxwell situations, e.g., Coulomb potential. They
also suggested that this decreasing phenomenon was associated with entropic properties
of the code: enforcing the decrease of the entropy in the numerical scheme would have a
stabilizing effect which prevents the Fisher information to fluctuate.
3.3. Consequence: decreasing of the Carlen-Carvalho !l*functional
In [121], Carlen and Carvalho introduced the function
:)~ ~ H(f) - H(Szf)
to measure the smoothness of the distribution function f, in the context of entropy-
entropy dissipation inequalities. Here (St)t>~o is as usual the adjoint Ornstein-Uhlenbeck
semigroup, i.e., the semigroup generated by the Fokker-Planck operator (161). Note that
if the moments of f are normalized by (177), then ~ is a nonnegative function. One of the
main points in the Carlen-Carvalho theorems 14 was to obtain a control of 7r for )~ close
to 0.
We now claim that for Maxwell collision kernel, thefunction ~ is pointwise nonincreas-
ing: for each value of )~, H (f) - H (Sz f) is nonincreasing as a function of time. To see this,
recall that in the case of Maxwellian collision kernel, the Boltzmann semigroup (Bt) com-
mutes with (St), as a consequence of Bobylev's lemma. 15 Therefore, the time-derivative
of n(f) - H(S~f) is
D(Sz f) - D(f).
To prove that D(Sz f) - D(f) is nonpositive, we just have to prove
d
--D(S~f) <~O.
d)~
But, 16 by commutation property, the dissipation along (St), of the dissipation of entropy
along (Bt), is also the dissipation along (Bt), of the dissipation of entropy along (St).
Hence,
d
d~.
dI
D(S~ f) -- -~ I (Bt f) <~O.
X=0 t=0
This proves the claim.
14Recall Section 4.4 in Chapter 2C.
15See Section 4.8 in Chapter 2A.
16We already made this remark in Section 4.4 of Chapter 2C.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-258-320.jpg)
![A review of mathematical topics in collisional kinetic theory 259
Assume that fo ~ L~+~ A H2+~(R N~ for some 6 > O. Then, there exist constants
C < +oo and ot > O, depending on f only via IIf0 IILI+~and IIf IIH2+~, such that
VNo~>I, Vt~>O, IIf (t) - gNo (t) IIt 1 ~ C
(1 --e-t) N~
NO Ol
Also, Ilf(t) - MIIc~ converges exponentially fast to O.
Part (i) of this theorem is from Toscani and Villani [427]. As for part (ii), the statement
about W2 is due to Tanaka [415], the one about d2 is in Toscani and Villani [427], the one
about I is from Villani [445]. Point (iii) is due to Tanaka for W, the same proof applies
for d2. Actually Tanaka's proof was given only under additional moment assumptions;
for the general result one needs tightness of the energy, which is proven in [225]. The
statement about the entropy is more delicate: in addition to the tightness of the energy, it
requires the monotonicity property of the function ~, as described in Section 3.3. These
two estimates make it possible to use the main result in Carlen and Carvalho [121] and
conclude that the relative entropy satisfies a closed differential equation which implies
its convergence to 0 at a computable rate. This argument is explicitly written in Carlen,
Carvalho and Wennberg [127] in the particular case when the collision kernel is constant
in co-representation.
Next, point (iv) is the main result of Carlen, Gabetta and Toscani [128], while point (v)
is the main result of Carlen, Carvalho and Gabetta [126]. Point (v) can be seen as a bound
(essentially optimal) on the error which is performed when replacing the solution of the
Boltzmann equation by a truncation of the Wild sum. Note that the result of exponential
convergence in (v) is much more general than the one in (iv), but the rate of convergence
is a priori worse.
We note that the proof of point (iv) uses the last part of (iii). Indeed, the convergence to
equilibrium is shown to be exponential only in a certain neighborhood of the equilibrium; 17
to make the constant C explicit it remains to estimate the time needed to enter such a
neighborhood, which is what entropy methods are able to do.
In conclusion, one can say that the theory of spatially homogeneous Maxwell molecules
is by now essentially complete. The links between information theory and kinetic theory
have been completely clarified in the last years, this being due in large part to the
contributions by Carlen and coworkers. Among the few questions still open, we mention
the classification of all nontrivial eternal solutions 18 - which certainly can be attacked
more efficiently in the Maxwellian case, thanks to the many additional tools available, as
demonstrated by the advances made by Bobylev and Cercignani [81] - and the problems
which are mentioned in the next two sections. Also, it would be extremely interesting to
know how point (iv) above generalizes to a spatially inhomogeneous setting, even from
the formal point of view, and even assuming on the solutions all the smoothness one can
dream of.
17Inparticular, because the bound (250) is only proven when f is close enough to M.
18See Section 2.9 in Chapter 2A.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-260-320.jpg)
![260 C. Villani
4.2. A remark on sub-additivity
An interesting problem is the classification of all Lyapunov functionals for the Boltzmann
equation. Recall the following result by McKean 19 [342]: for the Kac model, the entropy,
or H-functional is, up to addition of an affine function or multiplication by a constant, the
only Lyapunov functional of the form f A(f). McKean also conjectured that the Fisher
information would be the only Lyapunov functional of the form f A(f, Vf).
A related problem is to classify all functionals J, say convex and isotropic, which satisfy
J(Q+) <~J, in a way similar to (253), under the normalization (238). Such functionals
are particular Lyapunov functionals for the Boltzmann equation (240). In dimension 2 of
velocity space, Bobylev and Toscani [83] have obtained the following sufficient condition:
for all probability distributions f and g on R2, and for all )~e [0, 1],
J(f~ * gl-~.) ~<)~J(f) + (1 - )OJ(g). (255)
Whatever the dimension, this criterion is satisfied by all functionals that we have
encountered so far: H, I, d2(., M), W(., M) 2. However, nobody knows if it is sufficient in
dimension higher than 2.
As a consequence of our remarks on the Landau equation with Maxwellian collision
kernel [443] and the asymptotics of grazing collisions, any Lyapunov functional J has to
satisfy (255) in the particular case when f is radially symmetric and g is the Maxwellian
distribution with zero mean, and same energy as f.
4.3. Remark: McKean's conjectures
In his seminal 1965 work [341], McKean also formulated several conjectures. Even though
they all seem to be false, they have triggered interesting developments. Let us mention two
of these conjectures.
The "super-H theorem" postulates that the entropy is a completely monotone function
of time:
dH/dt <~O, (d2H)/(dt 2) >/0.... (-1)n(dnH)/(dt n) >/O.
For some time this was a popular subject among a certain group of physicists. This
conjecture is however false, as shown by Lieb [303] with a very simple argument.
Strangely, for the particular Bobylev-Krook-Wu explicit solutions, this "theorem" holds
true for n ~<101 and breaks downs afterwards [361].
The "McKean conjecture", strictly speaking. Let Ms stand for the Maxwellian
distribution with zero mean and 3 temperature, and consider the formal expansion
n(f , Ms) -- - Z n!
n=O
(256)
19This result is somewhat reminiscent of the axiomatic characterization of entropy by Shannon, see, for
instance, [156, pp. 42-43] and references therein.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-261-320.jpg)
![A review of mathematical topics in collisional kinetic theory 261
so that Io(f) -- -H(f), ll(f) -- I(f), I2(f) -- - ~ij f f[Oij(l~ 2, etc. Knowing
that dlo/dt >~0 and dI1/dt <<.O, McKean conjectured the more general inequality
(-1)ndln/dt >~O. This conjecture seems to be false in view of the formal study realized
by Ledoux [295] for the Fokker-Planck equation.
Keeping in mind that the entropy measures volume in infinite dimension, the successive
terms in (256) could be seen as infinite-dimensional analogues of the mixed volumes
arising in convex geometry. It is not even clear that they have alternate signs for n >~
1 .... However, this conjecture has inspired a few works in kinetic theory, see, for
instance, Gabetta [224], or the discussion of the Kac model in Toscani and Villani [428,
Section 7]. These ideas have also been used by Lions and Toscani [323] to establish certain
strengthened variants of the central limit theorem.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-262-320.jpg)
![Preface
The motion of fluids has intrigued scientists since antiquity and we may say that the
field of mathematical fluid dynamics originated more than two centuries ago. In 1755
Euler [2] gave a mathematical formulation of the principle of conservation of mass in
terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear
and angular momentum by PDEs. Material symmetry and frame invariance were used by
Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of
internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint-
Venant [11] and Stokes [12].
In the 19th century no sharp distinction was drawn between mathematicians and
physicists as we sometime see in more recent times. The formulation of the equations
of fluid motion could be considered as either mathematics or physics. The first work
in fluid dynamics that has a "modern" mathematical flavor may have been done by
Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous
problem. Riemann recognized the mathematical nature of the entropy. This notion led
him to his duality method for solving the non-characteristic Cauchy problem for linear
hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his
contemporaries. What we now call the Cauchy problem for a PDE and the search for its
solution did not have the significance that it is accorded nowadays. Even Poincar6 did not
raise that kind of question in his Th~orie des tourbillons.
For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly
accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the
Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28.
Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of
the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much
as Riemann, Leray developed new mathematical tools which proved to have independent
interest: e.g., weak solutions (that we now call Leray's solutions in this context) and
topological degree (a joint work with Schauder [4]).
Since the 1930s, the interest that mathematicians devote to fluid dynamics has
unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von
Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a
Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on
the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic
displays such a variety of models and questions that thousands of scientists, among them
many mathematicians, focus their research on fluid dynamics.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-264-320.jpg)
![A review of mathematical topics in collisional kinetic theory 265
The goal of this chapter is to present some of the main open problems in collisional kinetic
theory, then to discuss some of the new questions arising in two developing branches of
the field: the study of granular media on one hand, quantum kinetic theory on the other.
Other choices could have included semiconductors (whose modelling is very important for
industrial applications), modelling of biological interactions (in which problems have not
been very clearly identified up to now), the study of aerosols and sprays (which naturally
involve the coupling of kinetic equations with fluid mechanics), etc.
Also, we only discuss problems associated with the qualitative behavior of solutions, and
do not come back on less traditional issues like those which were presented in Section 2.9
of Chapter 2A.
Selecting "important" problems is always dangerous because of subjectivity of the
matter, and changes in mathematical trends and fashions. To illustrate this, let us quote
Kac himself [283, p. 178, footnote 5]: "Since the master equation 1 is truly descriptive of
the physical situation, and since existence and uniqueness of the solutions of the master
equation are almost trivial, the preoccupation with existence and uniqueness theorems
for the Boltzmann equation appears to be unjustified on grounds of physical interest and
importance."
1. Open problems in classical collisional kinetic theory
1.1. Strong solutions in a spatially inhomogeneous setting
The theory of the Cauchy problem for Boltzmann-like equations is by now fairly advanced
under the assumption of spatial homogeneity. For instance, in the case of hard potentials, it
seems reasonable to expect that this theory will soon be completed with the help of already
existing tools. On the other hand, essentially nothing is known concerning the general,
spatially inhomogeneous case in a non-perturbative context. Progress is badly needed on
the following issues:
- moment estimates,
- regularity estimates (propagation of regularity/singularity, regularization),
- strict positivity and lower bounds.
This lack of a priori estimates is a limiting factor in many branches of the field. A priori
estimates would enable one to
9 prove uniqueness of solutions, and energy conservation;
9 perform a simple treatment of boundary conditions (walls, etc.);
9 give estimates of speed of convergence to equilibrium along the lines presented in
Chapter 2C (for this one needs uniform estimates as times goes to +cx~);
9 justify the linearization procedure which is at the basis of so many practical
applications of kinetic theory, see, for instance, [148].
Such a priori estimates also would be very useful, even if not conclusive, to
9 prove the validity of the Landau approximation in plasma physics, viewed as a large-
time correction to the Vlasov-Poisson equation;
1SeeSection1.5below.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-265-320.jpg)
![266 C. Villani
9 prove the validity of the fluid approximation to the Boltzmann equation. For this,
local conservation laws seem to be the minimum one can ask for in order to prove the
hydrodynamic limit.2
Even in the more modest framework of solutions in the small, where smooth solutions
can often be built, many gaps remain, like the treatment of singular collision kernels, or the
derivation of uniform smoothness estimates as t --+ +o0.
1.2. Derivation issues
As already mentioned, Lanford's theorem is limited to a perturbative framework (small
solutions), and concerns only the hard-sphere interaction. The treatment of more general
interactions is almost completely open, and would be of considerable interest. As an
oustanding problem in the field is of course the formidable task of rigorously deriving
collisional kinetic equations for Coulomb interaction.
As for the problem of extending Lanford's theorem to a nonperturbative setting, one
of the main difficulties is certainly related to the fact that there is no good theory for the
Cauchy problem in the large - but we expect much, much more obstacles to overcome
here!
1.3. Role of the kinetic singularity
Let us consider a Boltzmann collision kernel, say of the form
B(v - v,, tr) = t/,(lv - v,[) b(cosO).
In Chapter 2B we have seen how the properties of the Boltzmann equation depend on
whether b is integrable or singular. On the other hand, what remains unclear even in the
spatially homogeneous case, is the influence of the kinetic collision kernel q~. When
is singular, does it induce blow-up effects, and in which sense? Does it help or harm
regularizing effects induced by an angular singularity?
The most important motivation for this problem comes from the modelling of Coulomb
collisions in plasma physics: the collision kernel given by the Rutherford formula presents
a singularity like Iv - v.1-3 in dimension N = 3. Here are two questions which arise
naturally.
(1) Consider the Boltzmann equation with truncated Rutherford collision kernel, of the
form Iv- v,l-3b(cosO)10~>~, which is physically unrealistic but used in certain modelling
papers [162]. This collision kernel presents a nonintegrable (borderline) kinetic singularity.
Does it entail that the equation induces smoothness, or even just compactifying effects?
Some formal arguments given in Alexandre and Villani [12] may support a positive answer,
but the situation seems very intricate. As explained in [12, Section 5], the geometry of the
2Thisis whatthe authorbelieved,untilveryrecentlysomeproofsofhydrodynamiclimitappeared[54,53,240,
245], coveringsituationsin whichlocalconservationlawsare not knownto holdfor fixedKnudsennumber,but
are asymptoticallyrecoveredin the limitwhenthe Knudsennumbergoesto 0!!](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-266-320.jpg)
![A review of mathematical topics in collisional kinetic theory 267
problem is "dual", in some sense, to the one which appears in the study of nonintegrable,
borderline angular singularities .... If the answer is negative, this suggests that such
collision kernels should be used with a lot of care!
(2) On the other hand, consider the Landau approximation for Coulomb collisions, or
Landau-Coulomb equation, which is more realistic from the physical point of view. Does
this equation have smooth solutions?
In the spatially homogeneous situation, the Landau-Coulomb equation can be rewritten
as
nrv2j -+- 87rf 2 t >~O, V6 R 3
Of
Ot -- ~ {lij OyiOy--~" ' '
ij
(257)
where
1 I l)iVJ1
{tij --~ -~[ ~ij ivl2 * f.
If f is smooth, then the matrix ({lij) is locally positive definite, but bounded, and (257) is
reminiscent of the nonlinear heat equation
0f
= Af + f2 (258)
at
which has been the object of a lot of studies [468] and generically blows up in finite time,
say in L~ norm. The common view about (258) is that the diffusive effects of the Laplace
operator are too weak to compensate for blow-up effects induced by the quadratic source
term. And if the diffusion matrix ({lij) is bounded, this suggests that (257) is no more
diffusive than (258).
Weak solutions to (257) have been built in Villani [446]; they satisfy the a priori estimate
v/-f 6 L2 (nvl) locally. This estimate is however, to the best of the knowledge of the author,
compatible with known a priori estimates for (258).
These considerations may suggest that blow-up in finite time may occur for solutions
of (257). If there is blow-up, then other questions will arise: how good is the Landau
approximation at a blow-up time? What happens to blow-up if the physical scales are such
that the Landau effects should only be felt as t --+ +cx~?
However, blow-up has never been reported by numerical analysts. And after seeing
some numerical simulations by E Filbet, the author has changed his mind on the subject,
to become convinced that blow-up should indeed not occur. All this calls for a wide
clarification.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-267-320.jpg)
![268 C. Villani
1.4. Improved entropy-entropy dissipation estimates
In this section we only consider very nice distribution functions, say smooth and rapidly
decaying, bounded below by a fixed Maxwellian. We saw in Chapter 2C that such
probability distributions satisfy entropy-entropy dissipation inequalities of the form
D(f) >~K H (flMf) ~,
where M f is the Maxwellian equilibrium associated with f, K and ot are positive
constants, H is the relative entropy functional, and D is the entropy dissipation for either
Boltzmann or Landau's equation. In several places do our results call for improvement:
Landau equation with hardpotentials. In the case of the Landau equation,
ot = 1 is admissible when qJ (Izl) ~ Izl2"
ot -- 1 + e (e arbitrarily small) is admissible when qJ(Izl) ~ Izl2+y, y < 0.
It is natural to conjecture that also c~= 1 be admissible for hard potentials (y > 0). More
generally, this should be true when !P(Izl) = Izl2~p(Izl) with 7~ continuous and uniformly
positive for Izl/> ~ > 0. This conjecture is backed by the spectral analysis of the linearized
Landau operator [161], and also by the similar situation appearing in Carrillo, McCann
and Villani [130] in the study of entropy-entropy dissipation inequalities for variants of
granular media models. At the moment, the best available exponent for hard potentials is
ct = 1 -+-2/y, from Desvillettes and Villani [183].
Boltzmann equation with hard potentials. In the case of the Boltzmann equation, c~ =
1 + e is admissible for Maxwellian or soft potentials. It is accordingly natural to think
that c~-- 1 + e is also admissible3 for hard potentials. Recall that Cercignani's conjecture
(or - 1) is false in most cases according to Bobylev and Cercignani [87].
Cercignani's conjecture revisited? Counterexamples in [87] leave room for Cercignani's
conjecture to hold true in two situations of interest:
9 when the collision kernel is noncut-off and presents an angular singularity. This would
be plausible since grazing collisions behave better with respect to large velocities, as
the example of the Landau equation demonstrates4;
9 when f ~ LP ((M f)- 1) for some p ~> 1. Of special interest are the cases p = 1
(cf. Bobylev's estimate for hard spheres, in Theorem 1(ii)); p = 2 (natural space for
linearization) and p --- ~ (when f/M f is bounded from above). Maybe a Maxwellian
bound from below is also needed for proving such theorems.
As we mentioned when discussing Cercignani's conjecture in Chapter 2C, about this
topic one also has to make the connection with the recent Ball and Barthe result about the
central limit theorem.
3As this review goes to print, the author just managed to prove precisely this result, under the assumption that
the density be bounded in all Sobolev spaces.
4Similar results in the theory of linear Markov processes would also be interesting.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-268-320.jpg)
![A review of mathematical topics in collisional kinetic" theory 269
1.5. Approach to equilibrium for Kac's master equation
A related topic is the Kac spectral gap problem and its entropy dissipation variant. This
subject is a little bit in digression with respect to those which we discussed so far, but we
wish to explain it briefly because of its intimate (and not well-known) connections with
Cercignani's conjecture. These connections were brought to our attention by E. Carlen.
In his famous paper [283], Kac introduced a stochastic model which he believed to
be a way of understanding the spatially homogeneous Boltzmann equation. His equation
models the behavior of n particles interacting through binary elastic collisions occurring at
random Poissonnian times, with collision parameter cr randomly chosen on the sphere. It
reads
Ot
1
dcr B(vi- 1)j, 0")[,/4/o.
j fn -- fn], (259)
where the summation runs over all pairs of distinct indices (i, j) in {1..... n}, and fn is
a symmetric probability distribution on the manifold (actually a sphere) of codimension
N + 1 in (RN) n defined by the relations
/7 /7
Z Ivi12 -- 2nE > O, ~ IJi -- ng E R N.
i=1 i=1
We use the notation f for the normalized integral on the sphere, ISN-1 [-1 f. Moreover the
linear operator ,A~ represents the result of the collision of the spheres with indices i and j,
ioJ
! !
.fit fn(Vl ..... Vn)- f (Vl ..... Vi..... Vj ..... Vn),
I Ui -'[- l)j Iui -- vjl
vi= 2 -+-~cr,
t l)i + Vj IIJi -- Uj[
vj -- 2 ~ c r .
As explained by Kac, the spatially homogeneous Boltzmann equation can be recovered,
at least formally, as the equation governing the evolution of the one-particle marginal of fn
in the limit n --+ +oo. In this limit, time has to be sped up by a factor n. See [283,412,256]
for a study of this and related subjects.
A simplified version, which is commonly called Kac's master equation, is given by
Ot -
- Lnfn -- -~ o dO [fn o R~ - fn], (260)
where fn is a probability distribution on the sphere in ]t{ n , defined by
ZvZ=2nE. (261)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-269-320.jpg)
![270 C. Villani
Moreover,
Rb v--(v, ..... ..... v,),
where
!
(v~, vj) = RO(1)i, Vj)
is obtained from (1)i, Vj) by a rotation of angle 0 in the (i, j) plane. Without loss of
generality, we set E = 1/2 in (261), so that the sphere has radius ~/-ff.With this choice, the
image measure of the uniform probability measure on the sphere, under projection onto
some axis of coordinate, becomes the standard Gaussian measure as n --+ c~ (Poincarr's
lemma, actually due to Maxwell). Moreover, we shall use the uniform probability measure
on the sphere as reference measure, so that probability distributions are normalized by
f fnda- 1.
~/-~Sn-1
Among the problems discussed by Kac is that of establishing an asymptotically sharp
lower bound on the spectral gap )~n of Ln as n --+ +cx~. Recently, Diaconis and Saloff-
Coste [189] proved )~n1 = O(n3), then Janvresse [281] proved Kac's conjecture that
)~n1 = O(n); she used Yau's so-called martingale method. Finally, a complete solution
was given very recently by Carlen, Carvalho and Loss [123], who managed to compute
the spectral gap by a quite unexpected method (also by induction on the dimension). This
work also extends to Equation (259) if the collision kernel B is Maxwellian.
Since time should be sped up by a factor n in the limit n ~ ~, the corresponding
evolution equation will satisfy estimates like
11A(t,.)- IIfn(0, .)- lllt2(4~sn-1 ) (262)
for some )~ > 0, which can be chosen uniform as n --+ cx~according to Janvresse's theorem.
Here 1 is the equilibrium state, i.e., the density of the uniform probability measure on the
sphere v/-nSn-1 . Inequality (262) conveys a feeling of uniform trend to equilibrium as
n --+ cx~,which was Kac's goal.
However, it is not very clear in which sense (262) is a uniform estimate. Since all
the functions fn'S are defined on different spaces, one should be careful in comparing
them. In particular, think that if fn satisfies the chaos property, then Ilfn IlL2 is roughly
of order C n for some constant C > 0 (which in general is not related to the L 2 (or
L2(M-I)) norm of the limit one-particle marginal f, see [283, Equation (6.44)]). And
Ilfn - 11122= Ilfn 1122- 1 is also of order C~. Having this in mind, it would be natural to
1In
compare distances in dimension n by the quantity II 9 IIL2 9But if we do so, we find
1/n ~ 1/n
, LZ(sn-1),](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-270-320.jpg)
![A review of mathematical topics in collisional kinetic theory 271
which does not behave well in the limits !
A way to circumvent the difficulty would be to compare all first marginals, which all
live in L 1(R), and prove that under some precise conditions on the sequence (fn),
ElX > O, Vn ~ 1, IIP, A(t, .)- MIIL= M_,> Ce-xt.
Now, a problem which looks more natural and more interesting in this context is the
problem of the entropy-entropy dissipation estimate for Kac's master equation. Again,
we state this problem assuming without loss of generality that E -- 1/2, so that ~ v2 -- n
in (261) and we use the uniform probability measure as reference measure for the definition
of the entropy:
H(fn) - f s.-1 f nlog f ndo'.
Note that H (1) -- 0.
PROBLEM. Find Kn optimal such that for all symmetric probability distribution fn on
~/-fiS~-l,
logfn ~> KnH(fn). (263)
If Kn 1 = O(n), then (263) entails the following entropy estimate for solutions of the
Kac equation:
H(fn(t, ")) - H(1) ~<e-Ut[H(fn(O, .))- H(1)],
for some # > 0. Since H (fn) typically is O(n), this would lead to the satisfactory estimate
H(fn(t"))- H(1) <~e-ut I H(fn(O"))- H(1)
I n n ' (264)
and also one-particle marginals of all fn'S could be compared easily as a consequence an
adequate chaos assumption for fn (0, .).
But from the counterexamples due to Bobylev and Cercignani [87], one expects
that Kn 1 --O(n) is impossible. Indeed, by passing to the limit as n --+ cx) in (263),
under a chaos assumption, one would have a proof of Cercignani's entropy dissipation
conjecture for Kac's model, which should be false (although this has never been checked
explicitly) .... On the other hand, the author [450, Annex III, Appendix B] was able to
prove Kn 1 -- O(n 2) by the same method as in Section 4.6 of Chapter 2C. This leads to two
open questions:
9 What is the optimal estimate?
5Or shouldthe relevantscalingbe fn ~ (1+h/V/if)| whichwouldmeanthatwe are interestedin fluctuations
of the equilibriumstate?This ansatzformallyleads to Ilfn- 11122= O(n) ....](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-271-320.jpg)
![A review of mathematical topics in collisional kinetic theory 273
the subject has grown so fast that some journals are now entirely devoted to it! And also the
number of involved physicists has become extremely large. Among the main motivations
are the understanding of how granular material behaves under shaking, how flows are
evolving or how to prevent them, how to facilitate mixing, how to prevent violent blow-up
of a silo, or avalanches, how matter aggregates in a newborn solar system, etc.
One popular model for these studies is a kinetic description of a system of particles
interacting like hard spheres, but with some energy loss due to friction. Friction is
a universal feature of granular material because of the roughness of the surface of
particles. Many studies have been based on variants of the Boltzmann or Enskog equations
which allow energy loss. This subject leads to huge difficulties in the modelling; see,
for instance, the nice review done by Cercignani [146] a few years ago. Also some
mathematical contributions have started to develop, most notably the works by Pulvirenti
and collaborators [70-72,68,69,395]. Here we point out some of the most fundamental
mathematical issues in the field. Since the physical literature is considerable, we only
give a very restricted choice of physicists' contributions. Thanks are due to E. Caglioti
for explaining us a lot about the subject and providing references.
2.1. Derivation issues: problems of separation of scales
The separation between microscopic and macroscopic scales in the study of granular media
is not at all so clear as in the classical situation, and this results in many problems when it
comes to derivation of the relevant equations.
To illustrate this, we mention the astonishing numerical experiments described in [237]:
a gas of inelastic particles is enclosed in a one-dimensional box with specular reflection
(elastic wall) on one end, Maxwellian re-emission (heating wall) on the other. It is found
that, basically, just one particle keeps all the energy, while all the others remain slow and
stay close to the elastic wall. In other words, the wall is unable to heat the gas, and moreover
it is plainly impossible to define meaningful macroscopic quantities!
Enskog equation. At the basis of the derivation of the Boltzmann equation, be it formal
or more rigorous, is the localization of collisions: the length scale for interaction is much
smaller than the length scale for spatial fluctuations of density. The Boltzmann-Grad limit
n --+ cx~, r --+ O, nr 2 --+ 1 (n = number of particles, r = radius of particles, dimension = 3)
is a way to formalize this for a gas of elastic hard spheres.
On the other hand, in the case of granular media, the size of particles is generally
not negligible in front of the typical spatial length. This is why many researchers use an
Enskog-like equation, with delocalized collisions: for instance,
~f
Ot
+v. Vxf=r2f•3 dv*fs2 dcr Iv - v*l
X [J G(x,x + rcr; p)f(t,x, f))f(t,x - rcr, fJ,)
- G(x,x - rcr; p)f(t,x, v)f(t,x - rcr, v,)]. (265)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-273-320.jpg)
![274 C. Villani
(See Cercignani [146] and Bobylev et al. [86].) Here r > 0 is the radius of particles, v, v,
are post-collisional velocities and ~, fi, are pre-collisional velocities, given by the formulas
_ v+ v, 1 -e 1 +e
v .... ~2-- --~-e (v- v,) + 4e Iv- v, la,
_ v+v, 1-e(v_v,)_ l+e
v,= ~ ~- 4e 4e Iv-v, la.
(266)
Moreover e is an inelasticity parameter: when e- 1 the preceding equations are the
usual equations of elastic collisions, while the case e = 0 correspond to sticky particles.
In general e may depend on Iv - v,I but we shall take it constant to simplify. Finally, to
complete the explanation of (265), J is the Jacobian associated to the transformation (266),
1 Iv- o,I
j = _ ~ (267)
e2 Iv- v,l'
and G is the famous but rather mysterious correlationfunction which appears in the Enskog
equation. Roughly speaking, G relates the 2-particle probability density with the 1-particle
probability density, as follows:
f2(t, x, v, y, w) = G(x, y; p(t, x), p(t, y)) f (t, x, v) f (t, y, w),
where p(t, x) -- f f dr. In the Boltzmann case this term did not appear because of the
chaos assumption .... Exactly which function G should be used is not clear and a little bit
controversial: see Cercignani [146] for a discussion.
If the reader finds the complexity of (265) rather frightening, we should add that we
did not take into account variables of internal rotation, which may possibly be important
in some situations since the particles are not perfectly spherical; see [146] for the
corresponding modifications.
There are no clear justifications for Equation (265), even at a formal level. In the limit of
rarefied gases with a chaos assumption, or in the spatially homogeneous case, one formally
recovers an inelastic Boltzmann equation, which is more simple: the collision operator just
reads, with obvious notations,
s f, --
Qe(f, f) - dr, dcr Iv - v,l(f f, - f f,).
3 2
(268)
This collision operator also has a nice weak formulation,
f Qe(f, f)q)= s ff,[q)'- q)]dvdv, dcr, (269)
3 x ]]~3x S 2](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-274-320.jpg)
![A review of mathematical topics in collisional kinetic theory 275
(with respect to v, v, taken as pre-collisional
where the post-collisional velocities v~, v,
velocities) are
V + 1 l+e
v' -- v, - e(v _ v,) + Iv - v, la,
------2--+ 4 4
v + v, 1- e.v v,) - l + e
v,- 2 ------4-- 4 Iv- v, lcr.
(270)
As in the elastic case, formula (269) can be symmetrized once more by exchange of v and
v,. Note that fi, fi, do not coincide with vf, v,f because collisions are not reversible. Also
note that
f Qe(f, f) dv - O, f Qe (f, f) v dv - O,
but
f Qe(f, f)lvl 2dr ~<0.
Now let us enumerate some more models. Starting from (268), many variants can be
obtained by
9 reducing the dimension of phase space by considering only 2-dimensional, or even
1-dimensional models;
9 adding a little bit of diffusion, which is presumably realistic in most situations (heat
bath, shaking...);
9 adding some drift term 0Vv. (f v), to model some linear friction acting on the system;
9 change the "hard-sphere-like" collision kernel Iv - v,I for Iv - v,I • with, say -1 ~<
y ~< 1. This provides an equation with some inelastic features, and which may be
easier to study .... The dimensional homogeneity of the equation can be preserved by
multiplying the collision operator by a suitable power of the temperature. For instance,
Bobylev, Carrillo and Gamba [86] have performed a very detailed study of the case
y = 0 ("pseudo-Maxwellian collision kernel") along the general lines of the theory
developed by Bobylev for elastic Maxwellian collisions;
9 only retain grazing collisions by an asymptotic procedure similar to the Landau
approximation7 [424,426]. Actually, some physicists mention that grazing collisions
do occur very frequently in granular material [236] .... The operator which pops out
of this limit procedure is of the form
QL(f, f) + Vv. [fVv(f * U)],
where U(z) is proportional to Izl3 (or more generally to Izly+2), and QL is an
elastic Landau operator with ~(Izl) proportional to Izl3 (or Izl• In particular,
7 See Section2.7 in Chapter2A.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-275-320.jpg)
![276 C. Villani
in dimension 1 of phase space, this elastic Landau operator vanishes and the resulting
collision operator is just a nonlinear friction operator
Vv . [fVv(f , W)].
The resulting evolution equation is
Of
0---[+ v . Vx f = Vv . [fVv(f * W)], (271)
with U (z) = Izl3. The very same equation can also be obtained as the mean-field limit
of a one-dimensional system of particles colliding inelastically, as first suggested (so it
seems) by McNamara and Young [343], and rigorously proven by Benedetto, Caglioti
and Pulvirenti [70]. Note that (271) is a Boltzmann-like, not an Enskog-like equation!
Of course, we have only presented some of the most mathematically oriented models. In
the physical literature one encounters dozens of other equations, derived from physical or
phenomenological principles, which we do not try to review.
Hydrodynamics. Another topic where the separation of scales is problematic is the
hydrodynamic limit. This subject is important for practical applications, but nobody
really knows, even at a formal level, what hydrodynamic equations should be used.
Due to the possibility of long-range correlations, there generally seems to be no clear
separation of scales between the kinetic and hydrodynamic regimes [236]. The Chapman-
Enskog expansion works terribly bad: each term of the series should be of order 1!
Resummation methods have been tried to get some meaningful fluid equations [236]. From
the mathematical viewpoint this procedure is rather esoteric, which suggests to look for
alternative methods ....
To add to the confusion, due to the tendency of granular media to cluster, it is not clear
what should be considered as local equilibrium! See the discussion about Homogeneous
Cooling States in Section 2.4 below.
Also, to be honest we should add that even in the most favorable situation, i.e., a
simplified model like (271), with some additional diffusion term to prevent clustering
as much as possible, and under the assumption of separation of scales, then limit
hydrodynamic equations can be written formally [69], but rigorous justification is also
an open problem, mainly due to the absence of Lyapunov functional ....
2.2. Spatial inhomogeneities
In the classical, elastic theory of the Boltzmann equation, spatial homogeneity is a
mathematical ad hoc assumption. However, it is not so unrealistic from the physical
viewpoint in the sense that it is supposed to be a stable property: weakly inhomogeneous
initial data should lead to weakly inhomogeneous solutions (recall [32]).
On the other hand, in the case of granular media, some physicists think that severe
inhomogeneities may develop from weakly inhomogeneous states, particularly because](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-276-320.jpg)
![A review of mathematical topics in collisional kinetic theory 277
of the possibility of collapse by loss of energy [198]. Also, numerical experiments seem
to indicate the unstability of a homogeneous description (see the references and comments
in [70, Section 4]). Besides shedding more doubts on mathematical studies based on the
assumption of spatial homogeneity, these remarks raise a very interesting mathematical
challenge, namely prove that weak inhomogeneity may break down in finite time for some
realistic initial configurations. It is not very clear whether this study should be performed
with a Boltzmann-like, or an Enskog-like equation ....
Some related considerations about blow-up: first of all, there are initial configurations of
n inelastic particles which lead to collapse in finite time, and they are not exceptional (some
nice examples are due to Benedetto and Caglioti [67]). Next, for the inelastic Boltzmann
equation, there is no entropy functional to prevent dramatic collapse, as in the elastic case.
Even the DiPema-Lions theory cannot be adapted to inelastic Boltzmann equations, so the
Cauchy problem in the large is completely open in this case!
We further note that Benedetto, Caglioti and Pulvirenti prove that there is no blow-up
for Equation (271) when W(z) = )~lzl3 with )~ very small; in this case Equation (271) can
be treated as a perturbation of vacuum. Of course it is precisely when )~ is of order 1
that one could expect blow-up effects. Also, Benedetto and Pulvirenti [73] study the one-
dimensional Boltzmann equation for a gas of inelastic particles with a velocity-dependent
inelasticity parameter e = e(lv - v.I) behaving like (1 + air - v.l• -1 for some a, y > 0,
and they show, by an adaptation of the one-dimensional techniques of Bony [95], that
blow-up does not occur in that situation.
2.3. Trend to equilibrium
For the moment, and in spite of our remarks in the previous paragraph, we restrict to
the spatially homogeneous setting (or say that we are only interested in trend to local
equilibrium, in some loose sense). We consider two cases:
(1) the inelastic Boltzmann collision operator alone; then, because of energy loss, the
equilibrium is a Dirac mass at some mean velocity;
(2) the inelastic Boltzmann collision operator together with some diffusion. Then there
is a nontrivial, smooth stationary state (see Gamba, Panferov and Villani [228]). It is not
explicit but some qualitative features can be studied about smoothness, tails, etc. The
same topic has also been recently studied by Carrillo and Illner in the case of the pseudo-
Maxwellian collision kernel.
In both cases entropy methods do not seem to apply because no relevant Lyapunov
functional has been identified (apart from the energy). What is worse, in the second
case, even uniqueness of the stationary state is an open problem. The study of trend to
equilibrium is therefore open.
However, such a study was successfully performed for two simplified models:
- in the pseudo-Maxwellian variant of the inelastic Boltzmann operator, with or without
diffusion [86,80]. Then the behavior of all moments can be computed, and trend to
equilibrium can be studied from the relaxation of moments;](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-277-320.jpg)
![278 C. Villani
- in the simplified model considered by Benedetto, Caglioti and Pulvirenti [70], or its
diffusive variant. In this case, the appearance of a new Lyapunov functional, with an
interaction energy
1 [ Iv - wl 3
- JR f (v)f (w)
2 • 3
dvdw
enables a study of rate of convergence by entropy dissipation methods. 8
In the spatially inhomogeneous case, the situation is still worse. In the non-diffusive
case, all states of the form p~(x)6o(v), for instance, are global equilibria. But are there
some preferred profiles p~ ? Even for the simplified Equation (271) or its diffusive variants,
the Lyapunov functional which worked fine in the spatially homogeneous context, now
fails to have any particular behavior. At a more technical level, the method of Desvillettes
and Villani [184] cannot be applied because of the non-smoothness of the equilibrium
distribution. On the whole, trend to equilibrium for granular media is a really challenging
problem.
2.4. Homogeneous Cooling States
For most physicists, a Dirac mass is not a relevant steady state, and the role played in the
classical theory by Maxwellian distributions should rather be held by particular solutions
which "attract" all other solutions. 9 They often agree to look for these particular solutions
in the self-similar form
1 (v-vo) Ru
fs(t, v) = ~ N (t----~
F or(t) ' t )O, v 9 . (272)
In the sequel, we set v0 = 0, i.e., restrict the discussion to centered probability distributions.
A solution of the spatially homogeneous inelastic Boltzmann equation which takes the
form (272) is called a Homogeneous Cooling State (HCS). Though the existence of HCS
is often taken for granted, it is in general a considerable act of faith. Sometimes HCS
are considered under some scaling where also the elasticity coefficient e goes to 1 as
t --+ +cx~ .... For physical studies of these questions one can consult [104,238].
For the pseudo-Maxwellian variant of the inelastic Boltzmann operator, Bobylev,
Carrillo and Gamba [86] have shown that HCS do not exist: one can construct a self-similar
distribution function which captures the behavior of all moments of all solutions, but this
distribution function is not nonnegativeI HCS exist only in the following weakened sense:
for any integer no, there exists a self-similar solution of the inelastic Boltzmann equation
which gives the fight behavior for all moments of order ~<no of all solutions. And yet, this
8See the discussionin Section6.3 of Chapter 2C.
9By the way, in the classical setting it was once conjectured that the Bobylev-Krook-Wu explicit solutions
would attract all solutions of the spatially homogeneousBoltzmann equation with Maxwellian molecules. But
this has been shownto be false ... except,in somesense,in the unphysicalregimeof negativetimes [81]!](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-278-320.jpg)
![A review of mathematical topics in collisional kinetic theory 279
weaker statement is still false for some particular values of the inelasticity parameter e.
See [86] for more details, in particular a discussion of the stability of the HCS description.
For model (271), in a spatially homogeneous setting, i.e.,
~f = Vv. [fVv(f 9 U)], (273)
Ot
with U(z) -Izl3/3, then HCS do exist and are just made of a combination of two Dirac
masses: up to some change of scales,
113 -+-3~].
f s(t, v)- -~ ~, z,
This solution is obtained via the search for steady states to the rescaled equation
~f
= Vv. (fVv(f * W)) - Vv. (fv). (274)
Ot
One finds that a distinguished steady state is Fs = (3 1 + 31)/2.
2 2
It is also true [70] that this HCS is a better approximation to solutions of (273), than just
the Dirac mass (at least if the initial datum has no singular part). This means that solutions
of (274) do converge towards the steady state Fs. But this approximation is in general
quite bad! It was shown by Caglioti and Villani [116] that the improvement in the rate of
convergence is essentially no better than logarithmic in time. For instance, if W stands for
the Wasserstein distance (244), then solutions of (273) satisfy
f+~ W(f(t), fs(t))dt = +cx~.
Since also W(f(t),3o)= O(1/t), this shows that the improvement in the rate of
convergence cannot be O(log l+e t), for any e > 0 - which means negligible by usual
standards, l~
The preceding considerations cast further doubts about the mathematical relevance of
HCS, which however are at the basis of several hydrodynamical equations for granular
media. Further clarification is still badly needed; an attempt is done in [86] for the
simplified pseudo-Maxwellian model.
This concludes our brief discussion of the kinetic theory of granular media. In the sequel,
we shall enter a completely different physical world.
3. Quantum kinetic theory
Very recently, Lu [328], Escobedo, and Mischler [210,211] have begun to apply the
techniques of the modem theory of the spatially homogeneous Boltzmann equation, to
l~ maybeshouldoneuse anevenweakernotionofdistanceto measurethe rateofconvergence??](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-279-320.jpg)
![280 C. Villani
quantum kinetic models, thus opening up the path to a promising new direction of research.
This is part of a general trend which has become increasingly active over last years: the
mathematical derivation and study of quantum statistical models. This also coincides with
a time when the interest of physicists in Bose condensation is enhanced by the possibility
of experiments with very cold atoms. Most of the explanations which follow come from
discussions with Escobedo and Mischler, and also from Lu's paper [328].
First of all, we should clarify the meaning of quantum kinetic theory: it does not rest on
the traditional quantum formalism (wave function, Wigner transforms, etc.). Instead, it is
rather a classical description of interacting particles with quantum features. This approach
was initiated by the physicists Nordheim, and Uehling and Uhlenbeck, in the thirties.
Thus, the basic equation still looks just like a Boltzmann equation:
~f
Ot
+ v(p). Vxf- Q(f, f), t ~>0, x elR 3 pER 3 (275)
Here p stands for the impulsion of the particle and v is the corresponding velocity: v(p) =
Vp E (p), where E (p) is the energy corresponding to the impulsion p. The unknown is a
time-dependent probability density on the phase space of positions and impulsions. When
dealing with massive particles, we shall consider a non-relativistic setting (to simplify) and
identify v with p. On the other hand, when dealing with photons, which have no mass, we
assume that the energy is proportional to IPl, and the velocity to p/lpl.
Now, all the quantum features are encoded at the level of the collision operator Q
in (275). One traditionally considers three types of particles:
, fermions, which satisfy Pauli's exclusion principle. In this case the transition from
state p~ to state p is easier if f (p) is low. Accordingly, the "Boltzmann-Fermi" collision
operator reads
QF(f, f)- fR3 dp, fs2 do- B(v-v,, o-)[f'fJ(1 + ef)(1 +ef,)
- ff,(1 + sf')(1 + el,)], (276)
where e is a negative constant, ll Up to change of units, we shall assume that e = -1.
Moreover, as in the classical case, ft= f(pl) and so on, and pl, p~ are given by the
formulas
, P+P. IP- P,]
p = + ~ o . ,
2 2
I P+P. IP- P.I
P*= 2 2 o."
(277)
llln physicalunits,e shouldbe -(h/m)3/g, whereh is Planck'sconstant,m themassofaparticleand g the so-
called"statisticalweight"ofthisspeciesofparticles.Forthederivationof (276)see Chapmanand Cowling[154,
Chapter 17].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-280-320.jpg)
![A review of mathematical topics in collisional kinetic theory 281
Equation (275) with Q = Q F will be called the Boltzmann-Fermi equation. It is
supplemented with the a priori bound
O~f~l [=-l/el;
, bosons, which, on the contrary to fermions, do like to cluster. The collision operator,
Q8, is just the same as (276), but now e = + 1. The corresponding equation will be called
the Boltzmann-Bose equation;
, photons, which are mass-free particles exchanging energy. Usually they are considered
only in interaction with bosons or fermions. For instance, here is the Boltzmann-Compton
model:
Qc(f, f) = fo~176 k2 + f) e-~ - f( k'2 + g')e-~:'] dk',
t~O,k~O. (278)
Here the phase space is just R+, the space of energies, because the distribution of photons
is assumed to be spatially homogeneous and isotropic. Thus the corresponding evolution
equation is just
af = Qc(f, f), t ~ O, k ~ O. (279)
0t
We quote from Escobedo and Mischler [211]: Equation (279) describes the behavior
of a low-energy, spatially homogeneous, isotropic photon gas interacting with a low-
temperature electron gas with Maxwellian distribution of velocities, via Compton scat-
tering. This model will be called Boltzmann-Compton.
We now survey some of the main problems in the field.
3.1. Derivation issues
The derivation of equations like Boltzmann-Fermi or Boltzmann-Bose is not a tidy
business (see Chapman and Cowling [154] ...). Therefore, the expected range of
applicability and the precise form of the equations are not so clear.
A better understood situation is the linear setting: description of the effect of a lattice
of quantum scatterers on a density of quantum particles. Not only is the exact equation
well understood, but also a theoretical basis, in the spirit of Lanford's theorem, can be
given. Starting from the many-body Schrtidinger equation as microscopic equation, Erd6s
and Yau [205,206] have been able to retrieve the expected linear Boltzmann equation as a
macroscopic description. Related works are performed in [132-134] .... In the sequel we
do not consider these issues and restrict the discussion to the nonlinear equations written
above. Here are a few problematic issues about them.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-281-320.jpg)
![282 c. Villani
Cross-sections. It seems, nobody really knows what precise form of the cross-section, or
equivalently of the collision kernel B(v - v,, or) in (275) (or b(k, U) in (278)) should be
used - except in some particular cases with photon interaction .... Some formulas can be
found in [154] but they are not very explicit. This makes it difficult to give an interpretation
of some of the mathematical results, as we shall see. It would be desirable to identify some
model collision kernels playing the same role as the ones associated with inverse-power
interactions in the classical theory. According to certain physicists, it would be not so bad
to understand the case of a simple hard-sphere collision kernel.
Grazing collisions. Some variants of these equations are obtained by a grazing collision
asymptotics. To this class belong the quantum Landau equation (see Lemou [298] and
references therein), or the well-known Kompaneets equationl2 [290],
O
fOlk20 f ]OF
Ot = 0--k ~ -4-(k2 - 2k)f + f2 =- Ok' t >~O, k >>.
O. (280)
A flux condition must be added at the boundary:
lim F(k) = 0. (281)
k--+0
Equation (280) describes the same kind of phenomena as the Boltzmann-Compton
equation, and can in fact be obtained from it by an asymptotic procedure similar
to the one leading from the Boltzmann to the Landau equation (see Escobedo and
Mischler [211]) under some assumptions on the initial datum. However, the validity of
this approximation cannot be universally true, because the Kompaneets equation has some
strange "blow-up" properties: Escobedo, Herrero and Velazquez [208] have shown that the
flux condition (281) may break down in finite time for arbitrarily small initial data.
Also the long-time behavior of the Kompaneets equation can be non-conventional; this
is consistent with the remark by Caflisch and Levermore [114] that for large enough mass
there are no stationary states ....
Besides physical interest, all these considerations illustrate the fact that the asymptotics
of grazing collisions may destroy (or create?) some important features of the models.
Consistency with classical mechanics. All these quantum models involve the Planck
constant as a parameter; of course when one lets the Planck constant go to 0 (which would
in fact be the formal consequence of a change in physical scales, from microscopic to
macroscopic), one expects to recover the Boltzmann-like equations of classical mechanics.
This can be justified in some cases, see, for instance, [196].
Hydrodynamics. Physicists expect that some hydrodynamic limit of the Boltzmann-Bose
equation leads to the Gross-Pitaevski, or Ginzburg-Landau, model (based on a cubic
nonlinear Schr6dinger equation) for the evolution of the Bose condensate 13 part. Again,
12Thisequationis oftenwrittenwithk2f as unknown.
13Seethe nextsection.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-282-320.jpg)
![A reviewof mathematical topics in collisionalkinetic theory 283
this would need clarification .... We note however that the justification of this limit would
look more interesting if the derivation of the Boltzmann-Bose equation was first put on a
more rigorous basis.
3.2. Trend to equilibrium
Equations such as Boltzmann-Fermi, Boltzmann-Bose or Boltzmann-Compton all satisfy
entropy principles, and equilibrium states are entropy minimizers 14:
(1) For the Boltzmann-Fermi equation, the entropy is
HBF(f)- f[flogf- (1 - f) log(1 - f)].
Equilibrium states are of the form
1
.T'(p) = (or > 0, fl E R) (282)
eulP-po I2+fi -n
t- 1
or
~'(p) = llp_pol~ R (R > 0).
A state like (282) is called a Fermi-Dirac distribution. Here P0 is the mean impulsion.
(2) For the Boltzmann-Bose equation, the entropy is
HBB(f) = f[flogf- (1 4- f)log(1 4- f)]
(here s = + 1) and the shape of equilibrium states depends on the temperature. There is a
critical condensation temperature Tc such that the equilibrium state/3 takes the form
1
-- (or > 0, /3 >~0) when T >~Tc, (283)
13(p) e~IP-Po12+fl -- 1
1
- + #6po (or > 0, # > 0) when T < Tc. (284)
B(p) e~ -po12 -- 1
These distributions are called Bose-Einstein distributions. The singular part of (284) is
called a Bose condensate.
(3) Finally, for the Boltzmann-Compton equation, the entropy is given by
f0 ~
Hsc(f)- [(k e + f)log(k e + f)- f log f- kf- k e log(k2)] dk,
14Underthe constraint0 ~<f ~<1for the Boltzmann-Fermimodel.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-283-320.jpg)
![284 C. Villani
and, according to Caflisch and Levermore [114], the minimizers are of the form
1
B(k) = ek+Z _ 1 + c~60,
where )~ and c~ are nonnegative numbers, at least one of them being 0. For )~ > 0 this is a
Bose distribution, for ~. = 0 it is called a Planck distribution.
As in the Boltzmann case, these distributions, obtained by a minimization principle, also
coincide with the probability distributions which make the collision operator vanish. There
are some technicalities associated with the fact that singular measures should be allowed:
they have recently been clarified independently by Escobedo and Mischler, and by Lu.
Now, let us consider the problem of convergence to equilibrium, in a spatially
homogeneous setting for simplicity. As far as soft methods (compactness and so on)
are concerned, Pauli's exclusion principle facilitates things a great deal because of the
additional L ~ bound. Therefore, convergence to equilibrium in a (very) weak sense is
not very difficult [211]. However, no constructive result in this direction has ever been
obtained, neither has any entropy-entropy dissipation inequality been established.
In the Bose case, this is an even more challenging problem since also soft methods
fail, due to the lack of a priori bounds. The entropy is now sublinear and fails to prevent
concentration, which is consistent with the fact that condensation may occur in the long-
time limit. Actually, as soon as T < Tc, a given solution cannot stay within a weakly
compact set of L 1 as t --+ +e~z .... Lu [328] has attacked this problem with the well-
developed tools of modern spatially homogeneous theory, and proven that
- when the temperature is very large (T >> Tc), solutions of the spatially homogeneous
Boltzmann-Bose equation are weakly compact in L 1 as t --+ +cxz, and converge
weakly towards a Bose distribution of the form (283);
- when the temperature is very low (T < Tc), solutions are not weakly compact in L 1,
but converge to equilibrium in the following extremely weak sense [328]: if (tn) is a
sequence of times going to infinity, then from f (tn, ") one can extract a subsequence
converging in biting-weak L 1 sense towards a Bose distribution of the form (284).
In this theorem, not only is biting-weak L 1 convergence a very weak notion (weaker
than distributional convergence), but also the limit may depend on the sequence (t,).
Furthermore, it is not known whether weak L 1 compactness as t ~ oo holds true when
T is greater than Tc, but not so large.
Lu's theorem is proven for isotropic homogeneous solutions. Isotropy should not be a
serious restriction, but seems compulsory to the present proof. What is more, Lu's work
relies on a strong cut-offassumption for the kernel B" essentially,
8(iv - v,i, coso) co(lo - o,i ,, Io - o,I), c > o, (285)
where 0 is as usual the deviation angle. This assumption enables a very good control of the
Q+ part, but may do some harm for other, yet to be found, a priori estimates.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-284-320.jpg)
![A review of mathematical topics in collisional kinetic theory 285
3.3. Condensation infinite time
Physical experiments with very cold atoms have recently become possible, and have
aroused a lot of interest. For instance, a few years ago it became possible to experimentally
create and study Bose condensates. Among other phenomena, physicists report the
formation of a condensate in finite time. However, Lu has proven that there is no finite
time clustering for the Boltzmann-Bose equation studied in [328]. This seems to leave
room for two possibilities, both of which may lead to exciting new research directions:
9 either the Boltzmann-Bose equation should be discarded for a more precise model
when trying to model Bose condensation;
9 or the Bose condensation is excluded by the strong cut-off assumption (285), which
penalizes interactions with v _~ v. (supposedly very important in condensation
effects). On this occasion we strongly feel the need to have a better idea of what
collision kernels would be physically realistic. Proving the possibility (or genericity)
of finite-time condensation for "bigger" collision kernels (say B = 1?) would be
a mathematical and physical breakthrough for the theory of the Boltzmann-Bose
model.
3.4. Spatial inhomogeneities
So far we have only considered spatially homogeneous quantum Boltzmann equations,
now what happens for spatially inhomogeneous data? Due to the additional L ~ bound,
the Boltzmann-Fermi model seems easier to study than the classical Boltzmann equation;
in particular existence results can be obtained without too much difficulty [196,309]. The
situation is completely different for the Boltzmann-Bose model, since one would like to
consider singular measures as possible data. A completely new mathematical theory would
have to be built! A particularly exciting problem would be the understanding of the space-
time evolution for the Bose condensate.
It was communicated to us by Lu that for small initial data in the whole of R 3, one can
prove that Bose-Einstein condensation never occurs .... This should be taken as a clue that
the underlying mathematical phenomena are very subtle.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-285-320.jpg)
![286 C. Villani
Bibliographical notes
General references. Standard references about the kinetic theory of rarefied gases and
the Boltzmann equation are the books by Boltzmann [93], Carleman [119], Chapman
and Cowling [154], Uhlenbeck and Ford [433], Truesdell and Muncaster [430], Cercig-
nani [141,148], Cercignani, Illner and Pulvirenti [149], as well as the survey paper by
Grad [250]. The book by Cercignani et al., with a very much mathematically oriented
spirit, may be the best mathematical reference for nonspecialists. The book by Uhlenbeck
and Ford is a bit outdated, but a pleasure to read. There is no up-to-date treatise which
would cover the huge progress accomplished in the theory of the Boltzmann equation over
the last ten years.
For people interested in more applied topics, and practical aspects of modelling by the
Boltzmann equation, Cercignani [148] is highly recommended. We may also suggest the
very recent book by Sone [407], which is closer to numerical simulations.
The book by Glassey [233] is a good reference for the general subject of the
Cauchy problem in kinetic theory (in particular for the Vlasov-Poisson and Vlasov-
Maxwell equations, and for the Boltzmann equation near equilibrium). Also the notes by
Bouchut [96] provide a compact introduction to the basic tools of modem kinetic theory,
like characteristics and velocity-averaging lemmas, with applications.
To the best of our knowledge, there is no mathematically-oriented exposition of the
kinetic theory of plasma physics. Among physicists' textbooks, Balescu [46] certainly has
the most rigorous presentation. The very clear survey by Decoster [160] gives an accurate
view of theoretical problems arising nowadays in applied plasma physics.
There are many, many general references about equilibrium and non-equilibrium
statistical physics; for instance, [49,227]. People who would like to know more about
information theory are advised to read the marvelous book by Cover and Thomas [156]. A
well-written and rather complete survey about logarithmic Sobolev inequalities and their
links with information theory is [16] (in french).
Historical references. The founding papers of modem kinetic theory were those of
James Clerk Maxwell [335,336] and Ludwig Boltzmann [92]. It is very impressive to
read Maxwell's paper [335] and see how he made up all computations from scratch! The
book [93] by Boltzmann has been a milestone in kinetic theory.
References about the controversy between Boltzmann and his peers can be found
in [149, p. 61], or Lebowitz [293]. Some very nice historical anecdotes can also be found
in Balian [49].
Certainly the two mathematicians who have most contributed to transform the study of
the Boltzmann equation into a mathematical field are Torsten Carleman in the thirties, and
Harold Grad after the Second World War.
Derivation of the Boltzmann equation. For this subject the best reference is certainly
Cercignani, Illner and Pulvirenti [149, Chapters 2 and 4]. A pedagogical discussion of
slightly simplified problems is performed in Pulvirenti [394]. Another excellent source
is the classical treatise by Spohn [410] about large particle systems. These authors
explain in detail why reversible microdynamics and irreversible macrodynamics are not](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-286-320.jpg)
![A review of mathematical topics in collisional kinetic theory 287
contradictory- a topic which was first developed in the famous work by Ehrenfest and
Ehrenfest [202], and later in the delightful book by Kac [284]. Further information on the
derivation of macroscopic dynamics from microscopic equations can be found in Kipnis
and Landim [287].
Hydrodynamic limits. A very nice review of rigorous results about the transition from
kinetic to hydrodynamic models is Golse [239]. No prerequisite in either kinetic theory, or
fluid mechanics is assumed from the reader. Note the discussion about ghost effects, which
is also performed in Sone's book [407]. The important advances which were accomplished
very recently by several teams, in particular, Golse and co-workers, were reviewed by the
author in [441 ].
There is a huge probabilistic literature devoted to the subject of hydrodynamic limits
for particle systems, starting from a vast program suggested by Morrey [352]. Entropy
methods were introduced into this field at the end of the eighties, see in particular the
founding works by Guo, Papanicolaou and Varadhan (the GPV method, [263]), and
Yau [466]. For a review on the results and methods, see the notes by Varadhan [439],
the recent survey by Yau [467] or again, the book by Kipnis and Landim [287].
Mathematical landmarks. Here are some of the most influential works in the mathemati-
cal theory of the Boltzmann equation.
The very first mathematical steps are due to Carleman [118,119] in the thirties. Not
only was Carleman the very first one to state and solve mathematical problems about the
Boltzmann equation (Cauchy problem, H theorem, trend to equilibrium), but he was also
very daring in his use of tools from pure mathematics of the time.
In the seventies, the remarkable work by Lanford [292] showed that the Boltzmann
equation could be rigorously derived from the laws of reversible mechanics, along the
lines first suggested by Grad [249]. This ended up a very old controversy and opened new
areas in the study of large particle systems. Yet much remains to be understood in the
Boltzmann-Grad limit.
At the end of the eighties, the classical paper by DiPerna and Lions [192] set up new
standards of mathematical level and dared to attack the problem of solutions in the large
for the full Boltzmann equation, which to this date has still received no satisfactory answer.
A synthetical review of this work can be found in G6rard [231 ].
Finally, we also mention the papers by McKean [341] in the mid-sixties, and Carlen
and Carvalho [121] in the early nineties, for their introduction of information theory in the
field, and the enormous influence that they had on research about the trend to equilibrium
for the Boltzmann equation.
Acknowledgements
It is a pleasure to thank D. Serre for his suggestion to write up this review, and E. Ghys for
his thorough reading of the first version of the manuscript. I also warmly thank L. Arkeryd,
E. Caglioti, C. Cercignani, L. Desvillettes, X. Lu, F. Malrieu, N. Masmoudi, M. Pulvirenti,
G. Toscani for providing remarks and corrections, and J.-E Coulombel for his job of](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-287-320.jpg)
![288 C. Villani
tracking misprints. The section about quantum kinetic theory would not have existed
without the constructive discussions which I had with M. Escobedo and S. Mischler.
Research by the author on the subjects which are described here was supported by the
European TMR "Asymptotic methods in kinetic theory", ERB FMBX-CT97-0157. Finally,
the bibliography was edited with a lot of help from the MathSciNet database.
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![Viscous and~orheat conducting compressiblefluids 309
1. Basic equations of mathematical fluid dynamics
1.1. Balance laws
Let S2 C R N be a domain in two- or three-dimensional space (N = 2, 3) filled with a fluid.
We shall assume the fluid is a continuous medium the state of which at a time t 6 I C R
and a spatial point x E I2 is characterized by the three fundamental macroscopic quantities;
the density 0 = O(t,x), the velocity u = u(t, x), and the temperature 0 = O(t,x). The
fluid motion is governed by a system of partial differential equations expressing the basic
principles of classical continuum mechanics.
Conservation of mass:
O0
Ot
+ div(0u) = 0 (1.1)
Balance of momentum (Newton's second law of motion):
O(ou)
Ot
+ div(ou | u) + Vp = div Z + of (1.2)
Conservation of energy (the first law of thermodynamics):
OE
Ot
+ div((E + p)u) - div(2Su) - div q + of. u (1.3)
Here p is the pressure, 27 denotes the viscous stress tensor, E stands for the specific energy,
q is the heatflux, and f denotes a given external force density.
We have chosen the spatial description where attention is focused on the present config-
uration of the fluid and the region of physical space currently occupied. This description
was introduced by d'Alembert and is usually called Eulerian in hydrodynamics. There is
an alternative way - the referential description - introduced in the eighteenth century by
Euler that is called Lagrangean. In this description the Cartesian coordinate X of the posi-
tion of the particle at the time t -- to is used as label for the particle X (see, e.g., Truesdell
and Rajagopal [105]).
1.2. Constitutive relations
The general system (1.1)-(1.3)of N + 2 equations must be complemented by constitutive
relations reflecting the diversity of materials in nature.
An important class of fluids that occupies a central place in fluid mechanics is the
linearly viscous or Newtonian fluid, whose viscous stress tensor I7 takes the form
57 -- Z(Vu) -- #(Vu + (Vu) t) + Adivuld,](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-308-320.jpg)
![310 E. Feireisl
where # and ~ are the viscosity coefficients assumed to be constant unless otherwise
specified.
The full stress characterized by the Cauchy stress tensor T is related to 27 by the Stokes
law
T=Z-pId,
where the pressure p = P(O, O) is a general function of the independent state variables O
and 0.
The specific energy E can be written in the form
E = Ekinetic + Einternal,
1
Ekinetic = =Olu] 2, ginternal = oe,
z
where e is the specific internal energy related to the density and the temperature by a
general constitutive law e = e(Q, 0).
In accordance with the basic principles of thermodynamics, we postulate the existence
of a new state variable- the specific entropy S = S(O, O) - satisfying
OS_ lOe OS=_l(Oe p)
00 -- 0 00' 0Q 0 00 O2 9 (1.4)
Consequently, one can replace (1.3) by the entropy equation
q) 27(Vu) 9Vu
Ot(QS) + div QSu + -~ - 0
q. V0
02
(1.5)
where, by virtue of the second law of thermodynamics, the right-hand side should be non-
negative which yields the restriction
2
~z~>0, z+~tt~>o together with q. VO ~ O.
We focus on viscous fluids assuming always
2
/z > 0, )~+ -:-:# ~>0. (1.6)
PC
Finally, the heat flux q is related to the temperature by the Fourier law
q = -x VO , tc >~O,
where the heat conduction coefficient K may depend on 0, O and even on V0 though it is
assumed constant in most of the cases we shall deal with.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-309-320.jpg)
![Viscous and~orheat conducting compressible fluids 311
1.3. Barotropic models
The flow is said to be barotropic if the pressure p depends solely on the density 0. There
are several situations when such a hypothesis seems appropriate. For instance, the ideal
gas consitutive relation for the pressure reads
p= (y -1)oe, e =cvO, cv > O, (1.7)
where ~, > 1 is the adiabatic constant. Accordingly, the entropy S takes the form
S = log(e) + (1 - y) log(o).
Substituting # = )~= x = 0 in (1.5) and assuming a spatially homogeneous distribution So
of the entropy at a time to E I we easily deduce S(t) = So for any t E I and, consequently,
P(o) =ao • a=(g-1)exp(So)>O.
Under such circumstances, Equations (1.1), (1.2) represent a closed system describing the
motion of an isentropic compressible viscous fluid.
A similar situation occurs in the isothermal case when we suppose O(t) -- 00 and (1.7)
reduces to
P(O) = rOoo, r > O.
For a general barotropic flow, the specific energy E can be taken in the form
1
E[ O, u]- ~O]u] 2 + P(O) (1.8)
z
with
e' (z)z - e(z) = p(z).
The energy of a barotropic flow satisfies the equality
OtE + div((E + p)u) -- div(Z:u) - r" Vu + of. u (1.9)
which is now a direct consequence of (1.1), (1.2).
From the mathematical point of view, the barotropic flows represent an interesting class
of problems for which an existence theory with basically no restriction on the size of data
is available (see Section 6 below).
1.4. Boundary conditions
To obtain mathematically well-posed problems, the equations introduced above must be
supplemented by initial and/or boundary conditions. The boundary 012 is assumed to](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-310-320.jpg)
![312 E. Feireisl
be an impermeable rigid wall, i.e., the fluid does not cross the boundary but may move
tangentially to the boundary. Accordingly, we require
u.n=O onlxO~, (1.10)
where n stands for the outer normal vector. For both experimental and mathematical
reasons, (1.10) should be accompanied by a condition for the tangential component of
the velocity. As observed in experiments with viscous fluids, the tangential component
approaches zero at the boundary to a high degree of precision. This can be expressed by
the no-slip boundary conditions:
u=0 on I x OS2. (1.11)
On the other hand, in vessels with frictionless boundary (cf. Ebin [25]), condition (1.10)
is usually complemented by the requirement that the tangetial component of the normal
stress is zero, which can be written in the form of the no-stick boundary conditions:
u.n=0, (rn) xn=0 onlx0~2. (1.12)
Similarly, one prescribes either the heat flux or the temperature. For a thermally insulated
boundary, the condition reads
q.n=O on/xOl2
while
O= Ob on/xOS2
when the boundary distribution of the temperature is known.
If S-2is unbounded, it is customary to prescribe also the limit values of the state variables
for large x 9 S2, e.g.,
~ 0~, u--+ u~, 0 ~ 0~ as Ix l ~ ~.
In the in-flow and/or out-flow problems, the homogeneous Dirichlet boundary condi-
tions (1.11) are replaced by a more general stipulation
U=Ub onlzOI2.
Moreover, the density distribution must be given on the in-flow part of the boundary, i.e.,
Q(t, x) -- Qb(X) for (t, x) 9 I x 012, Ub(X) 9n(x) < 0.
Other types of boundary conditions including unilateral constraints and free boundary
problems are treated in the monograph by Antontsev et al. [4, Chapters 1, 3].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-311-320.jpg)
![Viscousand~orheatconductingcompressiblefluids 313
1.5. Bibliographical comments
An elementary introduction to the mathematical theory of fluid mechanics can be found
in the book by Chorin and Marsden [12]. More extensive material is available in
the monographs by Batchelor [7], Meyer [77], Serrin [93], or Shapiro [94]. A more
recent treatment including the so-called alternative models is presented by Truesdell and
Rajagopal [105].
A rigorous mathematical justification of various models of viscous heat conducting
fluids is given by Silhav~ [96]. Mainly mathematical aspects of the problem are discussed
by Antontsev et al. [4], M~ilek et al. [68], and more recently by Lions [61,62].
2. Mathematical aspects of the problem
The first and most important criterion of applicability of any mathematical model is its
well-posedness. According to Hadamard, this issue comprises a thorough discussion of the
following topics.
9 Existence of solutions for given data. The data for the problem in question are usually
the values of the state variables Q, u, and 0 specified at a given time t = to and/or the
driving force f together with the boundary values of certain quantities as the case may
be. The problem is whether or not there exist solutions for any choice of the data on a
given time interval I.
9 Uniqueness. The model is to be deterministic, specifically, the time evolution of the
system for t > to must be uniquely determined by its state at the time to.
9 Stability. Small perturbations of the data should result in small variation of the
corresponding solution at least on a given compact time interval. On the other hand,
experience with much simpler systems of ordinary differential equations suggests that
chaotic behaviour may develop with growing time. Roughly speaking, the solutions
may behave in a drastically different way in the long run no matter how close they
might have been initially.
To begin, let us say honestly that a rigorous answer to most of the issues mentioned
above is very far from being complete. Global existence and uniqueness of solutions to the
system (1.1)-( 1.3) is still a major open problem and only partial results shed some light on
the amazing complexity of the problem.
In this introductory section, we review the presently available results on the existence
of classical as well as weak or distributional solutions to the full system of equations
of a compressible Newtonian and heat-conducting fluid. Equations (1.1)-(1.3) written in
Cartesian coordinates take the form
OkO O(~OU
j )
t =0;
at OXj
O(OUi) O(ouJu i)
Ot OXj
Op
Oxi
-[--~) aXj /~] -[- Ofi' i--1 ..... N;
(2.1)
(2.2)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-312-320.jpg)
![314 E. Feireisl
O(oO) +
Cv Ot
-~-~ OXj "
(2.3)
Here and always in what follows, the summation convention is used.
The term classical solution means that the state variables have as many derivatives
as necessary to give meaning to (2.1)-(2.3) on S2 • I and are continuous up to the
boundary 0S2 to satisfy the boundary conditions as the case may be. Usually, we make
no distinction between classical and strong solutions whose generalized derivatives are
locally integrable functions and satisfy the equations almost everywhere in the sense of
the Lebesgue measure. Typically, any strong solution is a classical one provided some
additional smoothness of the data is assumed.
Multiplying (2.1)-(2.3) by a compactly supported and smooth test function 9 and
integrating the resulting expressions by parts, we get the integral identities:
ffs~ 09 09 dxdt - 0; (2.4)
0-57 + Ouj Oxj
fl fl2 " 09 Oui 09
OUi 09 uj 09 + P -- lZ~~
--~ + OUt OXj ~ OXj OXj
OU
j O~
-- ()~ + lZ)OXj OXi ~-Ofi9 dx dt - 0, i = 1 ..... N; (2.5)
y,fo(ooo r
Cv -~ -l-oOuJ ) -- p )9 -- K~ ~
OXj OXj
O0 09
OXj OXj
+ -~ -~xk + ~xj 9 + )~ Oxj ,] 9dxdt-0. (2.6)
We shall say that Equations (2.1)-(2.3) hold in D I(I • 12) (in the sense of distributions)
or, equivalently, that 0, u, and 0 is a weak solution of the problem if the integral identities
(2.4)-(2.6) hold for any test function 9 6 D(I • S2). The symbol D(Q) denotes the space
of infinitely differentiable functions with compact support in an open set Q.
It is not difficult to observe that the local formulation (2.1)-(2.3) and the integral
formulation (2.4)-(2.6) are in fact equivalent provided the solution is smooth enough. On
the other hand, (2.4)-(2.6) make sense under much weaker assumptions, namely, when
the quantities 0, 0ui, 0uiuj, P, OUi/OXj,of i, 00, oOuj, pOuJ/Oxj, O0/OXi, IOui/Oxj[ 2,
i, j = 1..... N are locally integrable on I • S2. We shall always tacitly suppose that this
is the case whenever speaking about weak solutions.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-313-320.jpg)
![Viscous and/or heat conducting compressible fluids 315
2.1. Global existence for small and smooth data
Following the pioneering work of Matsumura and Nishida [72] we consider the system
(2.1)-(2.3) where
~z = ~z(o, 0), z = z(o,0)
are smooth functions satisfying # > O, )~+ 2/3/z ~>O; (2.7)
the pressure p is given by a general constitutive relation
Op Op
-- > O; (2.8)
p--p(o,O) and p, 00' O0
x=x(o,O), x>O. (2.9)
In addition and in accordance with (1.4), we suppose
Op(o,o)
p(o,O) = ~ 0 . (2.10)
O0
The problem (2.1)-(2.3) is complemented by the Dirichlet boundary conditions
uilos2=O, i=1,2,3, 010s~=0b, (2.11)
where 0b > 0; and the initial conditions
0(0, x) = Oo(x)> O, Ui (0, X) -- Uio(X), i -- 1, 2, 3,
O(O, x) = Oo(x), x ~ S2. (2.12)
The following global existence theorem holds.
THEOREM 2.1. Let I2 C R 3 be a domain with compact and smooth boundary. Let the
quantities #, )~, p, and tc comply with the hypotheses (2.7)-(2.10). Moreover, let the
i O0 belong to the Sobolev space W3'2(I-2) and satisfy the compatibility
initial data 00, uo,
conditions
i _ O, (90 -- Ob,
Uo
O
p(oo, 0o)
Oxi
0 OUio~ 0 ()~(00 00) -[-"lZ(OO, 00)) OXj ~]~] -- OOfi
-- OXj l's 00) OXj /] -[- ~Xi '](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-314-320.jpg)
![316 E. Feireisl
OuJo
p(oo, Oo)
00)
x(oo, Oo) +
OXj ~Xj
i =1,2,3,
2 ~-Txk+ Ox-Txj +)~(0~ 0~ ~xj '
on the boundary 0I-2. Finally, let fi __ OF/Oxi, i = 1, 2, 3 where F belongs to the Sobolev
space W5'2 (S2).
Then there exists e > 0 such that the initial-boundary value problem (2.1)-(2.3), (2.11),
(2.12) posseses a unique solution O, u, 0 on the time interval t ~ (0, oo) provided the initial
data satisfy
I100- 0 IIw3,2(s-2)+ Iluollw3,2(s2)+ I100-0bllw3,2(s2) + IIFIIws,2(x2)< s,
where
0 -- -~1 oodx.
Theorem 2.1 in its present form is taken over from Matsumura and Nishida [73] (cf.
also [72]). The proof, which is rather lengthy and technical, is based on a priori estimates
resulting from energy relations. Their method has been subsequently adapted by many
authors to attack various problems with non-homogeneous boundary conditions (cf., e.g.,
Valli and Zajaczkowski [109]) as well as the barotropic models (see Valli [108]). The
common feature of all these results is that they apply only to problems where the data
are small and regular.
The solutions the existence of which is claimed in Theorem 2.1 belong to the space
o, u, o e c([o, r]; W3'2(s?)),
where W k'p (~2) denotes the Sobolev space of functions whose derivatives up to order k
lie in the Lebesgue space L p (S-2) (for basic properties of Sobolev spaces see, e.g., the
monograph of Adams [1]). The scale Wk'2 forms a suitable function spaces framework
because of the variational structure of the problem. The a priori estimates are obtained
in the Hilbertian scale W k'2 in a very natural way via the "energy method" used in the
pioneering paper by Matsumura [71]. Assuming more regularity of the initial data and F
one could prove the same result with W3,2 replaced by W k'2 with k sufficiently large. It
follows then from the standard embedding theorems that the solution would be classical.
Alternatively, one can use the smoothing effect of the diffusion semigroup to deduce that
the solutions constructed in Theorem 2.1 are, in fact, classical for t > 0 (cf. Matsumura
and Nishida [72]).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-315-320.jpg)
![Viscous and~or heat conducting compressible fluids 317
2.2. Global existence of discontinuous solutions
Discontinuous solutions are fundamental both in the physical theory of nonequilibrium
thermodynamics and in the mathematical theory of models of inviscid fluids. It seems
natural, therefore, to have a rigorious mathematical theory for the system (2.1)-(2.3)
which would accommodate discontinuities in solutions. Of course, one has to abandon
the classical concept of solution as a differentiable function and turn to the weak solutions
which satisfy the integral identities (2.4)-(2.6).
It follows from (2.4)-(2.6) that the density 0, the momenta Qui, i -- 1, 2, 3, and the
specific internal energy Q0 considered as vector functions of time are weakly continuous,
i.e., the quantities
f~ Odpdx ' Is2 o u idpdx ' i -- 1 N,
[.
belong to C(I) for any fixed 4~e 79(S2).
f oOCdx
Consequently, it makes sense to prescribe the initial conditions even in the class of weak
solutions.
Pursuing this path Hoff [47] examined the system (2.1)-(2.3) on the whole space R3
where the pressure p and the internal energy e obey the ideal gas constitutive relations (1.7)
and the initial data 00, u0, 00 satisfy
00 - a ~ L ~ n L ~(R~),
00 - ~ ~ L:(R~),
uo ~ [ws'=(R~)] ~ se(1/3,1/2),
(2.13)
for certain positive constants ~, 0. The Sobolev spaces Ws'2(R 3) for a general real
parameter s may be defined in terms of the Fourier transform (see Adams [1]).
We report the following result (see [47, Theorem 1.1]).
THEOREM 2.2. Let 1-2 -- R 3. Assume that )~, lz and tc are constants satisfying
> 0, ~/3 < ~ + ~ < (~ + 1/3,/i3)u, ~ > 0.
Let the pressure p obey the ideal gas constitutive relation
P--(V- 1)CvQ0, Y > 1, Cv > 0, (2.14)
and the initial data Qo, uo, Oo satisfy (2.13) for certain positive constants ~, O.
Let positive constants 0 < Ql < Q < Q2, 0 < 02 < 01 < 0 be given. Finally, set f- 0
in (2.2).
Then there exists ~ > 0 depending on Qi, Oi, i - 1, 2, and s such that the initial-value
problem (2.1)-(2.3), (2.12) possesses a weak solution Q, u, 0 on the set (0, cx~) x R 3](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-316-320.jpg)
![318 E. Feireisl
provided
I1~0 - ~llt2nt~(R 3) + II00 -0llt2(e3) -+-Iluollws,2nt4<R3) < e,
ess inf00 >/01.
Moreover, the solution satisfies
Q1 ~<Q(t, x) ~<~02, O(t, x) >~02 for a.a. (t, x) E (0, c~) • R 3,
and
Q(t) --+ b, u(t) --+ 0, O(t) --+ 0 in L p (R 3) as t --~ cx~
for any 2 < p <<.cx~.
A similar result under slightly more restrictive hypotheses on the data can be proved for
12 = R2 (see Hoff [47, Theorem 1.1]).
The proof of Theorem 2.2 leans, among other things, on the regularity properties of the
quantity p - ()~+ 2#) divu termed the effective viscous flux. More specifically, this quantity
is shown to be free of jump discontinuities. This is the first indication of the important role
played by the effective viscous flux in the mathematical theory of compressible fluids. We
will address this issue in detail in Section 4.3.
2.3. Global existence in critical spaces
The solutions obtained in Theorem 2.2 solve the problem for a very general class of initial
data but are not known to be unique. On the other hand, Theorem 2.1 yields a unique
solution at the expense of higher regularity imposed on the data. A natural question to ask
is how far one can get from the hypotheses of Theorem 2.1 to those of Theorem 2.2 to save
uniqueness or, more precisely, what is a critical space of data for which the weak solutions
are unique.
Such a question was already studied for the incompressible Navier-Stokes equations
by Fujita and Kato [41]. The same problem for the full system (2.1)-(2.3) under rather
general constitutive relations has been addressed only recently by Danchin [16]. He obtains
existence and uniqueness of global solutions in a functional space setting invariant by the
natural scaling of the associated equations:
- ,(v2t, vx), -
where the pressure law p is changed to I)2p.
A functional space for the triple [0, u, 0] is termed critical if the associated norm is
invariant under the transformation
Lo, u, O] ~ [~o~,u~, 0~]](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-317-320.jpg)
![Viscousand~orheat conducting compressiblefluids 319
up to a constant independent of v. Accordingly, the well-posedness for the problem (2.1)-
(2.3) can be stated in terms of the Besov spaces B s (R3) whose exact definition goes
2,1
beyond the framework of the present paper (see [16]). Let us only remark that the final
result is of the same character as Theorem 2.1, namely, global existence and uniqueness
of (weak) solutions of the problem (2.1)-(2.3) for data which are a small perturbation of a
given equilibrium state.
2.4. Regularity vs. blow-up
Since the celebrated work of Leray, it has been a major open problem of mathematical fluid
mechanics to prove or disprove that regular solutions of the incompressible Navier-Stokes
equations in three space dimensions exist for all time. Clearly, the same problem for the
general system (2.1)-(2.3) seems even more delicate.
As a matter of fact, there is a negative result of XIN [110, Theorem 1.3]. He considers
the system (2.1)-(2.3) posed on the whole space R3 with zero thermal conductivity tc -- 0
and the initial density 00 compactly supported:
THEOREM 2.3. Let S-2 -- R 3 and m > 3 be a given number.
Consider the system (2.1)-(2.3) complemented by the initial conditions (2.12) where
the viscosity coefficients )~, lZ are constant and satisfy (1.6), and p obeys the constitutive
law (2.14). Moreover, let tc = O, f = O, and
~O0,U0, 00 E W m'2 (e3),
supp 00 compact in R 3 00/>0>0.
Then there is no solution of the initial value problem (2.1)-(2.3), (2.12) such that
~0,U, 0 E C 1([0, 0<~); wm'2(R3)).
It seems interesting to compare the conclusion of Theorem 2.3 with the existence result
of Theorem 2.1. Obviously, the above theorem does not seem to solve (in a negative way)
the question of regularity for the compressible Navier-Stokes equations because of the
hypothesis of compactness of the support of 00. We remark in this regard that the Navier-
Stokes system is a model of nondilute fluids in which the density is bounded below away
from zero. It is natural, therefore, to expect the problem to be ill-posed when vacuum
regions are present at the initial time.
2.5. Large data existence results
To begin with, one should say there are practically no global existence results for the full
system (2.1)-(2.3) when the data are allowed to be large. The question of local existence
of classical solutions for regular initial data was addressed by Nash [79]. There is no
indication, however, whether or not these solutions exist for all times.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-318-320.jpg)
![320 E. Feireisl
Note that the problem here is of different nature than for systems of nonlinear
conservation laws without diffusion terms. Indeed the equations (2.2), (2.3) are parabolic
in u, 0 respectively provided the density 0 is kept away from zero. Accordingly, one
can anticipate these state variables to be regular provided uniform estimates of 0 were
available.
On the other hand, the density solves the hyperbolic equation (2.1) which is, however,
only linear with respect to 0. Consequently, no shock waves should develop in 0 provided
they were not present initially and the velocity field u was sufficiently regular.
Formally, one can use the standard method of characteristics to deduce:
d
dt
--Q(t, X (t)) + Q(t, X (t))divu(t, X (t)) - 0,
where
X'(t) -u(t,X(t)), X (0) = X0 9 S2.
We end up in a "vicious circle" as we need uniform bounds on divu to estimates the
amplitude (and positivness) of Q but those are not available from the standard energy
estimates. As indicated by Choe and Jin [11, Theorems 1.3, 1.4], the following three
questions are intimately interrelated:
9 uniform (on compact time intervals) upper bounds on the density Q;
9 uniform boundedness below away from zero of Q;
9 uniform bounds on u.
Answering one of these questions would certainly lead to a rigorous large data existence
theory in the framework of distributional (weak) solutions for the problem (2.1)-(2.3) (cf.
also Lions [62]).
The above mentioned difficulties made several authors to search for a completely
different approach to the problem. Motivated by the pioneering work of DiPerna [22],
the theory of measure-valued solutions was developed by M~ilek et al. [68]. Roughly
speaking, the "value" of each state variable at a fixed point (t, x) is no longer a number (or
a finite component vector) but a probability measure (the Young measure) characterizing
possible oscillations in a sequence of approximate solutions used to construct this particular
variable. The numerical values of Q, u, 0 are centers of gravity of the corresponding
Young measures and the nonlinear constitutive relations are expressed in a very simple
way. These solutions are of course more general quantities than the distributional solutions
and coincide with them provided one can show that the Young measures are concentrated
at one point, i.e., they are Dirac masses for each value of the independent variables (t, x).
One can expect positive existence results in the class of measure valued solutions
whenever suitable a priori estimates are available so that the nonlinear compositions are
equi-integrable and consequently weakly compact in the space of Lebesgue integrable
functions. This is of course considerably less than compactness of the state variables in the
strong L l-topology- an indispensable ingredient of any existence proof of distributional
solutions.
The major shortcoming of measure-valued solutions is certainly the almost insurmount-
able problem of uniqueness solved only in the case of a scalar conservation law in [22].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-319-320.jpg)
![Viscous and~or heat conducting compressible fluids 321
This is, of course, the price to be paid for the relatively simple existence theory and one
might feel tempted to say it is the same situation as when the weak solutions were in-
troduced. However, this gap between existence and uniqueness, accepted for the weak
solutions, seems to be simply too large in the class of measure-valued solutions and the
approach is slowly being abandoned.
2.6. Bibliographical comments
Besides the results mentioned above, the small data existence problems were treated by
Solonnikov [97], Tani [101], Valli [108] and others. The existence theory in critical spaces
for barotropic flows was developed by Danchin [15]. As pointed out several times, the
main obstacle to obtain large data existence results is the lack of suitable a priori estimates.
Formal compactness results for the full system (1.1)-(1.3) were obtained by Lions [62] on
condition of uniform boundedness of all state variables.
3. The continuity equation and renormalized solutions
Motivated by the work of Kruzkhov on scalar conservation laws, DiPerna and Lions [23]
introduced the concept of renormalized solutions as a new class of solutions to general
linear transport equations. They play a similar role as the entropy solutions in the theory
of nonlinear conservation laws - they represent a class of physically relevant solutions in
which the corresponding initial value problems admit a unique solution.
Multiplying (1.1) by b'(Q), where b is a continuously differentiable function, we obtain
the identity
ab(Q)
Ot
4- div(b(Q)u) + (b' (Q)Q - b(o))div u -- O. (3.1)
Obviously, any strong (classical) solution of (1.1) satisfies automatically (3.1). For the
weak solutions, however, (3.1) represents an additional constraint which may not be always
satisfied. Following [23] we shall say that Q is a renormalized solution of (1.1) on the set
I x $2 if 0, u, Vu are locally integrable and (3.1) is satified in the sense of distributions (in
79'(I x s i.e., the integral identity
b(o)--~ 4- b(p)u 9V9 4- (b(o) - b'(o)Q) divu~o dx dt - 0 (3.2)
holds for any test function q99 79(1 x s and any b 9 C 1(R) such that
b'(z) = 0 for all z large enough, say, Izl/> M. (3.3)
Let us emphasize here that unlike the entropy solutions that can be characterized
as satisfying a certain type of admissible jump conditions on discontinuity curves, the
renormalized solutions characterize the so-called concentration phenomena (cf. Section 3.2
below).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-320-320.jpg)
![322 E. Feireisl
3.1. On continuity of the renormalized solutions
The renormalized solutions enjoy many remarkable properties most of which can be proved
by means of the regularization technique developed by DiPerna and Lions [23]. The
following auxilliary assertion is classical (cf. Lions [61, Lemma 3.2]).
LEMMA 3.1. Let 12 C R N be a domain and
u, Vu~L p
loc(ff2), Lqoc(n),
where
1 <~p,q <. oo, l/r-- lip 4- 1/q <~1.
Let Oe be a regularizing sequence, i.e., Oe ~ 7)(RN), Oe radially symmetric and radially
decreasing,
f/~ Oedx = 1,
N
Oe(x) --+ 0 as e --+ Ofor anyfixed x E R N {0}.
Then
[lOe 9 [div(cru)] - div([Oe 9 or]u)[[tr(g ) ~ c(g)llUllw,,p(g)ll~rlltq(g )
for any compact K C ~2 and
re = Oe * [div(o'u)] - div([Oe 9 o-]u) --+ 0 in L,roc(s as e --+ O,
where 9 stands for convolution on R N.
Now one can regularize (3.1), more precisely, take ~(x) = Oe(x - y) in (3.2) to deduce
OOe * b(o)
Ot
4-div([Oe 9 b(o)]u ) 4- Oe * [(b'(o)O - b(o))divu] -- re (3.4)
for t ~ I and x ~ s such that dist[x,012] > e. Here b is an arbitrary function
satisfying (3.3) and re(t) as in Lemma 3.1, i.e., rE --~ 0 in L~oc(I • 12) as e -+ 0 provided
u is locally integrable.
The first consequence of (3.4) is continuity in time of the renormalized solutions.
PROPOSITION 3.1. Let u, Vu be locally integrable on I • S-2 where I C R is an open time
interval and s C R N a domain. Let O - a locally integrable function - be a renormalized
solution of the continuity equation (2.1) on I • S2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-321-320.jpg)
![Viscous and~or heat conducting compressible fluids 323
Then for any compact B C F2 and any function b as in (3.3), the composition b(Q):
t 9 I w->b(Q)(t) is a continuous function oft with values in the Lebesgue space LI(K),
i.e.,
b(o) 9 C(J; Ll(B)) forany compact J C I.
Moreover, we have the following corollary.
COROLLARY 3.1. In addition to the hypotheses of Proposition 3.1, assume that I C R,
C R N are bounded; and
Q 9 Lc~(O, T; LP(~2)) fora certain p > 1,
u, Vu 9 L I(I • ,(-2).
Then 0 as a function of t 9 I is continuous with values in L 1(~2 )"
o c c(i; Ll(n)).
The proof of both Proposition 3.1 and Corollary 3.1 can be done via the regularization
technique as in [23].
3.2. Renormalized and weak solutions
Another conclusion which can be deduced from Lemma 3.1 and (3.4) is that the class
of weak and renormalized solutions coincide provided Q or Vu or both are sufficiently
integrable.
PROPOSITION 3.2. Assume
0 9 LlPoc(I x ~), u, Vu 9 Lqoc(I x ~),
where
1 ~ p,q <~ oo, 1/p+ 1/q <~ 1.
Then Q is a renormalized solution of (2.1) if and only if Q satisfies (2.1) in 73'(1 x F2),
i.e., the integral identity (2.4) holds for any test function cp 9 73(1 x F2).
Integrating (1.9) we can see that the typical regularity class for the velocity gradient is
Vu 9 L2oc(I x ~2). Accordingly, to apply Proposition 3.2, one needs 0 9 L2oc(I x ~2).
There is another reason why the density "should be" square integrable. The continuity
equation can be (formally) written in the form
Dto + odivu =0,
0
where Dt -- 0-t + u. VQ](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-322-320.jpg)
![324 E. Feireisl
is the so-called material derivative. The quantity 0 div u plays the role of a forcing term in
the above equation. Thus if we want to keep, at least in a certain weak sense, the structure
given by characteristics (cf. Section 2.5), we should have Qdiv u locally integrable. Taking
the square integrability of Vu for granted we are led to require 0 ~ L2oc (cf. also DiPerna
and Lions [23]). However, as we will see in Proposition 4.1 below, the square integrabilty
of the density is not necessary for a weak solution of (1.1) to be a renormalized one.
3.3. Renormalized solutions on domains with boundary
Assume that I2 C R N is a domain with Lipschitz boundary. As for the velocity field u,
we suppose u 6 L~oc(l • S-2), Vu 6 L~oc(l • S2 ) for a certain q > 1. Although u need
not be continuous, one can still consider the no-slip boundary conditions (1.11) in the
sense of traces. Accordingly, assuming (1.11) and extending u to be zero outside S2, one
has u E Wllo'q (RN). Equivalently, by virtue of the Hardy inequality (see, e.g., Opic and
Kufner [85]), one can replace (1.11) by the following stipulation:
lul
dist[x, 8S2]
Lqoc( r
Using (3.5) we can show a continuation theorem for renormalized solutions.
PROPOSITION 3.3. Let I-2 C R N be a Lipschitz domain and let u, Vu belong to L~oc(I •
I-2 )for a certain q > 1, and let (1.11) be satisfied in the sense of traces. Let Q be a
renormalized solution of (2.1) on I • I2.
Then ~ is a renormalized solution of (2.1) on I • R N provided 4, u are extended to be
zero outside I-2.
Proposition 3.3 together with Propositions 3.1, 3.2 yield an interesting corollary, namely,
the principle of total mass conservation for the weak solutions of (2.1). Consider a bounded
domain I2 C R N with Lipschitz boundary on which u satisfies the no-slip boundary
condition (1.11). Formally, one can integrate (2.1) over S-2to deduce
-~ o dx - O,
i.e., the total mass
m - fs2 odx
is a constant of motion. By virtue of Propositions 3.1-3.3, we have the same result for
distributional solutions:](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-323-320.jpg)
 - sup(lim sup{]Tk(on) - Tk(o)[[ Lp(Q))"
k>/1" n--+oc
Unlike the defect measures introduced by DiPerna and Majda [24], osc vanishes on any
set on which 0n tends to 0 strongly in the L l-topology regardless possible concentration
effects.
Now we shall address the following question: Under which conditions do the limit
functions 0, u solve (2.1)? Taking a function b as in (3.3) one deduces easily from (3.2),
(4.3), and (4.4) that
b(On)-+ b(o) inC(I P
" Lweak(~Q)), 1 ~<p < oc. (4.6)
Here and in what follows, we shall use the standard notation g(v) for a weak (L p) limit
of a sequence g(vn) where Vn tends weakly to v. The possibility to find a subsequence of
Vn such that the composition b(vn) is weakly convergent for any continuous b satisfying
certain growth conditions is the basic statement of the theory of Young measures (cf.
Tartar [102,103], Pedregal [88]). Such a limit, however, need not be unique unless the
convergence of Vn is strong.
A sequence vn converges to v in C (I" P
, Lweak(~'2)) if it is bounded in L~(I; LP(~-2)), the
function t ~-+fs2 Vn(t, x)dp(x)dx can be identified with a continuous function on I, and
fs? v~(t' x)dp(x) dx --+ fs? v(t, x)dp(x) dx uniformly with respect to t e I
for any test function 4~e D(s
By virtue of the Aubin-Lions lemma (see, e.g., Lions [59, Theorem 5.1], the
relations (4.3), (4.6) imply
b(On)Un --+ b(o)u weakly in Lqoc(I • s (4.7)
Indeed taking p large enough in (4.6) we get L p(~(2) compactly imbedded in W -l'q (~(-2) ;
whence
b(on) --+ b(o) in C(I; W -l'q (s
which together with (4.3) yields the desired conclusion. The distributions lying in the
"negative" Sobolev space W-1,q can be identified with generalized derivatives of vector
functions in Lq (see, e.g., Adams [1]). In particular, if we knew that 0n is bounded in
L~oc(I x s we could conclude that O, u solve (2.1) in the sense of distributions.
In a general case, we report the following result (cf. [29, Proposition 7.1]):](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-326-320.jpg)
 ~<c(Q) for any compact Q c I x I-2,
where
1 1
t <1.
P q
Then Q, u is a renormalized solution of (2.1) on I x 12.
The main advantage of Proposition 4.1 is that the sequence On itself need not be bounded
in LP. As we will see later, this is particularly convenient when barotropic fluids are studied
(cf. Proposition 5.3 below).
4.2. On compactness of solutions to the equations of motion
In this part, we shall assume that
IlUnlIL2(I• IIVUnlIL2(I• ~ C foralln-- 1,2 ..... (4.8)
In particular, the products uiuj, i, j = 1..... N, are bounded in L I(I, L2"/2(~)) where 2*
is the Sobolev exponent for the embedding W 1,2 C L2* to hold, i.e.,
2* is arbitrary finite for N -- 2
2N
and 2"= if N=3,4 .....
N-2
i J to be at least integrable, it is neccessary that
Consequently, for the cubic quantity QnUnU n
ess sup II (t ll c for a certain p ~>N / 2
tel
(4.9)
provided N > 2. Here again, we face one of the major obstacles to build up a rigorous
mathematical theory for the full system (2.1)-(2.3), namely, the lack of suitable a priori
estimates. The only available bounds on the density are those deduced from boundedness
of the total energy. In general, these "energy" estimates are not sufficient to get (4.9). Of
course, the barotropic case offers a considerable improvement as the energy is given by
formula (1.8) and, consequently, the desired estimates follow provided
y >~N/2.
The situation is more delicate in the physically relevant two-dimensional case. Here, the
Sobolev space W 1'2 is embedded in the Orlicz space L ~ generated be the function
q0(z) - exp(z e) - 1](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-327-320.jpg)
![Viscous and~or heat conducting compressible fluids 329
(see Adams [1]). Consequently, condition (4.9) should be replaced by
ess sup fs2 ~p(On)dx <<,c
tel
with qJ(z) ~>z log(z). (4.10)
Since On, Un satisfy also the continuity equation (4.1) we deduce from (4.9), (4.10)
respectively that
Onto inC(I; P
Lweak(I2)), p ~ N/2 if N -- 3..... (4.11)
or
On --+ O inC(I; q'
Lweak(S2)) for N -- 2. (4.12)
In both cases this implies compactness of Qn in L 2(I, W-1,2 (S-2)), and we get
OnUn --+ 0u weakly in, say, L j (I x s
i Yrepresents a more difficult problem. In
The weak compactness of the cubic term OnUnU n
addition to the above hypotheses, we assume the kinetic energy to be bounded uniformly
in n, i.e.,
o~lu~l 2 is bounded in L 1(I • s uniformly with respect to n.
Supposing (4.9) holds for p > N/2 we have, similarly as above,
2N
OnUn ~ 0u in C(I; Lreak(a"2)) for a certain r > N ~>2; (4.13)
N+2'
whence
i UJn~ ~OUtUj weaklyin, say, Ll(I x I2) i j = 1 ... N, N >~2.
D..nU n , , , ,
As a matter of fact, the result is not optimal for N -- 2; in that case one could use
directly (4.12) provided q~ was a function dominating z log(z).
Summing up the previous considerations we get the following conclusion:
PROPOSITION 4.2. Let the quantities Qn, Un satisfy the estimates (4.8), (4.9) with p >
N/2. Moreover, let the kinetic energy be bounded, specifically,
ess sup fs? On(t)lUn (t)12
tel
dx <, c foralln= l,2 ..... (4.14)
Finally, assume fn are bounded and
fn ~ f uniformly on I x I2. (4.15)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-328-320.jpg)
![330 E.Feireisl
Then the limitfunctions O, u, p, and f satisfy (2.1), (2.2) in 79t(I • $2), i.e., the integral
identities (2.4), (2.5) hold for any testfunction q9~ 79(1 • I-2).
Let us repeat once more that it is an open problem whether or not the bounds required for
the density component are really available. As we have seen in Section 2.5, uniform bounds
on the density are equivalent to uniform boundedness of u - a situation reminiscent of the
classical regularity problem for the incompressible Navier-Stokes equations.
4.3. On the effective viscous flux and its properties
Consider the quantity p - ()~4- 2#) div u called usually the effective viscous flux. Formally,
assuming all the functions in (2.2) smooth and vanishing for Ixl --+ oc we can compute
Pn - (~, + 2#)divun
= A -1 div(Onfn) - A -1 div(OnUn)t - "~i,j[OnUinuj ] (4.16)
(summation convention). The symbol T~i,j denotes the pseudodifferential operator
T~i,j - OxiA-10xj
or, in terms of symbols,
TP~,ij[V] -- ~--1 [~i~j .)L-.[V](~) ]
where .T denotes the Fourier transform in the x-variable.
The effective viscous flux enjoys certain weak compactness properties discovered by
Lions [62] which represent the key point in the global existence proof for barotropic flows.
Following [62] we can use (formally) (4.16) to obtain
(Pn - (~ + 2#)divun)b(On)
--b(On)A -1 div(Onfn) - Ot(b(On)A -1 div(OnUn))
4- b(On)t A-1 div(OnUn) - b(On)~i,j[On UnUn],ij (4.17)
where b is as in (3.3). One should keep in mind that 0n, Un here are defined on a bounded
domain I2 and, consequently, a localization procedure is needed to justify this argument.
Using Proposition 4.2 we can now pass to the limit for n --+ cxz in (4.16) and multiply
the resulting expression by b(o) to deduce
(p - (~ + 2#)divu)b(o)
= b(o)A -1 div(of)- Ot(b-~A -1 div(ou))
4- b(o)t A-1 div(ou) - b(o)T4.i,j[Ou iuJ]. (4.18)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-329-320.jpg)
![Viscousand~orheatconductingcompressiblefluids 331
Assuming, in addition to the hypotheses already made, that On, Un are also renormalized
solutions of (2.1), we can use (4.7), (4.18) together with the smoothing properties of A -1 ,
to pass to the limit in (4.17) for n --+ oo to obtain:
lim f fs2(p. - ()~ + 2u)divu,)b(o,)~odxdt
n--->(~
_ff
=nli~m~flf. b(On)(UinJ~i'j[OnuJ]-J~i'j[OnuiuJ])~~
- ff.
~( uiJ~i,j
[Ot/j] --J-~i,j
[ouit/J])~
0dx at (4.19)
for any test function q9e 79(1 • Y2).
It is a remarkable result of Lions [62] that the right-hand side of (4.19) is in fact zero. To
see this we offer two rather different techniques in hope to illuminate a bit the connection
of such a result with the theory of compensated compactness. Following the proof in [62]
one can make use of the regularity properties of the commutator
UiJ~i,j [~OUj ] -- "]-'~'i,j[Oui UJ]
discovered by Coifman and Meyer [13]. Specifically, this quantity belongs to the Sobolev
space W l'q provided ui e W 1'2 and Ouj e L r with r > 2 in which case 1/q- 1/p + 1/2.
Of course, this hypothesis requires 0 e LP with p > 3 for N >~3 which is too strong for
our purposes, but a simple interpolation argument shows one can treat the general case
0 E L p, p > N/2, by the same method.
Pursuing a different path we can write
f fs b(Qn)(uJJ"~i'j[QnUin]- T2~i'j[Qn i j
....])~ dx at
- f f..,:(x, (0.).,(o.-.)- v, (0.))dx ,,,
where the vector fields X j , Y, U, Vj are given by formulas
X[ (On) - (~ob(On)6j,k - 7~k,l[~Pb(o~)&,j]),
k _~2~k,i[On i
V/ (On) - T~k,l[~ob(On)31,j], j - 1..... N,
and ~i,j stands for the Kronecker symbol.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-330-320.jpg)
![332 E. Feireisl
Now, it is easy to check that
divXJ--divU--O and Y--V(A-l[div(OnUn)]),
V j - V(A -1 [div(~ob(On)&,j)]),
i.e.,
curl Y = curl V j -" O.
Applying the L P-L q version of Div-Curl Lemma of the compensated compactness
theory (cf. Murat [78] or Yi [111]) together with (4.6), (4.13), we conclude
xJ (On)" Y(OnUn) ~ ((flb(o)Sj,k - ~k,l[(flb(O)Sl,j]) "JP~.k,i[Oui]
in L2(I; W-I'2(~Q)),
and, similarly,
U(OnUn) "vJ (On) --+ (0 uk -- JP~.k,i[oui])T~k,l[(flb(O)Sl,j] in L2(I; W-1'2(;2))
provided p > N/2. This yields, similarly as above, the desired conclusion, namely, the
right-hand side of (4.19) equals zero.
Thus we have obtained the following important result (see [62]):
PROPOSITION 4.3. Let the quantities On, Un, and fn satisfy the hypotheses (4.8), (4.9)
for p > N/2, togetherwith (4.14), (4.15). Let, moreover,
IlpnllL~(Z• ~ c fora certain r > 1.
Then we have
lim fifs2(Pn-()~+2lz)divun)b(Qn)qgdxdt
11---+oo
=ffs(P-(X+2.)clivu)b(V)~~
for any b satisfying (3.3) and any testfunction ~pE 7)(I x ~).
4.4. Bibliographical remarks
The theory of compensated compactness has played a crucial role in the development
of the first large data existence results for systems of nonlinear conservation laws (see
Dafermos [14], DiPerna [21], Tartar [102]). A good survey on weak convergence methods
can be found in the monograph by Evans [26]. One of the well-known results is the so-
called Div-Curl Lemma refered to above (cf. Murat [78]):](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-331-320.jpg)
![~scous and~orheat conducting compressiblefluids 333
LEMMA 4.1. Let Un, Vn be two sequences of vector functions defined on some open set
Q c R u such that
Un --+ U weakly in L p (Q), Vn --+ V weakly in L q (Q)"
and
div Un precompact in W- l, p (Q), curl Vn precompact in W- 1,q (Q),
where
1 1
1 <p,q <ec, -+-~<1.
P q
Then
Un . Vn --+ U. V in 79'(Q).
Note that the situation in Proposition 4.3 is particularly simple as div Un = curl Vn = 0
and the proof of Lemma 4.1 is elementary.
There is yet another way to show Proposition 4.3 presented in [27, Lemma 5].
The defect measures similar to osc were introduced by DiPerna and Majda [24] in their
study of the Euler equations.
5. Mathematical theory of barotropic flows
We review the recent development of the mathematical theory of barotropic flows,
specifically, we shall discuss some large data existence and related results originated by
the pioneering work of Lions [62]. Accordingly, the crucial hypothesis we cannot dispense
with is that the pressure p and the density 0 are functionally dependent and the relation
between them is given by formula
P--P(O)
with p-[0, ec) --~ [0, ec) - a nondecreasing and continuous function. (5.1)
As a matter of fact, most of the results will be stated for the simpler isentropic pressure-
density relation
P(O)--ao • a>O, y~>l, (5.2)
and possible generalizations discussed afterwards.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-332-320.jpg)
![334 E. Feireisl
The temperature 0 being eliminated from the pressure constitutive law, the system (2.1)-
(2.3) reduces to
O0
Ot + div(ou) = 0, (5.3)
OOu
Ot + div(ou | u) + Vp(o) =/zAu + (Z +/.t)V(divu) + of. (5.4)
The spatial variable x will belong to a regular boundeddomain ~2 C RN, N = 2, 3, and
the velocity u will satisfy the no-slip boundary conditions
ulas2 =0. (5.5)
Taking (formally) the scalar product of (5.4) with u and integrating by parts we obtain
the energy inequality:
d fs E(t) dx -+-fs2lZlVul2
dt + 0~ +/z) Idivul 2dx ~<fr2 of. u dx, (5.6)
where the specific energy E satisfies (1.8). If p is given by (5.2), we have
1~ e + o log(0)
E=~
1 a
for 9/= 1, E- x01ul 2 + 0 •
y-1
z
if y> 1.
As already agreed on in Section 1, the fluids under consideration are viscous, i.e.,
/z>0 and )~+/z~>0.
Note that the restrictions imposed on )~allow for all physically relevant situations.
In what follows, we consider thefiniteenergyweaksolutionsof the problem (5.3)-(5.5)
on the set I x s more specifically, 0, u will meet the following set of conditions:
9 the density 0 and the velocity u satisfy
0~>0, o ~L~176L• (~)), u ~ L2(I; [W0'2(I2)]N);
9 the specific energy E belongs to L~oc(l; L1 (~)) and the energy inequality (5.6) holds
in D' (I), i.e.,
fl OtTr(fs~Edx) dt - f Tr(fs21zlvule+ ()~+lz)ldivulZdx)dt
~fTrfs2of'udxdt
holds for any function ~ E D(I), ~ >~0;](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-333-320.jpg)
![Viscous and~or heat conducting compressible fluids 335
9 the functions 0, u extended to be zero outside C2 solve the continuity equation (5.3)
in D'(I x R N) (cf. (2.4)); moreover, (5.3) is satisfied in the sense of renormalized
solutions, i.e., (3.2) holds for any b as in (3.3);
9 the equations of motion (5.4) are satisfied in 79'(1 x S-2)(cf. (2.5)).
As the reader will have noticed in Section 4, the value of the adiabatic constant V will
play an important role in the analysis. In most cases, we shall assume
V > N/2,
where N = 2, 3 are the physically relevant situations.
The external force density f is assumed to be a bounded and measurable function such
that
ess sup [f(t,x)[<~F.
tEI, xEU2
In what follows, we shall give an outline of the large data existence results in the class
of finite energy weak solutions. We shall also discuss the long-time behaviour and related
asymptotic problems. To this end, we pursue the classical scheme for solving nonlinear
problems:
9 First of all, we find a priori estimates, i.e., the bounds imposed formally on any
classical solution and depending only on the data (cf. Sections 5.1, 5.2).
9 Given a family of solutions satisfying the bounds induced by a priori estimates, we
examine the question of compactness, i.e., whether or not any accummulation point
of this family in suitable topologies is again a solution of the original problem (see
Sections 5.3, 5.4).
9 Finally, one has to find a suitable approximation scheme solvable, say, by a classical
fixed-point technique, and compatible with both the estimates and compactness
properties mentioned above (Section 5.5).
To conclude this introduction, let us note that any finite energy weak solution satisfies
0 E C(I; • L ~ ,
Lweak(S2)) (-1C(I; (.C2)) 1 <~ot < V,
2V
Qu E C(I" L•
, weak(n)) (5.7)
provided y > N/2 (cf. Proposition 3.1). In particular, the density and the momenta are
well defined at any specific time t E I. Moreover, the total mass
m -- fs~ 0 dx is independent of t E I; (5.8)
and the total energy E defined for any t E I by formula
g(t) - g[0, (0u)] (t) = [ 1 I(0u)l 2
Jo(t)>0 2 0
a
- ~ ( t ) + 0 • (t) dx (5.9)
y-1
is a lower semi-continuous function of t E I (see [27, Corollary 2]).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-334-320.jpg)
.
t--+inf{l}+
The bound (5.10), which can be easily obtained combining the energy inequality (5.6)
and the Gronwall lemma, can be viewed as an a priori estimate though it holds for any
finite energy weak solution of the problem. It is not difficult to see that similar results can
be derived provided p is given by a general constitutive relation (5.1) and satisfies suitable
growth conditions for large values of the density. On the other hand, as already mentioned
in Section 2.5, uniform a priori estimates of Q seems to be out of reach of the standard
techniques and represent a major open problem of the present theory.
5.2. Pressure estimates for isentropic flows
By virtue of (5.10), the isentropic pressure p(Q) belongs automatically to the set L 1 (I x 12)
at least for bounded time intervals I. On the other hand, the weak compactness results like
Proposition 4.3 require p in a weakly complete (reflexive) space L r (I x ~2) with r > 1.
Such a bound is indeed available as the following result shows:
PROPOSITION 5.2. Assume S-2 C R N, N >~2, is a bounded Lipschitz domain. Let O, u be
a finite energy weak solution to the problem (5.3)-(5.5) on I x I-2 where the isentropic
pressure p is given by (5.2) with y > N/2. Let
0<rl<min ~,~ --~--1
be given. Denote by m = fs~ Qdx the (conserved) total mass and let F = ess sup/• if[.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-335-320.jpg)
![Viscous and~orheat conducting compressiblefluids 337
Thenfor any bounded time interval J C I, we have
V+I
fJf~ OY+rJdxdt<'c(m'F'rl'lJl)(1-+-sup'Y(t)) •
" t e J (5.11,
A local version of the above estimates was obtained by Lions [62]. In fact, the bounds
on r/in Proposition 5.2 are not optimal. Similarly as in the local case (see [62]), one could
verify the best values for r/:
2
O<r/~y-1.
However, for further purposes, it is convenient to have the integrals containing 07 bounded
in terms of the total mass m equivalent to the Ll-norm rather than the total energy s
proportional to the L • of O.
The validity of (5.11) up to the boundary of I-2 was proved in [39] by means of a
multiplier technique. More specifically, the main idea is to take the quantities
~i(t,x) -- 1/e(t)~i[b~e * Tk(~Or/)], i -- 1 ..... N,
as test functions for (2.5). Here
c D(1), 0~< ~ ~< 1, Z lOtaPldt <<.2,
O~(x) is a regularizing sequence as in Lemma 3.1, Tk are the cut-off operators introduced
in (4.5), and, most importantly, the symbol B stands for an inverse of the div operator, i.e.,
v = B[g] solves the equation
div v -- g - ~ g dx, vlos2 =0. (5.12)
Equation (5.12) has been studied by many authors. Here, we have adopted the approach
which is essentially due to Bogovskii [8]. It can be shown that the problem (5.12) admits a
solutions operator B: g ~ v enjoying the following properties:
9 B- [B1 ..... BN] is a bounded linear operator from LP(S2) into [w;'P(s2)] N,
specifically,
c(p)llglltp(~) for any 1 < p < ec.
9 The function v = B[g] solves the problem (5.12).
9 If, moreover, g ~ LP(S-2) can be written in the form g =divh where h 6 [Lr(S-2)]N,
h- n = 0 on 0s then
[ILr( c(p,r) llhllLr( ).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-336-320.jpg)
![338 E.Feireisl
The proof of the existence of the operator/3 as well as the above properties can be found
in Galdi [42] or Borchers and Sohr [9].
An alternative approach to show (5.11) based on properties of the Stokes operator was
proposed by Lions [63].
The estimates (5.8), (5.10), (5.11) are the only available for the problem (5.3)-
(5.5) unless some smallness assumptions are imposed on the data, i.e., on go, m,
and f. Thus in accordance with Proposition 4.2, the main stumbling block to show
compactness of solutions is the pressure term. Indeed the above estimates guarantee
only weak compactness of the density 0 while for p = P(O) to be compact we need
strong compactness of 0 in L p. As we shall see later, the way out this vicious circle is
provided by Proposition 4.3, namely, by the compactness properties of the effective viscous
flux.
5.3. Density oscillations for barotropic flows
We show how Proposition 4.3 can be used to describe the amplitude of density oscillations
for barotropic flows discussed in Section 4.1.
In addition to (5.1), we suppose
p(O) = O, P(O) ~ 0 for 0 i> O, p convex on [0, co).
It is easy to verify that
p(y)-p(z))p(y-z) for all0~<z~<y
which yields immediately
(p(y) - p(z))(Tk(y) -- Tk(z))
) P(lT~(y)- Z (z l)I rk(z~] for all y, z ~>0. (5.13)
Under the hypotheses (and notation) of Proposition 4.3, we have p = P(O) and we can
write
lim fQ P(On)Tk(On) - P(O) Tk(o) dxdt
n--'+O0
-- nlirn fQ (P(On) - P(O))(Tk(On) -- Tk(o)) dx at
+ fQ (P(O) - P(O))(T~(o) - Tk(o) ) dx at (5.14)
for any bounded Q c I x I2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-337-320.jpg)
 intro-
duced in Section 4.1. In particular, the following result holds (see [29, Proposition 6.1]).
PROPOSITION 5.3. Let S-2 C R N, N ~ 2 be a bounded Lipschitz domain and I C R a
bounded interval. Let the pressure p be given by the formula (5.1), p(O) = O, p convex,
and
P(O) ~ ao • a > O, Y > N/2.
Let On, Un be a sequence of finite energy weak solutions of the problem (5.3)-(5.5) with
f = fn and such that
mn--f On<~m,
limsup C[On,(OnUn)](t)<~CO,
t-+inf{l }+
and
ess sup If~l~ F
1•163
independently of n.
Then
osc• - o](Q) ~<c(Q)(sup Ildivu, IIL2(Q~)1/y
"n>/1](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-338-320.jpg)
![340 E. Feireisl
for any weak limit O of the sequence On and any bounded Q c I x 12.
As a straightforward consequence of Propositions 4.1, 5.3 we get the following:
COROLLARY 5.1. Under the hypotheses of Proposition 5.3, let
On ---§ O
Un --~U
weakly star in L ~ (I; L • (I-2)),
weakly in L2(I; W1'2($2)).
Then O, u solve (5.3) in the sense of renormalized solutions, i.e., Equation (3.2) holds
for any b satisfying (3.3).
5.4. Propagation of oscillations
For simplicity, we suppose the pressure p is given by the isentropic constitutive
relation (5.2) with y > N/2. Similarly as in Proposition 5.3, let On, Un be a sequence
of finite energy weak solutions of (5.3)-(5.5) on some bounded time interval I such that
limsup $[0~, (O~u~)] ~ Co,
t---~inf{l}+
Ilfn []L~176
• ~ F
uniformly in n.
The issue we want to address now is the time propagation of oscillations in the density
component. To begin with, it seems worth-observing that any reasonable solution operator
we could associate with the finite energy weak solutions cannot be compact with respect
to O-This is due to the hyperbolic character of the continuity equation (5.3). In accordance
with the observations made by Lions [60], the oscillations should propagate in time.
Serre [92] studied this phenomenon and showed the amplitude of the Young measures
associated to the sequence On(t) is a non-increasing function of time. His proof is complete
in the dimension N -- 1 and formal for N >~2 taking the conclusion of Proposition 4.3 for
granted. Having proved Proposition 4.3 Lions [62] completed the proof for N ~>2. The
fact that oscillations cannot be created in On unless they were present initially plays the
crucial role in the existence theory developed in [62].
Here we go a step further by showing that the amplitude of possible oscillations decays
with time at uniform rate depending solely on the value of the initial energy g0 (see [37]). In
particular, the time images of bounded energy initial data are asymptotically compact with
respect to the density component. This is precisely what is needed to develop a meaningful
dynamical systems theory associated to the problem.
In accordance with our hypotheses, we can show
On "-+ O in C(I; •
Lweak(~Q)) ,
Tk(On) ~ Tk(O) in C(I; L~weak(f2))for any a t> 1, k/> 1,
Un --~ u weakly in L2(I; Wo'2(ff2)).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-339-320.jpg)
 -- fs? v(t, x) dx, where v - 0 log(0) - 0 log(0). (5.17)
By virtue of Corollary 5.1, both On and 0 are renormalized solutions of (5.3) and we
have
; ot 5"2 ,
Onlog(On)-~ol~ inC(J Lweak()) l~<ct<y,
0 log(0) E C(J; Lweak~ (if2)) , 1 ~<ot < 9/.
Consequently, dft[0n - 0] is a continuous function of t 6 I.
Mainly for technical reasons, we are not able to deal directly with the function dft. We
consider instead a family of approximate functions:
z log(z)
Lk(z) -- ok
Tk(s)/s e
Zlog(k) + z J~" ds
forO ~<z ~<k,
forz ) k.
It is easy to observe that Lk(z) -- ~kZ + bk(z) where bk satisfy (3.3) and
L'k(z)z- L~(z)= T~(z).
Since both On, ~oare renormalized solutions of (5.3) on I x R 3, we deduce
fs2(Lk(On) -- Lk(O))(t2) dx - Js2 (Lk(On) -- Lk(o))(tl) dx
f
f
-- Tk (0) div u - Tk(0) div Un dx dt
+ (Tk(O) -- Tk(On)) divun dx dt
for any t~, t2 E I.
Letting n ~ oo and using Proposition 4.3 together with (5.15), we obtain
fn (Lk(O) -- Lk(o))(t2) dx - fn (Lk(o) - Lk(o)) (tl) dx
a lim sup ITk(On) -- Tk(o)l • dx dt
t X+21z n~oc
<~ (Tk(O) -- Tk(O)) divudx dt for any tl ~ t2, tl, t2 e I. (5.18)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-340-320.jpg)
 for k --+ oo
while
II o1(1 x n))
y-1
2y
By virtue of Proposition 5.3, the right-hand side of the above inequality tends to zero for
k --+ oo and so does the right-hand side of (5.18).
Finally, we have
lim sup [Tk(On)- Tk(0)1• dx dt
/7-+00
ot-y+l f lt2
>~ 1$21 '~ limsup IIT (On)- r*(0)ll '+'
L~(n) dx dt,
/7-----~(x)
and (5.18) yields:
dft[o/7 -- 0](t2) --dft[On -- 0](tl)
( ~-~+') f'~ •
+ all2[)~
+ 2#
~ limsUPn____~oo [[0n - 0liLt(n) dx dt ~<0 (5.19)
for any tl ~<t2, tl, t2 6 I and 1 ~<c~ < y.
To conclude, we shall need the following auxiliary result (cf. [28, Lemma 2.1]).
LEMMA 5.1. Given ~ E (1, y) there exists c = c(a) such that
z log(z) - y log(y) ~< (1 + log+(y))(z - y)+ c(ot)(lz - y11/2 + [z - yl ~)
for any y, z ~ O.
In accordance with Lemma 5.1, we can write
n Onl~ - fn (1 + log+(o))(On - o) dx
2~- 1
~<c(cO (IS21-~ lion - oIIZL
~ '~
(n) + lion - ollL,~(n))
which together with (5.19) yields
L
t2
dft[0/7 - O](t2) -dft[0/7 - 0](tl) + q~(dft[0/7 - 0](t))dt ~<0,](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-341-320.jpg)
<~s
t ~ inf{ I }+
IlfnIIL~(Z~n) ~ F
independently of n. Let O be a weak limit of the sequence On.
Then
dft[On - O](t2) <~X(t2 - tl) for any tl, t2 9 I, tl ~ t2,
where X is the unique solution of the initial-value problem
X'(t) + q~(X (t)) - 0, x(O)=dft[Qn-Q](tl)
and ci9 is a fixed function satisfying (5.20).
It can be shown that q~ has a polynomial growth for values close to zero and,
consequently, the quantity dft[on - O](t) behaves like t -# for a certain fi > 0 when
t ---~ ~.
5.5. Approximate solutions
The a priori estimates derived in Sections 5.1, 5.2 together with the compactness results
in Propositions 4.2, 5.4 form a suitable platform for a larga data existence theory for the
problem (5.3)-(5.5). The final task, as usual, is to find a suitable approximation scheme
compatible with both a priori estimates and the compactness results claimed above.
Needless to say there are many ways to do it. Here we pursue the approach of [32] and
consider the approximate problem:
OQ
Ot
-F div(Qu) = e AQ, (5.21)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-342-320.jpg)
![344 E, Feireisl
~0u
+ div(ou | u) + Vp(0) + ~V0 ~ + ~Vu. V0
8t
=/xAu + (~ + #)V divu + 0f (5.22)
complemented by the boundary conditions
ulos2 : XT09nlos2:0. (5.23)
The parameters e > 0, ~ > 0 are "small" and fl > 0 "large". The system (5.21)-(5.23)
can be solved by means of the standard Faedo-Galerkin method to obtain approximate
solutions 0e,~, ue,~ (cf. [32, Proposition 2.1]). Then one can pass to the limit, first for
e ~ 0 and then for ~ ~ 0, to obtain a finite energy weak solution of the problem (5.3)-
(5.5) (see [32]).
The reason for introducing two parameters e and ~ is that the energy estimates presented
in Section 5.1 and the pressure estimates in Section 5.2 are compatible only if fl > N.
An alternative approach is the approximation scheme introduced by Lions [62] or the
method of time-discretization based on solving a family of stationary problems (see also
Lions [62]).
6. Barotropic flows: large data existence results
The mathematical theory presented in Section 5 can be used to obtain rigorous existence
results for barotropic flows with essentially no restriction on the size of the data. We
start with a very particular case posed in two space dimension where one can show even
existence of strong (classical) solutions.
6.1. Global existence of classical solutions
The result we are going to present is due to Vaigant and Kazhikhov [107]. Consider the
system
Ot
+ div(0u) = 0, (6.1)
O(ou)
~t
+ div(0u | u) + aVo• =/zAu + V(0~(O) + #)divu), (6,2)
where (t,x) a (0, T) x R2. The functions 0, u are for simplicity considered spatially
periodic, i.e.,
O(t, x 4- o9) = O(t, x), u(t, x 4- o9) = u(t, x). (6.3)
The problem is complemented by the initial conditions
0(0, x) = Oo(x) ~>0 > 0, u(0, x) -- u0(x). (6.4)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-343-320.jpg)
![Viscous and~orheat conducting compressible fluids 345
Under the hypotheses
/z>O, a>O, y>~O, and )~(o)--bo ~, b>O, /3>~3, (6.5)
Vaigant and Kazhikhov [107] proved the following result.
THEOREM 6.1. In addition to the hypotheses (6.5), let
00 E LpC~r(R2), UO E Wle2(R2).
Then the initial-value problem (6.1)-(6.4) possesses a global (T = oo) weak solution.
The continuity equation (6.1) holds in 79f((0, T) • R2) and the equations of motion (6.2)
are satisfied a.a. on (0, T) x R 2.
If, moreover,
00 E Wlerq (R2), U0 E Wp2e
q (e 2) for some q > 2,
then there is a unique strong solution satisfying the equations a.e. on (0, T) • R2.
Finally, if
(7l+ce
00 E vper (R2) , (-72+ot
u0 E vper (R 2) for some ot > O,
then the strong solution is classical (smooth).
Theorem 6.1 is a remarkable result since it solves both the problem of existence and
uniqueness as well as regularity of solutions. The obvious restrictions of applicability
are due to the rather unnatural hypotheses (6.5), i.e., the viscosity coefficient # must be
constant while )~ depends on 0 in a very specific way.
The proof of Theorem 6.1 is based on very strong a priori estimates - much better than
presented in Sections 5.1, 5.2. These estimates are available thanks to the particular form
of the constitutive relations and the fact the problem is posed in two space dimensions.
6.2. Global existence of weak solutions
We consider the problem (5.3)-(5.5) posed on a bounded regular domain 12 C R N,
N = 2, 3. We prescribe the initial conditions
p(O) = Oo >/O, (pu)(O) = qo (6.6)
satisfying a compatibility condition
q0(x) = 0 whenever Q0(x) = 0 (6.7)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-344-320.jpg)
![346 E. Feireisl
and such that
00 6 L•163 Iq~ 6 Ll(~). (6.8)
00
The assumption (6.8) is nothing else but the requirement the initial data to be of finite
energy.
The following theorem asserts the existence of the finite energy weak solutions to the
problem (5.3)-(5.5), (6.6) introduced in Section 5.
THEOREM 6.2. Let s C R N, N -- 2, 3, be a bounded regular domain and T > 0 given.
Consider the system (5.3), (5.4) complemented by (5.5), (6.6), where p is given by the
isentropic constitutive law (5.2) with
N
Y>2'
f is a bounded measurable function on (0, T) x X-2,and the initial data 00, q0 satisfy (6.7),
(6.8).
Then the problem (5.3)-(5.5) posseses a finite energy weak solution O, u on (0, T) x s
satisfying the initial conditions (6.6).
As already remarked in (5.7), both the density 0 and the momenta (0u) are continuous
functions of t with respect to the L P-weak topology, and, consequently, the initial
conditions (6.6) make sense.
The existence result stated in Theorem 6.2 was first proved by Lions [62] for y ~>3/2
if N = 2 and y ~>9/5 for N = 3. The proof needs some modifications presented in [39]
and [63] to accommodate the Dirichlet boundary conditions. The present version including
the full range of y > N/2 was shown in [32, Theorem 1.1].
Given the weak compactness results, namely Propositions 4.1, 4.2, for solutions
of (5.3), (5.4) respectively, the main ingredient of the proof of Theorem 6.2 is the strong
compactness of the density stated in Proposition 5.4, the proof of which requires, among
other things, convexity of the pressure. It is easy to see, however, that the same result can
be obtained for a general barotropic pressure p as in (5.1) that can be written in the form
P(O) =ao • + po(O), a>0, y>N/2,
where P0 is a globally Lipschitz function. Note that the proof for the range ?, ~> 9/5,
N = 3, t' ~>3/2, N = 2 can be modified to include a general constitutive law (5.1) where
P(O) ~ ao • for all 0 large enough (cf. Lions [62]).
It seems interesting to note that the physically relevant isothermal case where y = 1
seems to be completely open even if N = 2. The only large data existence result is that
of Hoff [46] where the initial data (as well as the solutions) are radially symmetric. The
general case g ~> 1, N = 3 for radially symmetric data was solved only recently by Jiang
and Zhang [55].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-345-320.jpg)
![Viscous and~orheat conducting compressiblefluids 347
6.3. Time-periodic solutions
Similarly as above, we consider the system (5.3)-(5.5) driven by a volume force f which is
periodic in time, i.e., f is a bounded measurable vector function on R • I2 satisfying
f(t + co, x) = f(t, x) for a.a. t 6 R, x E s
for a certain period co > 0. We are interested in the existence of a finite energy weak
solution O, u enjoying the same property, i.e.,
O(t + co) = 0(t), (0u)(t + co) = (0u)(t) for all t E R
and such that
]~0dx =m,
where m is a given positive total mass.
There are three main obstacles making this problem rather delicate. Given the existence
results for the initial-boundary value problem presented above, only weak solutions are
available, for which the question of uniqueness is highly nontrivial and far from being
solved. This excludes all the so-called indirect methods based on fixed-point arguments
for the corresponding period map. While the former difficulty might seem only technical,
there is another feature of the problem, mentioned already in Section 5.4 namely, there
is no "solution operator" or "period map" which would be compact due to possible time
propagation of oscillations in the density. Last but not the least, fixing the total mass m, we
have to look for solutions lying on a sphere in the space L 1 which excludes the possibility
of using any fixed-point technique in a direct fashion.
In the light of the above arguments, the only possibility to get positive results is to work
directly in the space of periodic solutions that means to consider a genuine boundary-value
problem for the evolutionary system (5.3), (5.4). This approach has been used in [31]
to prove the existence of the time periodic solutions to (5.3), (5.4) on a cube in R3
complemented by the no-stick boundary conditions (1.12). Combining the method of [31]
with the existence theory [32] one can prove the following result.
THEOREM 6.3. Let ~ C R N, N = 2, 3, be a bounded regular domain. Consider the
problem (5.3)-(5.5) where p is given by the isentropic constitutive law (5.2) with
y>5/3 if N = 3, y>l for N = 2,
and f is a bounded measurable function on R x I2 such that
f(t + co, x) = f(t, x) for a.a. t E R, x E I-2 and a certain co > O.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-346-320.jpg)
![348 E. Feireisl
Then, given m > 0, there exists a finite energy weak solution O, u of (5.3)-(5.5) on
R x ~ such that
O(t + co) = O(t), (Ou)(t + w) = (0u)(t) for all t ~ R
and
fs odx =m.
The condition y > 5/3 in the three-dimensional case seems rather strange compared
with y > 3/2 required for solving the initial-value problem. This is related to the problem
of ultimate boundedness or resonance phenomena for global in time solutions. We will
discuss this interesting topic in the next section.
6.4. Counter-examples to global existence
It is not clear to which extent the hypothesis y > N/2 is really necessary for global
existence results. Several attempts have been made to show that the barotropic model does
not admit globally defined strong or even weak solutions but the results are still not very
convincing in either positive or negative sense.
Following the method of Vaigant [106], Desjardins [19, Proposition 1] studied the
integrability properties of the density Q in the system (5.3)-(5.5).
PROPOSITION 6.1. Let ~2 = B(1) C R 3 be a unit ball and let p satisfy (5.2) with
1 < y <3. Let
q>
lly -2
2y
Then there exist f 6 L 1(0, T; L 7=-r(S2)) and a globally defined weak solution ~, u of
(5.3)-(5.5) such that
1~(/, x)I q dx dt = cx~.
The weakness of this result stems from the necessity to use the forcing term f which is
singular at t = T. It is still an open problem whether or not the uniform upper bounds on
the density can be obtained independently of the choice of y.
6.5. Possible generalization
We shall comment shortly on possible improvements of Theorem 6.1 lying in the scope of
the present theory.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-347-320.jpg)
![Viscous and~orheat conducting compressiblefluids 349
To begin with, Theorem 6.1 still holds when 12 is a general (not necessarily bounded)
domain with compact boundary on which the no-slip boundary conditions for the velocity
are prescribed. As far as the other boundary conditions discussed in Section 1.4 are
concerned, the possibility to show positive existence results seems to be closely related
to the question of the boundary estimates of the pressure discussed in Section 5.2.
Similarly, the hypothesis that f is bounded can be replaced by a more general condition
2V
f 6 Ll(I; L• (l-2)).
Other possibilities and suggestions are discussed by Lions [62].
7. Barotropic flows: asymptotic properties
Similarly as in the preceding section, we focus on the system (5.2)-(5.5) considered on a
bounded regular domain S-2C R N, N = 2, 3. We shall assume that the driving force f is a
bounded measurable function defined, for simplicity, for all t 6 R, x 6 12 such that
[f(t,x)[~<F fora.a, t6R, x6S-2. (7.1)
In accordance with Section 5, the total energy defined as
fo 1 IQul 2
S[O, 0u](t) -
(t)>0 2 0
a
- ~(t) + 0 • (t) dx
y-1
is a lower-semicontinuous function of t.
7.1. Bounded absorbing balls and stationary solutions
We shall address the problem of ultimate boundedness of global in time finite energy weak
solutions, the existence of which is guaranteed by Theorem 6.2. We shall see that the
total energy S is the right quantity to play the role of a "norm" in these considerations.
If the driving force f is uniformly bounded as in (7.1), the "dynamical system" generated
by the finite energy weak solutions of the problem (5.3)-(5.5) is ultimately bounded or
dissipative in the sense of Levinson with respect to the energy "norm" provided that the
adiabatic constant satisfies 9/> 1 for N = 2 and y > 5/3 if N = 3. Specifically, we report
the following result (see [38, Theorem 1.1]), the proof of which is based on the pressure
estimates obtained in Proposition 5.2:
THEOREM 7.1. Let 12 C R N, N = 2, 3, be a bounded Lipschitz domain and I C R
an interval such that inf{I} > -o~. Consider the system (5.3)-(5.5) with the isentropic
pressure p given by (5.2) with
y > 1 ifN=2, y >5/3 forN=3, (7.2)
and f satisfying (7.1).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-348-320.jpg)
 ~<Coc for all t E I, t > r + inf{I}
for any O, u - a finite energy weak solution of the problem (5.3)-(5.5) - satisfying
lim sup C[O,u](t) ~<Co,
t~inf{1}+
fs2 Qdx - m.
It seems interesting to compare Theorem 7.1 with the result of Lions [62, Theorem 6.7]
on the existence of stationary solutions of (5.3)-(5.5) to shed some light on the role of the
hypothesis (7.2).
THEOREM 7.2. Let s C R N, N = 2, 3, be a bounded regular domain, f = f(x) a
function belonging to Lc~(I-2), and m > O. Assume p = P(O) is given by (5.2) with y
satisfying (7.2).
Then there exists a pair of functions Q - Q(x) E LP(~), p > y, u = u(x) E wl'2(12)
solving the stationary problem
div(ou) = O,
div(ou | u) + aVo • =/zAu + ()~+ #)V divu + of,
f s2 Qdx = m
in 79'(I2).
As we will see later, Theorem 7.2 can be deduced from Proposition 5.4, Theorem 6.3,
and Theorem 7.1.
The property stated in Theorem 7.1 is evidence of the dissipative nature of the
system (5.3), (5.4). In finite-dimensional setting, J.E. Billoti and J.P. LaSalle proposed it as
a definition of dissipativity. Unfortunately, however, some difficulties inherent to infinite-
dimensional dynamical systems make it, in that case, less appropriate.
7.2. Complete bounded trajectories
~(R;L ~
We suppose that the driving force f belongs to ~-- a bounded subset of Lloc (S'2)).
To bypass the possible problem of non-uniqueness of finite energy weak solutions, we
introduce a quantity U(t0, t) playing the role of the evolution operator related to the](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-349-320.jpg)
![Viscous and~or heat conducting compressible fluids 351
problem (5.3)-(5.5).
[
U[s t) = [ [O(t), (Ou)(t)] [O, u is a finite energy
~](t0, weak solution
of the problem (5.3)-(5.5) defined on an open interval I,
(to, t]CI, with f E U
and such that limsupg[o, u](t) ~<go}.
t--->to+
We start with the concept of the so-called short trajectory in the spirit of M~ilek and
Neeas [67]:
US[Co, f'](to, t) = {[Q(t + v), (Qu)(t + v)], v e [0, 1]IQ, u is a finite energy
weak solution of the problem (5.3)-(5.5) on an
open interval I, (to, t + 1] C I,
with f e ~-, and such that limsups ~<go[.
t--+to /
The following result can be viewed as a corollary of Proposition 5.4 and Theorem 7.1
(cf. [37, Theorem 1.1] or [27, Proposition 10]).
PROPOSITION 7.1. Let $'2 C R N, N = 2, 3, be a bounded domain with Lipschitz
boundary. Let the pressure p be given by (5.2) with
y > l for N = 2, Y>5/3 if N = 3.
Let .~ be bounded in L~ x $-2). Consider a sequence [On, (OnUn)] E US[Co, .T'](a, tn)
for a certain tn --+ ec.
Then there is a subsequence (not relabeled) such that
On -+ 0 in L • ((0, 1) x S'2) and in C([O, 1]; L~(I-2)) for 1 <<.ot < y,
~o,,u~ ~ (Ou)
2y
, L y+l 2V
in LP((O, 1) • $2) andin C([0, 1]" weak(S2))forany 1 ~ p < -7~' and
cO[On, (OnUn)] -+ s (Ou)] in L 1(0, 1),
where Q, u is a finite energy weak solution of the problem (5.3)-(5.5) defined on the whole
real line I = R such that s e L ~ (R) and with f e ~+ where
U+ = {f If = lim hn (" + rn) weak star in L ~ (R • S-2)
~-n---->~
for a certain hn e ~ and rn --+ ec }.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-350-320.jpg)
![352 E. Feireisl
Proposition 7.1 shows the importance of the complete bounded trajectories, i.e., the finite
energy weak solutions defined on I = R whose total energy s is uniformly bounded on R.
Let us define
Asia] : {[~)(r), (•u)(r)], r ~ [0, 11 I~),u is a finite energy weak solution
of the problem (5.3)-(5.5) on the interval I = R,
with f 6 .T"+ and s (Qu)] E L ~ (R)}.
The next statement is a straighforward consequence of Proposition 7.1 (see also [28,
Theorem 3.1 ]).
THEOREM 7.3. Let 12 C R N be a bounded Lipschitz domain. Let p be given by (5.2) with
y > l for N = 2, y>5/3 if N = 3.
Let ~ be a bounded subset of L e~(R x 12).
Then the set As[f] is compactin L• 1) x 12) x [LP((0, 1) x 12)]3 and
[ inf (11~ -
sup ~ Ilt•215
[O,Ou]EU[s I_[~,~fi ]~.As [.T"]
+ II - 0 as t --+ oc
for any 1 <<.p < 2y/(y + 1).
Theorem 7.3 says that the set As[f] is a global attractor on the space of "short"
trajectories. This is a result in the spirit of Mdlek and Ne~as [67] or Sell [89].
In particular, the set A s[,T'] is compact non-empty provided ~ is non-empty. Consider
the special case when f = f(x) is a driving force independent of time. Accordingly, we can
take
y = ~+ = {f}.
By virtue of Theorem 6.3, the problem (5.3)-(5.5) possesses a time-periodic solution ~)n,
Un for any period ~On= 2-n such that
f s2 Qn dx = m.
Moreover, Theorem 7.1 implies that the restriction 0n, OnUn to the time interval [0, 1]
belongs to .As . As .As is compact, the sequence On, un has an accummulation point which
is a complete global solution of (5.3)-(5.5). Moreover, this solution is clearly independent
of t, i.e., it is a stationary solution of a given total mass m. In other words, we have proved
Theorem 7.2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-351-320.jpg)
![Viscousand~orheatconductingcompressiblefluids 353
7.3. Potentialflows
We shall examine the flows driven by a potential force, i.e., we assume
f = f(x) -- VF(x),
where F is a Lipschitz continuous function.
In this case, the term on the right-hand side of the energy inequality (5.6) can be rewritten
as
f~ d~
(Ou).VFdx= dt' where ~ (t) = fs~ 0 F dx,
and, consequently, (5.6) reads as follows:
d(g(t)dt - 7-[(t)) + fs~~lVul2 + (~ + 2/z)ldivu[2 dx dt ~<0. (7.3)
We denote
s = ess lim [s - 7-t(t)].
t---~~
By virtue of (7.3) and the Poincar6 inequality, the integral
f~ Ilull2~,2(s~)dt is convergent, (7.4)
in particular,
fT T+I
lim ~kin(t) dt= 0,
T--+ oo
and
&in -- ~ 0[u] 2dx,
f T+Ifs a Oz
lim -- 0 F dx dt= gT-/oc. (7.5)
r~ec y- 1
Similarly as in Proposition 7.1, one can show that any sequence tn ~ oo contains a
subsequence such that
ftntn+l II0(t) - 0, IIL• --+ 0,
where, in view of (7.4), (7.5), Os is a solution of the stationary problem
aVOZs = OsVF in X2, fs? 0s dx = m,
f~ a
y 1
Ozs - OsF dx = g~.
(7.6)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-352-320.jpg)
![354 E. Feireisl
Consequently, it is of interest to study the structure of the set of the static solutions, i.e.,
the solutions of the problem (7.6); in particular, whether or not they form a discrete set.
If this is the case, any finite energy weak solution of (5.3)-(5.5) is convergent to a static
state. A partial answer was obtained in the case of potentials with at most two "peaks" ([36,
Theorem 1.1] and [33, Theorem 1.2]).
THEOREM 7.4. Let I-2 C R N be an arbitrary domain.
(i) Assume F is locally Lipschitz continuous on S2 and such that all the upper level sets
[F > k] = {x 9 t"2 I F(x) > k}
are connected in f2 for any k.
Then given m > O, the problem (7.6) possesses at most one nonnegative solution Qs.
(ii) If F is locally Lipschitz continuous and I-2 can be decomposed as
~Q -- ff21Uf22, /21N f22 -- ~,
where if2 i are two subdomains (one of them possibly empty) so that
[F > k] (q if2 i is connected in ~(2i for i = 1, 2for any k E R, (7.7)
then, given m, s the problem (7.6) admits at most two distinct non-negative solutions.
Making use of Theorem 7.4, one can show the following result on stabilization of global
solutions for potential flows (cf. [34, Theorem 1.1], [27, Theorem 15]).
THEOREM 7.5. Let S2 C R N, N = 2, 3, be a bounded Lipschitz domain. Let the pressure
p satisfy the constitutive relation (5.2) with y > N/2. Let f = fix) = VF(x) where F
is globally Lipschitz potential on S-2.Moreover, assume that S2 can be decomposed as in
Theorem 7.4 so that (7.7) holds.
Then for any finite energy weak solution O, u of the problem (5.3)-(5.5) defined on a
time interval I = (to, oo), there exists a solution Os of the stationary problem (7.6) such
that
Q(t) --+ Qs strongly in L y (I2) as t --+ c~,
fs2 1 fo I~u12
Ekin(t) dx-- ~ (t)>0 Q
~ (t) dx --+0 as t --+ cx~.
The conclusion of Theorem 7.5 still holds if Y2 is a general (not necessarily
bounded) domain with compact boundary provided F satisfies the stronger hypothesis of
Theorem 7.4, namely, all upper level sets [F > k] must be connected. Similar problems
on the exterior of an open ball and for radially symmetric solutions were investigated
by Matu~fi-Ne~asovfi et al. [76]. Related results can be found in Novotn~ and Stra~kraba
[83,84].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-353-320.jpg)
![Viscous and~or heat conducting compressible fluids 355
It is an interesting open problem if the conclusion of Theorem 7.5 still holds when the
hypothesis on the upper level sets of F is relaxed. If ~2 C R N is a bounded domain and
the potential F nontrivial (nonconstant), there always exists an m - the total mass - small
in comparison with F such that the solutions of the static problem (7.6) contain vacuum
zones (cf. [34, Section 5]). Thus for any nonconstant F the global solutions approach
rest states with vacuum regions as time goes to infinity. One should note in this context
there are many formal results on convergence of isentropic flows to a stationary state under
various hypotheses including uniform (in time) boundedness of the density away from zero
(see, e.g., Padula [86]). As we have just observed, this can be rigorously verified only for
solutions representing small perturbations of strictly positive rest states.
7.4. Highly oscillating external forces
There seems to be a common belief that highly oscillating driving forces of zero integral
mean do not influence the long-time dynamics of dissipative systems. Averaging a function
over a short time interval should be considered analogous to making a macroscopic
measurement in a physical experiment. The result of such an experiment being close to
zero, the effect on the solutions to a sufficiently robust dynamical systems, if any, should
be negligible at least in the long run. From the mathematical point of view, these ideas
have been made precise by Chepyzhov and Vishik [10] dealing with trajectory attractors
of evolution equations. They showed that the trajectory attractors of certain dissipative
dynamical systems perturbed by a highly oscillating forcing term are the same as for the
unperturbed system. Their results apply to a vast set of equations including the damped
wave equations and the Navier-Stokes equations for incompressible fluids. Our goal now
is to present comparable results for the problem of isentropic compressible flows dynamics.
Highly oscillating sequences converge in the weak topology, i.e., the topology of
convergence of integral means. Consider a ball Bc of radius G centered at zero in the
space L ~ ((0, 1) x 12). The weak-star topology on BG is metrizable and we denote the
corresponding metric de. We report the following result (see [30, Theorem 1.2]).
THEOREM 7.6. Assume S-2 C R N, N = 2, 3 is a bounded Lipschitz domain. Consider the
system (5.3)-(5.5) where the pressure p is given by (5.2) with
Y> 1 for N = 2, y>5/3 if N = 3,
and
f(t, x) = VF(x) + g(t, x),
where F is globally Lipschitz continuous and such that the upper level sets [F > k] are
connected for any k.
Then given G > O, e > 0 there exists 6 = 6(G, e) > 0 such that
limsup[ll0(t)- QsIlL• [Iou(t)II,, s2 ] <
l----~OO](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-354-320.jpg)
![356 E. Feireisl
for any finite energy weak solution Q, u of the problem (5.3)-(5.5) provided
lim sup I[g(t)1[ L~(~, ~)• < G,
/---+OO
lim sup dG[g(t + s)ls~[0,1], 0] < 6.
l---->OO
Here Qs is the unique solution of the stationary problem (7.6).
7.5. Attractors
For a general dynamical system a set ,A is called a global attractor if it is compact,
attracting all trajectories, and minimal in the sense of inclusion in the class of sets having
the first two properties. The theory of attractors for incompressible flows is well developed.
We refer the reader to the monographs of Babin and Vishik [5], Hale [44], and Temam [104]
for this interesting subject. A global or universal attractor describes all possible dynamics
of a given system, and, as an aspect of dissipativity, the attractor usually has a finite fractal
dimension.
There seems to be at least one essential problem to develop a sensible dynamical systems
theory for compressible fluids, namely, the finite energy weak solutions we deal with are
not known to be uniquely determined by the initial data. On the other hand, the notion
of global attractor itself does not require uniqueness or even the existence of a "solution
semigroup" and plausible results in this respect can be obtained.
Let
r = {[Q(0), (Qu)(0)] IQ, u is a finite energy weak solution
of the problem (5.3)-(5.5) on I = R with f e 9
r+ and s E L~ (R) }.
Roughly speaking, the set .A contains all global and globally bounded trajectories where
global means defined on the whole time axis R.
The next statement shows that ,A[~] is a global attractor in the sense of Foias and
Temam [40] (cf. [28, Theorem 4.1]).
THEOREM 7.7. Let ~ C R N, N = 2, 3, be a bounded Lipschitz domain, and let p be
given by (5.2) with
y > l ifN=2, y>5/3 forN=3.
Let F be a bounded subset of L ~ (R x S2).
2y
L • (S2) and
Then r v] is compact in Lc~(S2) x weak
su, r
[Q,Qu] E U [Eo,.T'] (to,t) Lm~fi],A[7]
as t --+ cxz
2y
for any 1 <~~ < y and any ~ e [L • (~)]3
0o) 1)]](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-355-320.jpg)
![Viscous and~orheat conducting compressible fluids 357
The apparent shortcomming of this result is that A is only a "weak" attractor with respect
to the momentum component. Pursuing the idea of Ball [6], one can show a stronger result
on condition that some additional smoothness of A is known (see [27, Theorem 17]).
THEOREM 7.8. In addition to the hypotheses of Theorem 7.7, assume the total energy
C defined by (5.9) and considered as a function the density O and the momenta Ou is
(sequentially) continuous on A[F], specifically, for any sequence
[On, qn] e .A[f] such that On --~ 0 in Ll(I2), qn --~ q weakly in Ll(S-2)
one requires
C[On, qn] ~ E[O, q].
Then
[
sup inf (110 - ~llt~(~> + II0u- ~llt'(~))] 0
[O,Ou]EU[Co,F](to,t) t.[~,~fi ]E.A[.T']
as t --+ cx~.
7.6. Bibliographical remarks
The existence of global attractors for the problem (5.3)-(5.5) with 9/= 1 and N = 1 was
studied by Hoff and Ziane [50,51 ]. In this case, any forcing term f is of potential type so
the only situation which is not covered by Section 7.3 is the case when f is time dependent.
Similar results for the full system (1.1)-(1.3), still in one space dimension, were obtained
recently by Zheng and Qin [112].
8. Compressible-incompressible limits
It is well-accepted in fluid mechanics that one can derive formally incompressible models
as Navier-Stokes equations from compressible ones. Such a situation can be expected
when letting the Mach number go to zero in the isentropic compressible Navier-Stokes
equations. Following Lions and Masmoudi [64] we consider a system
OOe
Ot
+ div(oEue) = 0, (8.1)
80~ue
Ot
a
+ div(0eue | ue) + ~SV0ff = #eAue + ()~e+ #e)V divue (8.2)
complemented by the initial conditions
o~(o) - o ~ o, (o~u~)(o) - q~ (8.3)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-356-320.jpg)
![358 E. Feireisl
satisfying (6.7). We shall always assume
/zs --~ # > O, Xs --~ I. > -# as s --~ O.
8.1. The spatially periodic case
In addition to the above hypotheses, assume the initial data are spatially periodic as in (6.3).
Moreover, let
qs
U0 weakly in 2 N
-- Lper(R ) as s ~ 0, (8.4)
and
fo~Ol..,
f~ON
Iq~[2
oo
1 O(mO)z_l (mO)•
-t- ~-~[(0~ • - YOs + (y - 1) dx ~<c (8.5)
where
0
m e -- (.Oi
-1
f0 )1 f0 ~
9.. o~ 1 as s--+ 0
independently of s.
Let us denote, as usual, the total energy
f{o 1 lOsUs 12
a
nt- 62(y -- 1) (0s)• dx.
The following result is due to Lions and Masmoudi [64]"
THEOREM 8.1. Assume y > N/2. Let Os, us be a (spatially periodic) finite energy weak
solution of the problem (8.1)-(8.3) on the time interval (0, r where the data satisfy (8.4),
(8.5). Moreover, let
f0 O)1 f00)N1Iqs 12
ess sup g[0s, (0sus)] ~< ...
t--+0+ 2 0~
a
- 1) (o~
Then
0s ~ 1 in C([0, T]; L~er(R2))
and us is bounded in L2(0, T; Wpl~2(y2)) for arbitrary T > O.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-357-320.jpg)
![Viscous and~or heat conducting compressible fluids 359
Moreover, passing to a subsequence as the case may be we have
ue ~ U weakly in L2(0, T; Wple2(R2)),
where U solves the incompressible Navier-Stokes equations
OtU + div(U | U) =/zAU + V P, div U = 0 (8.6)
with the initial condition U(0) = 79U0 where 79 is theprojection on the space of divergence-
free functions.
8.2. Dirichlet boundary conditions
Now we focus on the system (8.1)-(8.3) posed on a bounded domain S-2 C R N and
complemented by the no-slip boundary conditions for the velocity:
u~10~ =0. (8.7)
Consider the following (overdetermined) problem:
-Ar = v4~ in S-2, V~ 9nlas2 = 0, 45 constant on 012. (8.8)
A solution of (8.8) is trivial if v = 0 and 45 is a constant. The domain 12 will be said to
satisfy condition (H) if all solutions of (8.8) are trivial.
The following result was proved by Desjardins et al. [20]:
THEOREM 8.2. Let 1-2 C R N, N = 2, 3, be a bounded regular domain. In addition to the
hypotheses of Theorem 8.1, assume that ue satisfies the no-slip condition (8.7).
Then Qe tends to 1 strongly in C([0, T]; L• and, passing to a subsequence if
necessary,
ue --~ U weakly in L 2((0, T) x $2)
for all T > 0 and the convergence is strong if 1-2 satisfies condition (H). In addition,
U satisfies the incompressible Navier-Stokes system (8.6) complemented by the no-slip
boundary conditions on 012 and with U(0) -- 79U0.
8.3. The case Fn ~ cx~
Let us consider the isentropic system in the case when F -- Fn -+ ~. We follow the
presentation of Lions and Masmoudi [65].
Let 12 C R 3 be a bounded regular domain. Consider the system
OtOn + div(OnUn) = 0, (8.9)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-358-320.jpg)
![360 E. Feireisl
Ot(OnUn) "+"div(OnUn | Un) q'-aVOZn" =/ZAUn -k-()~ -q-/z)V divun (8.10)
with the no-slip boundary conditions for the velocity
u,, I~s2 =0 (8.11)
and complemented by the initial conditions
o (o) = o~ >I o, (OnUn)(0) : qn, (8.12)
where
Iq.I 2
Io~ L• (~) <<-Ctn, On bounded in L 1(~),
Oo
bounded in L 1(if2), (8.13)
independently of n.
We are interested in the limit of the sequence On, Un of finite energy weak solutions
of the problem (8.9)-(8.12) when Yn ~ c~. To this end, let us first formulate the limit
problem:
Otto "+-div(0u) = 0, 0 ~<0 ~< 1, (8.14)
0t(0u) + div(0u | u) + V79 =/zAu + (~. +/z)V divu, (8.15)
div u = 0 a.a. on the set {0 = 1}, (8.16)
79=0 a.a. on{o<l}, 79~>0 a.a. on{o=l}. (8.17)
The following result is due to Lions and Masmoudi [65].
THEOREM 8.3. Let ~ C R 3 be a bounded regular domain. Let ~On,Un be a sequence of
finite energy weak solutions of the problem (8.9)-(8.12) on (0, T) x I-2 where the data
satisfy (8.13). Let 0~ converge weakly to some 0 ~ and qn converge weakly to q.
Then On, Un contain subsequences such that
[On - 11+ --~ 0 in Lc~(O, T; L~(I2))forany 1 <~~ < c~,
On ~ ~0 weakly star in L ~ (0, T; L ~(12)), 1 <<.ot < cx~,
where
0~<0~<1.
Moreover, OZn
n is bounded in LI((0, T) • 12)and
OZn
" --+ 79 weakly star in M ((0, T) x ~).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-359-320.jpg)
![Viscous and~or heat conducting compressible fluids 361
If in addition, Qo __+Qo strongly in L1 (S2), then Q, u, 79 solve theproblem (8.14)-(8.17)
in 79'((0, T) x I2) where u is a weak limit of un in L 2(0, T, Wo '2(I2)).
Here A//denotes the space of Radon measures.
9. Other topics, directions, alternative models
9.1. Models in one space dimension
In the above analysis, we have systematicaly and deliberately avoided the case of one space
dimension. Note that for compressible fluids such a situation can be physically relevant as
well as interesting. From the mathematical point of view, these problems exhibit a rather
different character due to the particularly simple topological structure of the underlying
spatial domain.
The question of global existence is largely settled in the case of one space dimension.
The basic result in this direction is that of Kazhikhov [56], a more extensive material can be
found in the monograph of Antontsev et al. [4]. The discontinuous (weak) solutions were
studied by Hoff [45], Serre [90,91] and Shelukhin [95]. The results are quite satisfactory
with respect to the criteria of well-posedness discussed in Section 2. Jiang [54] proved
global existence for the full system in one space dimension when the viscosity coefficients
depend on the density. Probably the most general result as well as an extensive list of
relevant literature is contained in the recent paper by Amosov [2].
There is a vast amount of literature concerning the qualitative properties of solutions.
Stragkraba [98,99], Stra~kraba and Valli [100] and Zlotnik [113] studied the long time-
behaviour of global solutions in the barotropic case driven by a nonzero external force.
Similar results for the full system were obtained in [35]. More information can be found
in Amosov and Zlotnik [3], Hsiao and Luo [53], Matsumura and Yanagi [74] and many
others. A complete list of references goes beyond the scope of the present paper.
9.2. Multi-dimensional diffusion waves
A more detailed description of the long-time behaviour for the barotropic case in several
space dimensions was obtained by Hoff and Zumbrun [52].
Following their presentation we consider the system (5.3), (5.4) with f = 0 on the whole
space 12 = R3. The initial data
Q(O) = po, (Qu)(O) = qo (9.1)
are smooth and close to the constant state Q* -- 1, q0 = 0.
Under these circumstances, the problem (5.3), (5.4) admits a global solution and the
following theorem holds.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-360-320.jpg)
![362 E. Feireisl
THEOREM 9.1. Assume that the initial data satisfy
Iloo - lllt~nwl+d,2(R35+ Ilqollt~nw~+d,2(R3) < e,
where e > 0 is sufficiently small and d ~ 3 is an integer
Then the initial-value problem (5.3), (5.4), (9.1) possesses a global solution 0, u
satisfying
II~x~<~<t>- 1>11~<~ + II~x~<~u><t>ll~<~
c(d)e [ (1 + t) -r~,p for 2 <. p <. oo,
<.
I (1 -+-t) -r~,p+I/p-1/2 if l <. p < 2
for any multi-index loci ~ (d - 3)/2 where
ra,p --1~1/2 + 3/2(1 - 1/p).
Theorem 9.1 shows that perturbations of the constant state decay at the rate of a heat
kernel for p >~2 but less rapidly if p < 2; in fact, the bound may even grow with time in
the latter case.
A more detailed picture of the long-time behaviour in the LP-norm for p ~>2 is provided
by the following result.
THEOREM 9.2. Under the assumptions of Theorem 9.1, we have
II~x~<O- 1)(t) II/~p<~ + II~x~(<ou><t>- ~# 9 [Pqo]) IILp<~)
<<.c(d)e(1 + t) -r~,p+l/p-1/2,
p >/2, where P is the projection on the space of divergence free functions and 1Ca is the
standard heat kernel, i.e., the fundamental solution of the problem
3tV -- lzAv = O.
Thus the dynamics in LP, p > 2 is dominated by a term with constant density and a non-
constant divergence-free momentum field decaying at the rate of the heat kernel. In other
words, for p > 2, all smooth, small amplitude solutions are asymptotically incompressible.
Finally, consider an auxiliary problem:
1
OtQ + div(ou) - ~()~ + 2/z)A0,
d.1,
1
Ot(• + p'(1)VQ -/zA(Qu) H 2)~V div(ou).
(9.2)
Let us denote
u(t) = [o(t), (ou)(t)]](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-361-320.jpg)
![Viscous and~or heat conducting compressible fluids 363
the solution of the linear problem (9.2) with the initial data
0(0) = Oo - 1, (Ou)(O) = qo.
The long-time dynamics in LP, p < 2, is described as follows.
THEOREM 9.3. Under the hypotheses of Theorem 9.1, we have
IIa~([o(t)- 1, (Ou)(t)]- U(t))I]LP(R3 )
<, c(1, o-)e(1 + t) -r~'p+3/4(2/p-1)-l/2+~ l~<p<2,
for any positive or.
All results in this part are taken over from [52].
9.3. Energy decay of solutions on unbounded domains
Various authors have considered the long-time behaviour of solutions on unbounded
domains.
Following Kobayashi and Shibata [58] we consider the full system (2.1)-(2.3)on an
exterior domain ~2 C R3 where the pressure p = P(O, O) is given by a general constitutive
law conform with the basic thermodynamical principles expressed in (1.4).
As for the boundary conditions, we take
ulas~ -0, Olas~ --Ob,
lim u(t,x)=0, lim O(t,x)=Ob.
Ixl~ Ixl~
Assuming the initial data
o(O) = oo, u(O) = uo, 0(0) = Oo
are closed to a constant state [~, 0, 0b], Kobayashi and Shibata [58, Theorem 2] show the
following decay rates:
I1~(')- ~IIL~(~)+ II"(t)II L~(~)+ Ilo(t)- Obll~(~) ~ C' -~/~,
II~(')-~1 L~(~ + II"(t)II L~(~> + Ilo(t)- ObllL~(~)~ C' -~/~
Related results were obtained by Deckelnick [17], Kobayashi [57], Padula [87] and many
others.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-362-320.jpg)
![364 E. Feireisl
9.4. Alternative models
Up to now, we have considered only Newtonian fluids where the viscous stress tensor
27 was a linear function of the velocity gradient Vu. However, some experimental results
show that in nature there exist stronger dissipative mechanisms not captured by the classical
Stokes law. Let us shortly discuss this interesting and rapidly developing area of modern
mathematical physics which gives an alterantive and, given the enormous amount of
open problems in the classical theory, mathematically attractive way to describe the fluid
motion.
In the linear theory of multipolarfluids, the constitutive laws, in particular, the viscous
stress tensor 27 depend not only on the first spatial gradients of the velocity field u but also
on the higher order gradients up to order 2k - 1 for the so-called k-polar fluids.
In the work of Neras and Silhav~ [82], an axiomatic theory of viscous multipolar fluids
was developed in the framework of the theory of elastic non-viscous multipolar materials
due to Green and Rivlin [43]. Accordingly, the viscous stress tensor Z: takes a general
form:
k-1
-- ~-~(--1)J(/~jAJ(Vu + (Vu)t)+ )~jAJ divuId)
j=O
+ co(Vu + (Vu) t) + f divuld, (9.3)
where, in the nonlinear component,
/~ - ~(IVul, divu, det(Vu)), ~o: ~o(IVul, divu, det(Vu)).
The existence of the so-called measure-valued solutions of the initial value problem for
isothermal flows, i.e., for the system (2.1), (2.2) with 27 given as in (9.3) and the pressure
satisfying p = rOoo, was proved by Matu~fi-Nerasov~i and Novotn3~ [75].
The weak solutions for linear multipolar fluids were obtained in a series of papers by
Neras et al. [81,80].
Recently, new results concerning the so-called power-law fluids, i.e., when k = 1
in (9.3), were shown by Mamontov [69,70].
10. Conclusion
Despite the enormous progress during the last two decades, we still seem to be very
far from a satisfactory rigorous mathematical theory of viscous compressible and/or heat
conducting fluids. There are good an bad news according to the degree of complexity of
the problems considered but we still wait, for instance, for a large data existence result for,
say, the isothermal flow in two and three space dimensions. On the point of conclusion, let
us discuss shortly the major mathematcal difficulties presently encountered.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-363-320.jpg)
![Viscousand~orheatconductingcompressiblefluids 365
10.1. Local existence and uniqueness, small data results
As we have seen in Section 2, the initial value problem for the full system (1.1)-(1.3)
complemented by physically relevant constitutive relations admits a unique global in time
classical solution. From the mathematical point of view, this is nothing less or more than
to say that the linearized system is well-posed. This fact seems to be the primary criterion
of applicability of any mathematical model. Indeed there is only a little to say should the
linearized problem be ill-posed. However, there still can remain an essential gap between
"linear" and "nonlinear" provided there is no dissipative meachanism present as it is the
case for nonlinear hyperbolic systems. The possibility to construct classical though only
"small" solutions reveals the dissipative character of the problem, namely, the effect of the
diffusion terms present in the parabolic equations (1.2), (1.3).
Another aspect of dissipativity is the existence of bounded absorbing sets discussed in
Section 7.1 and the existence of global attractors mentioned in Section 7.5. Although we
still do not know if the attractor has a finite fractal dimension, there are strong indications
(cf. Hoff and Ziane [50]) it might be the case.
10.2. Density estimates
Unlike (1.2), (1.3), the continuity equation (1.1) governing the time evolution of the density
is hyperbolic and linear with respect to Q. As a consequence, one cannot expect any
smoothing effect as for parabolic problems or compactification phenomena as it is the
case for genuinely nonlinear hyperbolic equations. We have made it clear several times in
this paper that the major obstacle to develop a rigorous large data theory for our problem
is the lack of a priori estimates on the density Q.
The density being a non-negative function there are two aspects of the problem-
boundedness from below away from zero and uniform upper bounds. Let us remark that the
system (1.1)-(1.3) and, in particular, the constitutive relations for Newtonian fluids hold
for nondilute fluids with no vacuum zones.
Let us review the results of Desjardins [18] illuminating the role of upper bounds
on Q in the well-posedness problem. Consider the isentropic model represented by the
system (5.3), (5.4) where the pressure p satisfies (5.2) with y > 1. For simplicity, we
consider the case of spatially periodic boundary conditions in two space dimensions. The
following result is proved by Desjardins [18, Theorem 2].
THEOREM 10.1. Consider the system (5.3), (5.4) where
P(0)=a0 • a>0, Y >1,
in two space dimensions and with spatially periodic data
Q(0) /> 0 E L~er(R2), u(O) E wpler
2(R2), f~ 0.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-364-320.jpg)
![366 E. Feireisl
Then there exists To > 0 and a weak solution O, u of the problem such thatfor all T < To
0 E g ~ (0, T; Lp~er(R2))
and
X//-O0tUE t2(0, T; Lp2er(R2)),
p -()~ + 2#) divu E L2(0, T; W~er2(R2)),
Vtt E Lee(0, T; Lp2er(R2)),
vu L2(o,r; %2r (R2)),
where 79 denotes the projection on the space of divergence free functions. Moreover, the
regularity properties stated above hold as long as
ii ii
sup ]]O(t) llL~ < (X).
tE[0,T]
Desjardins [18] proved also that the weak solutions constructed in Theorem 10.1 enjoy
the weak-strong uniqueness property well-known from the theory of incompressible flows.
Specifically, the above weak solution coincides with a strong one as long as the latter exists
(cf. [18, Theorem 3]).
The lower bounds on the density represent an equally delicate issue. As we have
seen in Section 7.3, one cannot avoid vacuum states provided we accept the isentropic
model as an adequate description for the long time behaviour of solutions. On the other
hand, the density should remain strictly positive for any finite time t provided its initial
distribution enjoys this property. Unfortunately, however, this is not known in the class of
weak solutions provided N ~>2. To reveal the pathological character of the problem when
vacua are present, we follow Liu et al. [66] and consider the isentropic model in one space
dimension where the initial distribution of the density is given as
o(O) = Oo(x - 2r) + Oo(x + 2r),
where O0 is a compactly supported smooth function with support contained in the ball
{Ix l < r }. Should the model correspond to physical intuition, one would expect, at least on
a short time interval, the solution to be given as
O(t, x) = ~(t, x - 2r) + ~(t, x + 2r),
u(t, x) = fi(t, x - 2r) + fi(t, x + 2r),
where ~, fi solve the problem for the initial data ~(0) = O0. However, as shown in [66],
this is not the case. Of course, this apparent difficulty is due to the discrepancy between](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-365-320.jpg)
![Viscous and~or heat conducting compressible fuids 367
the finite speed of propagation property which holds for the hyperbolic equation (5.3) and
the instantaneous propagation due to the diffusion character of (5.4) (for other unusual
features of the problem we refer also to Hoff and Serre [48]). In fact, one should consider
the viscosity coefficients/z and )~ depending on the density Q in this case (see Jiang [54]).
The formation or rather non-formation of vacua has been studied in a recent paper by
Hoff and Smoller [49]. They prove that the weak solutions of the Navier-Stokes equations
for compressible fluid flows in one space dimension do not exhibit vacuum states in a
finite time provided that no vacuum is present initially under fairly general conditions on
the data. Unfortunately, however, such a result is not known in higher space dimensions
even when the data exhibit some sort of symmetry, say, they are radially symmetric with
respect to origin.
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![Preface
The motion of fluids has intrigued scientists since antiquity and we may say that the
field of mathematical fluid dynamics originated more than two centuries ago. In 1755
Euler [2] gave a mathematical formulation of the principle of conservation of mass in
terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear
and angular momentum by PDEs. Material symmetry and frame invariance were used by
Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of
internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint-
Venant [11] and Stokes [12].
In the 19th century no sharp distinction was drawn between mathematicians and
physicists as we sometime see in more recent times. The formulation of the equations
of fluid motion could be considered as either mathematics or physics. The first work
in fluid dynamics that has a "modern" mathematical flavor may have been done by
Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous
problem. Riemann recognized the mathematical nature of the entropy. This notion led
him to his duality method for solving the non-characteristic Cauchy problem for linear
hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his
contemporaries. What we now call the Cauchy problem for a PDE and the search for its
solution did not have the significance that it is accorded nowadays. Even Poincar6 did not
raise that kind of question in his Th~orie des tourbillons.
For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly
accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the
Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28.
Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of
the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much
as Riemann, Leray developed new mathematical tools which proved to have independent
interest: e.g., weak solutions (that we now call Leray's solutions in this context) and
topological degree (a joint work with Schauder [4]).
Since the 1930s, the interest that mathematicians devote to fluid dynamics has
unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von
Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a
Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on
the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic
displays such a variety of models and questions that thousands of scientists, among them
many mathematicians, focus their research on fluid dynamics.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-372-320.jpg)
![376 H. Fan and M. Slemrod
where co is the assumed constant specific heat at constant volume. Then the specific
internal energy e is given by
e = -a/w = cvO + constant
and the specific total energy E in (1.1) is
2/2.
E = u2/2 -k- e + Aw x
That is, the total energy is the sum of contributions from kinetic energy, internal energy,
and interfacial energy.
The isothermal version of (1.1) is clearly
Ow Ou
Ot Ox '
Ou mO{
at = Ox -p(w) + s-- -
OU 62A 02113
Ox -ff~x2'l
(1.2)
while its inviscid case is covered by
Ow Ou
Ot Ox
=0,
Ou 0
+ ----p(w) = o.
O---f Ox - -
(1.3)
The constitutive equation for a van der Waals fluid at fixed temperature below the critical
temperature
8a
Oc -- 27Rb
has the shape depicted in Figure 1.
In the isothermal case, we are interested in subcritical temperatures and hence assume
p E C 1(R) and
p'(w) <0 ifw ~t [a, fl],
p'(w)>O ifw~(a, fl).
(1.4)
The regions w < ot and (w > r) correspond, for van der Waals fluids, liquid and vapor
phase region respectively. The line joining (m, p(m, T)) and (M, p(M, T)) is called
Maxwell line where two equilibrium phases can coexist. The region ot < w < fl is called
the spinodal region. If the fluid ever enters the spinodal region, the fluid will quickly
decompose to liquid or vapor or their mixture. In other words, the spinodal region is a
highly unstable region. To see this intuitively, we consider a ball of such fluid with w in
spinodal region, see Figure 2. Pressures in the ball and its surrounding medium are set](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-374-320.jpg)
![378 H. FanandM. Slemrod
Systems (1.1), (1.2), (1.3) coupled with (1.4) not only serve as prototype models for
studying the dynamics of phase transitions, but also are interesting mathematical objects
in its own fight. For example, system (1.2) are of hyperbolic-elliptic mixed type with
ot < w < 13as its elliptic region. It is well known that the initial value problems for elliptic
systems are ill-posed. Systems of hyperbolic type have been extensively studied. The
presence of both hyperbolic and elliptic region in (1.2) certainly leads to new phenomena
and new issues. The isothermal system (1.2), having physical background and being one
of the simplest systems of conservation laws of hyperbolic-elliptic mixed type, certainly
qualifies to be a prototype model for studying such systems.
In this paper, we shall review some recent results related to (1.1), (1.2), (1.3) with (1.4).
Although we tried our best to cover as much related results as possible, it is possible that
we missed some.
The rest of this paper is arranged as follows: in Section 2, we derive Equation (1.1).
In Section 3, we review some results on the initial value problem of (1.3) and related
admissibility criteria. In Section 4, we recall the proof of the existence of solutions of (1.3)
satisfying the traveling wave criterion via the vanishing similarity-viscosity approach.
Although these results and proofs appeared in our earlier works, we present here a revised
version which is more readable.
2. The equations of motion
We consider the one-dimensional motion of fluid processing a free energy
62A(Ow) 2
f (w, O)= fo(w, O) + --~ -~x " (2.1)
Here w is the specific volume, 0 the absolute temperature, A > 0 a constant, and x the
Lagrangian coordinate. The term
where e > 0 is a small parameter, is the specific interfacial energy introduced by
Korteweg [52]. The graph of f0 as a function of w for fixed 0 will vary smoothly from
a single well potential for 0 > 0crit to double well potential for 0 < 0crit. The 0crit is called
the critical temperature. Discussions of such free energy formulations may be found in [3,
10,13,14,12,25,35,66,67,78].
The stress corresponding to the free energy (2.1) is given by
Of Ofo (W, O) -- ,f,2AO2W (2.2)
T- Ow = Ow Ox
---g"](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-376-320.jpg)
![Dynamicflowswithliquid~vaporphasetransitions 379
Note that there is no viscous force in (2.2). Addition of a viscous stress term gives us the
stress of the form
OU 02W
T = -p(w, O) + e Ox eZA OX2 (2.3)
suggested by Korteweg's theory of capillarity [52]. In (2.3), u(x, t) denotes the velocity of
the fluid, e > 0 is the viscosity and p - Ofo/Ow is the pressure.
The one-dimensional balance laws of mass and linear momentum are easily written
down:
Ow Ou
= (mass balance), (2.4a)
Ot Ox
Ou OT
= (linear momentum balance). (2.4b)
Ot Ox
The equation for balance of energy is more subtle. While a thorough examination of the
energy equation appears in Dunn and Serrin [25] it is the conceptually simple approach
of [32] we recall here. Let e(w, O) denote the internal energy. Felderhof's postulate
is that the internal energy is influenced only by the component of internal stress r =
au i.e., the balance of energy is given by
-p(w,O) + s-~,
De Ou Oh
= r t , (2.4c)
Ot Ox Ox
where h is the heat flux. Unlike Equations (2.4a, b), Equation (2.4c) is not in divergence
form. To alleviate this difficulty we consider the specific total energy
E -~ -F e (w , O) + ---f- -~x
made up the specific kinetic, internal, and interfacial energy. Now compute the time rate of
change of E"
DE
Ot
OU 2W 02W
=u-- -Fet + e2A ~ ~
Ot Ox OxOt
OT Ou 22W OU
OX OX OX2
= U F T---- + eZA
Ox
Oh e2 O(OwOu)
F~x+ A~x -~-xOx '
where we have used the relation T -- r - e 2 A 02w We easily see that the balance of energy
o-T"
can be written as
OE 0 (uT)+eeA O (OuOw) Oh (2.5)
at =0-7 +0-7](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-377-320.jpg)
![380 H.FanandM.Slemrod
Ou Ow represents the "interstitial working" [25]. For simplicity we constitute
The term eZA ~
00 where xe > 0 is the (assumed constant) thermal conductivity.
h by Fourier's law: h - x e~
Then we may collect the balance laws and write them as
Ow Ou
= (mass), (2.6a)
Ot Ox
OU 0[ OU 021/3]
Ot = O---X --p(w, O) + e O---X-- ezA ~X 2 (linear momentum), (2.6b)
=~ u-p+e~-e2A
Ot Ox / L Ox OxZJ
+ ~2A ~~ +~~ (energy). (2.6c)
The isothermal case of (2.6) is
Ow Ou
= (2.7a)
0t 0x'
Ou 0 { Ou e2A O2to}
Ot = O---x -p(w) + e O---~- ~x 2 . (2.7b)
3. Initial value problems of the inviscid system (1.3) and admissibility criteria
In this section, we recall recent results on the initial value problems of inviscid system (1.3).
Most results on the initial value problem of (1.3) are on Riemann problems. The
Riemann problem of (1.3) is the initial value problem
tot -- Ux -- O,
ut+p(W)x--O,
(u-, v-l,
(u(x, 0), v(x, 0)) = (u+, v+),
ifx <0,
if x>0.
(3.1)
Through the study of the Riemann problem, we gain understanding on the behavior of
solutions of (1.3). Based on knowledge about solutions to Riemann problems, Glimm's
scheme can be used to construct solutions of (1.3) for general initial data.
Compare to the viscous system (1.2), the inviscid system (1.3), as an approximation
of (1.2), offer the following advantages: the structure of solutions are clearer. Solutions of
Riemann problems may be constructed by solving a few algebraic equations. However,
these advantages come with a price to pay: solutions of initial value problems of the](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-378-320.jpg)
![Dynamic flows with liquid~vapor phase transitions 381
inviscid system (1.3) are usually weak solutions with jump discontinuities. Such solutions
are nonunique unless further restrictions on weak solutions are applied. These restrictions
are called admissibility criteria. The admissibility criterion should pick "good" solutions
suitable for the problem under consideration: here we are considering phase transitions
modeled by (1.2). The inviscid system (1.3) is used as an approximation of (1.2). Thus,
to make solutions of (1.3) to mimic those of (1.2), it is natural to require that admissible
solutions of (1.3) to be e --+ 0+ limits of solutions of (1.2) with the same initial value. This
is called the vanishing viscosity criterion.
However, enforcing the vanishing viscosity criterion is usually very difficult and
expensive. For example, to implement the vanishing viscosity criterion, one have to be
able to (a) prove that solutions of (1.3) satisfying the criterion exist and (b) verify whether a
given solution of (1.3) satisfy the criterion or not. These tasks are usually very difficult and
expensive. So far, the part (a) is carried out for strictly hyperbolic systems of conservation
laws [11,24,23]. For the (b), some results and techniques are given in [34] for piece-wise
smooth solutions, with small shocks, of stricly hyperbolic systems.
Thus, simpler admissibility criteria are called for. An internal layer asymptotic analysis
on solutions of the viscous system (1.2) indicates that jump discontinuities of solutions
of (1.3) must have traveling wave profiles in order for the solution of (1.3) to approximate
that of (1.2). Traveling waves of (1.2) are solutions which are functions of the form
g(x - st), where the constant s is the speed of the traveling wave. The traveling wave
equations corresponding to (1.1) with ~ = (x - st)/e, w = w(~), u = u(~), 0 =0(~) are
dw
d~
dl)
A~
d~
dO
K m
d~
m - - - l } ~
-- --S2(tO -- 113_) -- p(to, O) -+- p(w_, 0_) - sv,
-- --s{(e(w,O) -e(w_,O_))
s 2 Asv 2
- --(to - to_) 2
2 2
(w,u,O)(-~)=(w_,u_,O_),
(w, u, 0)(+c~) - (w+, u+, 0+),
p(w_,O_)(w-w_)},
(3.2)
where s is the speed of the traveling wave. For the isothermal case (1.2), above becomes
d//3
d~
dl)
d~
-- --S2(tO -- l/3_) -+- p(w_) - p(w) - sv,
(w, u)(-~) - (w_, u_), (w, u)(+~) = (w+, u+).
(3.3)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-379-320.jpg)
![P t
I
I
382 H. Fan and M. Slemrod
w_ ~2(w_,~) w3(W_,~) ~ (w_,s)
Fig. 3.
~--- W
A shock solution of (1.3)
[ (u+, w+),
(u w)(x t)-- I
' ' I (u_, w_),
if x-st >0,
if x - st < 0, (3.4)
where s is the speed of the shock, is said to have a traveling wave profile if the traveling
wave equation (3.3) has a solution. This leads to the traveling wave admissibility criterion:
Traveling wave criterion states that a shock (3.4) is admissible if the system of traveling
wave equations (3.3) has a solution. When (3.3) has a solution, we also say that there is a
connection between (w_, u_) and (w+, u+). If all singular points of a solution of (1.3) are
jump discontinuities and these jump discontinuities are admissible by the traveling wave
criterion, we say that the solution is admissible by the traveling wave criterion.
The solvability of the traveling wave equation (3.3). We are particularly interested in
the case w_ < or, w+ >/~, since this data involves phase changes. Indeed, solutions of
Riemann problems of (1.3) cannot take values inside the spinodal region (or,/~) and hence
must have a shock jumping over the spinodal region, at least for the case A >~ 1/4 [27,
Lemma 2.3(i)]. The solvability of the connecting orbit problems (3.3) were studied by
Slemrod [74-76] and Hagan and Slemrod [40], Hagan and Serrin [39] and Shearer [70-
72]. Let w_ ~<ot and s >~0. For simplicity, we assume the ray starting from (w_, p(w_)),
with slope -s 2, to the right can intersect the graph of p at most at three points (cf. Figure 3).
We denote the w-coordinates of these points by
w2(w-,s), w3(w-,s) and w4(w-,s),](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-380-320.jpg)
![Dynamicflows with liquid~vaporphase transitions 383
respectively. Points w_ and wk(w_, s), k = 2, 3, 4, are equilibrium points of (3.2). w_
and w3 (w_, s) are saddle points of (3.2) while w4(w-, s) is a node of (3.2). By [27],
Riemann solvers of (1.3) cannot have values in the spinodal region (or, 13) at least for the
case A >~ 1/4, thus, traveling waves connecting w_ and wz(w-, s) is ofno use in this case.
Now we consider the existence of a solution of (3.2) connecting w_ and w3(w-, s),
i.e., w(-oo) = w_, w(+oo) = w3(w_,s). For w_ E [y,m], if there is a g ~> 0 such
that the signed area between the graph of p and the chord connecting (w_, p(w_)) and
(w3(w_, s), p(w3(w_, s))) is 0 (cf. Figure 3), then there is a speed s*/> 0 such that
0 <~s* ~<g (3.5)
and the problem (3.2) with s = s*, W2 = W3(W-,S*) has a solution, which satisfies
~ (~) > 0 and is a saddle-saddle connection, i.e.,
0 ~<s* < v/-P'(W_), s* < v/-P'(W3(W_, s*)). (3.6)
We note that this saddle-saddle connection accounts for the usual liquid-vapor phase
transitions, including the coexistence of two phase equilibria in the case s = 0. In (3.4),
equality holds if and only if g = 0. Furthermore, for any 0 < s < s* the trajectory of (3.2)
emanating from (w_, 0) will overshoot w3 (w_, s) and flow to (w4 (w_, s), 0) as ~ --+ cx~.
In other words, for all w2 > w4(w-, s*), there is a traveling wave solution of (3.2).
Furthermore, this traveling wave solution is a saddle-node connection, i.e.,
v/--p'(w_) > s > v/--p'(w4(w_,s)). (3.7)
These statements were proved in Hagan and Slemrod's paper [40]. Grinfeld [36] proved
that if g exists, then for any N 6 Z+, there is a number AN > 0 such that for all A ~> AN,
the system (3.2) has N saddle-saddle connection solutions, wj (~), j = O, 1, 2..... N - 1,
such that wj (~) intersects w~= 0-axis traversely j times. If g does not exist but
so := min(s: w3(s, w_))> O,
then there is at least one saddle-saddle connection for all A > 0.
If p(w) further satisfies
p"(w) (w - w0) > 0 for w 5~ w0, (3.8)
for some w0 E (c~,fl) then, for Y ~< w_ ~< m, there is a unique speed s* /> 0 such
that w_ can be connected to w3(w_,s*) by a traveling wave solution of (3.2) with
w2 = w3 (w_, s*), which is a saddle-saddle connection [69-71 ]. We notice that when (3.8)
holds there is no w4(w_, s) for w_ 6 (y, m]. In fact, uniqueness of s* holds for all w_ < ~.
In the above paragraph, w_ is fixed. However, if we fix w+ = w3 (w_, s), there can be
two w_ < ot such that there are connections with s > 0 between w_ and w3 [8].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-381-320.jpg)
![384 H. Fan and M. Slemrod
We note when p(w) is a cubic polynomial, we can have an explicit solution for (3.2).
Let
m+M)
p(w) = Po - Pl (w - m)(w - M) w 2 ' (3.9)
where m and M are Maxwell constants. Then a solution of (3.2) is (cf. [79,80])
w_-q- w+ w+-- w_ tanh(v/pl w+-w_ )
w(~) = 2 -I- 2 2A 2 (~ - ~o) 9 (3.10)
For each w_ fixed, w+ in (2.11) is determined by equations:
3(1 -6A)(2y - z + 1)2 -I-Z2 -- 1,
OSy=(m-w+)/(m-m),
z--(w+-w_)/(m-m).
(3.11)
The number of solutions of (3.11) ranges from zero to two. When (3.11) has two solutions,
we get two solutions of (3.2) of the form (3.10); one of them has positive speed and the
other negative. This is, of course, consistent with Theorem 3.1. In fact, the nonuniqueness
of traveling waves connecting a fixed w3 to some w_ is true in general [8].
In addition, Grinfeld [37] and Mischaikow [59] conducted studies on the full sys-
tem (2.6) using Conley's index theory.
Stability of traveling waves is an important topic for (1.2). In fact, having a stable
or metastable shock profile, which is a traveling wave solution of (1.2), is a necessary
condition for the shock of (1.3) to be admissible. Hoff and Khodja [47] proved the dynamic
stability of certain steady-state solutions of the Navier-Stokes equations for compressible
van der Waals fluids
Vt -- ttx = 0,
ut + p(v e)x = [e(X)Ux/V] X ~
(u2/2 + e)t + [up(v, e)]x -- [e(x)uux/V + 1.(x)T(v, e)x/V]x.
(3.12)
The steady-state solutions consist of two constant states, corresponding to different phases,
separated by a convecting phase boundary. They showed that such solutions are non-
linearly stable in the sense that, for nearby, perturbed initial data, the Navier-Stokes system
has a global solution that tends to the steady-state solution uniformly as time goes to
infinity.
Benzoni-Gavage [7] studied the linear stability of planar phase jumps satisfying the
traveling wave criterion (3.3) in Eulerian coordinates with viscosity neglected, called
capilarity admissibility criterion. She showed that the such phase boundaries are linearly
stable. Although neglecting the viscosity is unphysical, such a result served as the base
from which she studied the case when the viscosity is small [9] to yield similar results.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-382-320.jpg)
![Dynamic flows with liquid~vapor phase transitions 385
Zumbrun [85] proved the linear stability of slow heteroclinic traveling waves of (2.7)
under localized perturbation. He also showed that homoclinic traveling waves near
Maxwell line involving multiple phase transitions are exponentially unstable. This implies
that the slow heteroclinic traveling waves of (2.7) are stable if they are monotone. The
method used are spectrum analysis framework [33,86] and some energy estimates.
Motion of phase boundary under perturbation and the effect of boundary conditions
of (1.2) was studied by Chen and Wang [15]. The initial data is a perturbation of
the stationary phase boundary, the Maxwell line. They found the ordinary differential
equations describing the motion of the phase boundary under perturbation by an asymptotic
expansion and a matching analysis. They conclude that the phase boundary will approach
a well defined location as time goes to infinity.
Existence of solutions of the Riemann problem for (1.3) satisfying the traveling wave
criterion. One method for solving (3.1) is construction of wave and shock curves that are
admissible according to some criteria and then construct a wave fan of centered waves and
shocks that matches the initial data. Being constructive, this approach yields very detailed
structure of the solutions if successfully carried out. The difficulty is that it is hard to know
all the admissible shocks to enable such a construction.
When A --0 in (3.3), James [49] considered the Riemann problem. Shearer [69],
Hsiao [48] proved the existence of solutions of the Riemann problem. In this case, a phase
boundary is admissible if and only if the speed of the phase boundary is 0. The uniqueness
of such Riemann solutions is proved by Hsiao [48]. See also [50,51 ].
In the case A > 0, the only stationary phase boundary is the one connecting (m, 0) and
(M, 0) [74]. This is in perfect agreement with the Maxwell equal area rule. When the
Riemann data are in different phase region, e.g., w_ < ot and w+ >/3, Shearer [72] proved
that solutions of Riemann problem exist if Iw- - ml + Iw+ - MI + [u+ - u-I is small,
where m, M are the Maxwell line constants. He first studied the behavior of traveling
waves near Maxwell line, then constructed the Riemann solvers accordingly. To extend his
approach to more general Riemann data, one will have to know explicitly, for any given wl,
what w2 can be connected to Wl by a traveling wave. This is almost impossible in general.
Another approach is to construct the solution of (1.3) as the e ~ 0+ limit of the viscous
system (1.2), or simply that of the solutions of
Wt ~ Ux -- 8Wxx,
(3.13)
ut + p(w)x = eUxx,
with the same initial data. Although this approach has been carried out successfully for
strictly hyperbolic 2 • 2 systems [24,23] for hyperbolic-elliptic mixed type system (1.3),
this approach seems quite difficult at present. Thus, Slemrod [77] and Fan [26,28,29] used
the vanishing similarity viscosity approach pioneered by Dafermos [20] and Tupciev [81].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-383-320.jpg)
![386 H. Fan and M. Slemrod
iT
Metastablevapor
i,
(a)
tT
x/t= x/t=c
Liquid ~
Metastablevapor Metastablevapor
(u_,v_) ~u+<u_,v+=v_)
(b)
Fig. 4.
The idea of this approach is to construct the weak solution of (3.1) as the e ~ 0+ limit of
the solution of
tOt -- Ux = etWxx,
ut + p(w)x = etUxx,
(u(x, o), v(x, o)) - /
(u_, l)_),
(u+, v+),
/
ifx <0,
ifx >0.
(3.14)
This approach enables us to establish the existence of weak solutions of (3.1) for general
Riemann data not in the spinodal region. The condition required is p(w) ~ -4-c~ as w --+
qxc~. Although the form of the viscosity used in (3.12) is often criticized as unphysical,
it turns out that solutions constructed through e ~ 0+ limiting process of (3.12) are
also admissible by the traveling wave criterion derived from (3.11) [26], see Section 4.3.
Furthermore, when p" (w)(w - w0) < 0 for w # w0 6 (or, 13) and with w_ < c~ < 13 < w+
(or w+ < ot < 13 < w_), the solution of the Riemann problem admissible by the traveling
wave criterion is unique. Thus, under above condition, if one obtains a solution via the
vanishing viscosity method (3.12), it would be the same as what we obtained by the
vanishing similarity viscosity approach (3.12).
When the Riemann data are on the same side of the spinodal region, w+ < m (or
w+ > M), Shearer [71] showed that when the Riemann data Iw+ - m[ (or Iw+ - MI)
are small, then for some u+, there are at least two solutions for the Riemann problem. An
example of the nonuniqueness is illustrated by Figure 3.
Shearer's result raised the question that which of the two solutions is physically "good".
We think that both solutions are good, but at different times: consider the shock tube
experiment corresponding to the Riemann data (w+, u• depicted in Figure 3. We note
that -u+ = u_ > u. = 0 and fl < w_ = w+ < M. This data describe the shock tube
experiment where vapor moves from both sides towards the center x = 0 where the fluid
is at rest. The pressure at the center part increases due to the compression from both sides.
The data in Figure 3 are such that the pressure at the center part is above the equilibrium
pressure and hence the vapor in the middle part is metastable vapor. The metastable vapor
will stay as vapor until enough liquid drops are initiated due to random fluctuation. Thus,
in the early stage, the first solution in Figure 4(a) is the good solution. As time increases,](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-384-320.jpg)
![Dynamicflows withliquid~vaporphase transitions 387
enough many liquid drops are initiated in metastable vapor, more likely in the center part
where pressure is higher. Then phase changes occur rapidly via the growth of these liquid
drops. In this late stage, we expect to see the second solution, shown in Figure 4(b). We
note that the center of the wave in the early time of the second stage may be different
from that of the first solution due to the randomness in location of liquid drop initiated.
From the above consideration, we see that both solutions in Figure 3 are "good", but at
different times. In fact, the two-phase-boundary solution depicted in Figure 4(b) is visually
quite stable in numerical simulations once it is initiated [73]. As to when the solution in
Figure 4(a) changes to that in Figure 4(b), the viscosity method does not provide answer.
This is because system (3.1) with higher-order derivative terms, such as (1.2) is based
on interfacial energy. The Maxwell equal area rule is derived from the consideration
of interfacial energy which is the reason why viscosity-capillarity type of high-order
derivative terms in (1.2) agrees with the Maxwell equal area rule. However, such high-order
derivative terms do not cover the mechanism for creating of liquid drops due to random
fluctuations. Thus, (1.2) is designed to describe the motion of phase boundaries, not the
initiation of new phases. This is why some selection criteria, called initiation criteria, are
used to help to decide when new phases are initiated in many classical theory of quasi-
static problems, cf. [38]. We imagine that the transition from the solution without phase
boundary to the one with two phase boundary is a dynamic process that takes some time
for enough many nuclei of new phase to form and grow to complete the phase change.
Typically, nucleation process are slow in metastable states unless near the Wilson line, or
spinodal limits, which is why we have metastable states. Proper choice of initiation criteria
used in (1.3) to correctly describe the physical process is an open problem. Initiation
criteria should reflect (a) that if the fluid is in metastable state, then after a sufficiently
long time, nucleation process will initiate enough many fluid drops of the stable phase and
eventually, the fluid change to the stable phase, and (b) that the further away the metastable
fluid is from the equilibrium, the faster the new, stable phase will be initiated. From above
consideration, we see that initiation criteria must involve the time spend in metastable state
and the distance of the pressure from the equilibrium pressure. So far, initiation criteria
used in most works on (3.1) do not include above factors. Rather, these criteria typically
specify that if the distance of the pressure from the equilibrium pressure is less than a fixed
barrier, then the one-phase solution is picked, otherwise the two-phase solution is picked.
This ignored the fact that no matter how close the pressure is to the equilibrium pressure,
transition to the two-phase solution will happen later, even though slower.
Kinetic relation admissibility criteria. Above considerations demonstrate that the sys-
tem (3.1) is not complete by itself. It must be augmented by some selection criteria that
helps to find the solution relevant to the problem under consideration. For phase transi-
tion problems, such criteria may depend on materials in a complicated way. This leads
to another point of view on admissibility criteria for phase boundaries: rather than trac-
ing the admissibility criteria for phase boundaries to something more elementary such as
viscosity and capillarity, one can consider admissibility criteria as constitutive relations
controlling the speed of the phase boundary, determined by the materials and to be mea-
sured in laboratory. Such restriction are considered by Truskinovskii [79] and Abeyaratne](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-385-320.jpg)
![388 H. Fan and M. Slemrod
and Knowles [1] and are called kinetic relations, [1]. Consider an interval [x1, X2] of fluids
in the Lagrangian coordinate. The total mechanical energy associated with the interval is
E(t) -- -e(w)(x, t) + -~U2(X,t) dx,
1
(3.15)
where P = fo p(r/)dr/. A calculation based on (1.3) shows that
d
p(Wl)Ul -- p(w2)u2 -;TE(t) -- f (wl, w2)s(t) (3.16)
at
if (u, w)(x, t) is a shock solution of (3.1) and X2 > s(t) > Xl. Here in (3.16),
2 1
f(Wl, W2) -- -- p(r/) dr/-+- ~ (p(w2) -F p(wl))(w2 - Wl).
1
(3.17)
We can see that the left-hand side of (3.16) is the excess of rate of work of the external
forces over the rate of increase of mechanical energy. Since the motion (3.1) described is
isothermal, the well known Clausius-Duhem inequality requires that the instantaneous rate
of mechanical dissipation to satisfy
f (wl, W2)S ~ O, (3.18)
which is the classical entropy criterion. A subsonic phase boundary {(Ul, W2), (U2, //)2); S}
is called admissible by kinetic relation criterion if, besides the entropy criterion (3.18) and
Rankine-Hugoniot condition, the function f defined by (3.17), also satisfies
f(wl, W2) -- qg(S) (3.19)
for a function ~0predetermined by the material. The kinetic relation criterion (3.19) actually
is a stricter version of the entropy criterion:
OtE(u, w) + OxF(u, w) = lZ <~O, (3.20)
where/~ is a given nonpositive measure [55].
Abeyaratne and Knowles [1] implemented the kinetic relation criterion onto trilinear
materials, i.e., the function p(w) is three-piecewise linear and of the shape depicted in
Figure 1. They proved that the Riemann solver satisfying their kinetic relation, the initiation
criterion, and the entropy inequality is unique. Later in [2], they extended above result
to nonisothermal case with heat conduction taken into consideration. The speed of the
phase boundary is not constant. They found an integro-differential equation for the speed.
LeFloch [55] proved the L 1 continuous dependence of the Riemann solver, admissible
by kinetic relation criterion, on the Riemann data. Abeyaratne and Knowles [1] also
showed that, at least for trilinear materials, the traveling wave criterion and the entropy
rate criterion, etc., when applied to subsonic phase boundaries, is a kind of kinetic relation](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-386-320.jpg)
![Dynamicflows with liquid~vaporphase transitions 389
criterion. Later, Fan [30] showed that traveling wave criteria are kinetic relation criteria
if p(w) is symmetric around the point (w* = (or +/3)/2, p(w*)). Natalini and Tang [60]
considered some discrete kinetic models with the objective of providing a practical tool
encompassing various kinetic relations for the phase boundaries.
Similar to Shearer's results on nonuniqueness of the Riemann solver, the Riemann
problem (3.1) has two solutions admissible by kinetic relation criterion for some Riemann
data. One solution entirely lies in one of the phase region {(u, w); w < c~} (or {(u, w);
w > t3}) with no phase boundary while the other solution has two phase boundaries and
hence takes values in both phase regions. For convenience, we shall call the first solution
the one-phase solution and the latter the two-phase solution. In fact, this nonuniqueness
phenomenon is common for a large class of local admissibility criteria which are local
restrictions on points of discontinuities of solutions of (3.1) [30].
To handle this nonuniqueness, Abeyaratne and Knowles [1] used an initiation criterion
which specifies a critical value for a function h and assert that phase transitions will occur
and hence the two-phase solution is "good" if the value of h exceeds the critical value.
Otherwise, the one-phase solution should be picked.
LeFloch [55] considered the (1.3) with initial data being a BV perturbation of an
admissible phase boundary. He constructed solutions of (3.1) 1 and (3.1)2 with p being
trilinear. The selection criteria used are the kinetic relation criterion, initiation criterion
and entropy inequality. He constructed solutions by Glimm's scheme and proved these
solutions are admissible by the above selection criteria. In recent papers Bedjaoui
and LeFloch has investigated the relation between the kinetic relation and viscosity-
capillarity [5,6]. The instability from Glimm scheme when applied to (1.3) is discussed
by Pego and Serre [63].
Asakura [4] studied the Cauchy problem for (1.3) with initial data being Maxwell
stationary phase boundary plus a small perturbation. He showed that there exists a global
in time propagating phase boundary which is admissible in the sense that it satisfies the
kinetic relation criterion; the states outside the phase boundary tend to the Maxwell states
as time goes to infinity.
Colombo and Corli [16] constructed Riemann semigroup of (1.3) admissible by
aS-relation, a generalized kinetic relation criterion. In particular, this result allows the
authors to build a complete theory of existence (via front tracking) and continuous
dependence on the initial data of the solutions. Related papers have been done by Corli
[17-19].
Entropy rate criteria. Another interesting admissibility criterion is the entropy rate
criterion proposed by Dafermos [21]. This criterion asserts that the weak solution of (3.1)
which dissipates the total entropy the fastest is the admissible solution. The total entropy
is typically the total mechanical energy:
E(t) .= -p(w) + ~u +/'(wo)- u2 dx, (3.21)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-387-320.jpg)
![390 H. Fan and M. Slemrod
where (w0, u0) is the initial value. The dissipation of the total energy when the solutions
are piecewise smooth is measured by
dE(t)
dt
-
shocks
(3.22)
where f is given in (3.17). Dafermos [22] further justified this admissibility criterion by
proving that in strictly hyperbolic systems, wave fans satisfying Liu's shock admissibility
criterion [56] consisting of rarefaction waves and shocks of moderate strength do maximize
the rate of entropy production. In elastodynamics, this statement holds for arbitrary shock
strength. Hattori [41,42,44] and Pence [64] applied this criterion in their study of Riemann
problems of systems of conservation laws of mixed type. Hattori [43] further studied
initial value problems of (1.3) using the entropy rate admissibility criterion and Glimm's
scheme. He proved the existence of weak solutions when initial data is a BV perturbation
of Riemann data and the perturbation is compactly supported.
Among admissibility criteria mentioned in the above, the vanishing viscosity crite-
ria (1.3) and (3.13) and the entropy rate criterion are of global nature. Others, such as
traveling wave criteria and the kinetic relation criterion are local restrictions at points of
jump discontinuity. It is a hope that these local restrictions can characterize completely
admissible solutions of (3.1). If this hope fails, more conditions, probably conditions of
global nature, should be imposed. Thus it is important to experiment with various criteria
with global authority, especially those motivated by physics. For example, when the Rie-
mann solvers admissible by a local admissibility criteria are not unique, which one does a
global admissibility criteria pick? A comparison of the effect of vanishing viscosity crite-
rion (3.13), entropy rate criterion and traveling wave criterion derived from (3.13) in the
context of (3.1) is made in [30]. In [30], initial data is chosen such that there are two solu-
tions of (3.1), admissible by the traveling wave criterion, one being the one-phase solution
and the other the two-phase solution. It is found that only the one-phase solution is admis-
sible by the vanishing viscosity criterion (3.13), at least for the special pressure function
p given in [30]. However, the entropy rate criterion picks the two-phase solution. We note
that by the vanishing viscosity criterion, the two-phase solution cannot be initiated from
the initial data in (3.1). Once the two-phase solution is initiated, it is quite stable, at least
visually in numerical simulations [73].
More results on the viscous system (1.2).
boundary value problem
Hattori and Mischaikow [46] studied the initial
Utt = O'(Ux)x -t- 13Uxxt - 17Uxxxx,
u(0, t) -- 0, cr(Ux(1, t)) + VUxt(1, t) - rlUxxx(1, t) = P,
Uxx(1,t) --Uxx(O,t) =0.
(3.23)
Note that Equation (3.23)1 can be tranformed to (1.2) under the variable change v = ut,
w = Ux. They proved the existence and uniqueness of the global solution of (4.1) under
certain growth condition on or(q). They also proved the existence of global compact](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-388-320.jpg)
![Dynamicflows with liquid~vaporphase transitions 391
attractor. They obtained the complete bifurcation diagram. The existence and large time
behavior of initial boundary value problem of (4.1)l with r/-- 0 is studied by Pego [62].
He showed that discontinuities of the solution are stationary, that the energy of the system
cannot be minimized as t --+ oo, among other things.
Hattori and Li [45] studied the initial value problems of a fluid dynamic model for
materials of Korteweg type in two-dimension:
fit -I- (pU)x + (pV)y = O,
(pu)t + (puZ)x + (pUV)y + p(p)x = (Tll)x + (T12)y,
(pv)t + (puv)x + (pvZ)y + p(p)y = (T21)x + (T22)y.
(3.24)
They proved the existence of the unique local solution. Their proof does not depend on the
monotonicity of the pressure function and hence can be used for van der Waals pressure.
Nicolaenko [61] showed the existence of inertial manifolds for (1.2). The proof used the
slightly dissipative Hamiltonian structure of the system.
Milani, Eden and Nicolaenko [58] established the existence of local attractors and of
exponential attractors of finite fractal dimension. This showed that even in regions of mixed
type, the initial value problem exhibits finite-dimensional dynamical behavior.
Serre [68] used a very interesting approach computing the formal oscillatory limit, in
the spirit of Whitham [84], of the thermo-visco-capillarity system as the small parameters
tends to zero. He then obtained a system of modulation equations for the limiting motion.
4. Existence of solutions of the Riemann problem (3.1)
In this section, we recall our proofs for the following results on the existence of solutions
of (3.1) in [77,28]. Their approach is to use the vanishing similarity viscosity (3.14) to
establish the existence of weak solutions of (3.1) that are also admissible by traveling
wave criterion derived from (3.13).
Reasonable shock admissibility criteria should be compatible with translations and
dilations of coordinates, under which the system is invariant. Dafermos [22] argued that
admissibility should be tested in the framework of Riemann problem, i.e., in the context
of solutions of the form U (x, t) = V(x/t) which represent wave fans emanating from the
origin at time t = 0. Thus we utilize the system (3.14), which is invariant under translations
and dilations of coordinates, to handle the Riemann problem (3.1). This approach has been
pursued by many authors in their stydies of Riemann problems [20,54,27,53,81,77,82]. For
convenience, we shall call solutions of (3.1)constructed in this way admissible according
to the similarity viscosity criterion. The main results are as follows:
THEOREM 4.1. Assume in (3.1) that w+ q~[c~,3] and
p(w) ~ -+-oo as w --+ Too. (4.1)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-389-320.jpg)
![392 H. Fan and M. Slemrod
Then there is a sequence {en}, en --+ O+ as n --+ cxz such that the solution of (3.14)
with initial data (3.1)3 converges almost everywhere to a weak solution u(x, t) of (3.1).
Furthermore this solution also satisfies the traveling wave criterion derived from (3.13).
The structure of solutions of (3.1) when w_ < c~ </5 < w+ constructed in Slemrod [77]
and [26,28,29] by the similarity viscosity approach is as follows: each of these solutions
can be embedded on a continuous curve in (u, w) the phase plane. Solutions must have a
phase boundary, i.e., w(~) r (or,/5) for any ~ e I~. Solutions consist of two wave fans:
< 0 the first kind wave fan and ~ > 0 the second kind wave fan. A first (second)
kind wave fan consists of 1-shocks and (2-shocks) and 1-simple waves (2-simple waves)
and possibly the phase boundary and constant states. ~ = 0 is either a constant state or
the phase boundary (cf. [26]). Most of above results are generalized by Lee [54] to the
system
ut -- f (V)x = O,
vt - g(u)x =0,
(4.2)
where f is strictly increasing and convex, and g is increasing (and either concave or con-
vex) except in a finite interval where it is decreasing and so the system is hyperbolic-elliptic
mixed type.
Under stricter restrictions on p(w), we have the following uniqueness results:
THEOREM 4.2. Assume conditions in Theorem 4.1. If w_ < ot < ~ < w+ (or w+ < ot <
< w_) and that
p" (w) > O for w < ot, and p" (w) < O for w > fl, (4.3)
then
(i) the solution of (1.3) satisfying the traveling wave criterion based on (1.2) is unique,
and
(ii) the solution (u ~, wE)(x, t) of (3.14) converges almost everywhere to the unique
solution of (1.3)as e --+ 0+.
The statement (i) of Theorem 4.2 is proved in [27]. The statement (ii) follows
immediately from (i) and Theorem 4.1.
The rest of Section 4 is devoted to the proof of Theorem 4.1.
4.1. Existence of solutions of the Riemann problem (1.3)
To take the advantage of the invariance of (3.14) under dilatation of coordinates, we make
variable change ~ = x/t in (3.14). A simple computation shows that (3.14) reduces to the](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-390-320.jpg)
![Dynamic flows with liquid~vapor phase transitions 393
following system
eu"=-~u' + p(w)',
8W tt -- __~ W f m U',
(U, 113)(--00) = (U_, W_), (u, w)(+oo) = (u+, w+).
(4.1.0)
Our program for proving Theorem 4.1 is to show that there is a solution of (4.1.1) with total
variation bounded uniformly in e > 0. Then the first statement in Theorem 4.1 follows. The
proof for the second statement of Theorem 4.1 will be given in Section 4.3.
To this end, we consider, instead of (4.1.0), the following altered system
eu"= -~u' + #p(w)',
8W II -- --~ W I _ l,U I,
(u(+L), w(+L))- (u• w•
(4.1.1)
where L > 1, 0 ~ # ~<1.
LEMMA 4.1.1 [77]. Let (ue(~), we(~)) be the solution of (4.1.0). Then one of the
following holds on any subinterval (a, b) for which p' (we(~)) < 0.
(1) Both ue (~) and we (~) are monotone on (a, b).
(2) One of the ue(~) and we(~) is a strictly increasing (decreasing)function with
no critical point on (a, b) while the other has at most one critical point that is
necessarily a local maximum (minimum)point.
(3) If the criticalpoint in (2) is of w(~), then the condition p'(w(~)) < 0 can be relaxed
to p'(w(~)) <, Ofor ~ E (a, b).
Now, we rewrite Lemma 2.2 of [77] which describes the shape of a solution of (4.1.0)
in the elliptic region {(u, w) 6 IR2: w E (or,fl)}.
LEMMA 4.1.2. Let (u(~), w(~)) be a solution of (4.1.1)with lZ > O. Then on any interval
(11,12) C (-L,L) for which p'(w(~)) > 0 the graph of u(~) versus w(~) is convex at
points where w' (~) > 0 and concave at points where w~(~) < O.
By considering (4.1.1), the existence of the connecting orbit problem (4.1.0) can be
proved, as shown in the following theorem.
THEOREM 4.1.3. Suppose u_ < u+ and w+ < or. Then there is a solution of (4.1.0)
satisfying that
(U(~l), W(~l)) -7(=(u(~2), to(~2)) for any ~l, ~2 E (--00, -+-(~), ~1 zik ~2
and w(s~l)/> tb := max(w_, w+) (4.1.2)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-391-320.jpg)
![394 H. Fan and M. Slemrod
and that
there are at most two disjoint open intervals (a, b) such that (4.1.3)
w(~) ~ (if), ~) for ~ ~ (a, b) (4.1.3a)
and
either w(a) -- if;, w(b) -- ot or w(a) -- or, w(b) -- (v, (4.1.3b)
provided that the possible solution of (4.1.1) satisfying (4.1.2) and (4.1.3) is bounded in
CI([-L,+L]),
for some M > 0 independent of tx ~ [0, 1] and L > 1.
(4.1.3c)
PROOF. We rewrite (4.1.1) as
ey"(~) = lzf (y)' - ~y'(~), (4.1.4)
where
y(~) _ (u(~))
w(~) ' -u(~) "
Multiplying (4.1.4) by the factor exp(-~2/(2e)) and integrate twice, we can rewrite (4.1.4)
as the integral equation:
y(~) = y(-L) + z(y) exp dr + -- f(y(r)) dr
L -~8 8 L
f_f ,,
tx ~ r rfty(r)~ex p drd(,
8 L L 2e
(4.1.5a)
where
z(r) = 1 { --f f(Y(r)) dr
f-LLexp(@2)d~ y(+L)_ y(_L)_ # L
8 L
+lz f_L f_
L L rf(Y(r)) exp( r dr d(
2e
= zl (Y) -k-lZz2(Y). (4.1.5b)
Choose
r/6 (~b,or). (4.1.6)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-392-320.jpg)
![Dynamic flows with liquid~vaporphase transitions 395
We are interested in those functions (u(~), w(~)) e C 1([-L, +L]; R 2) satisfying
and w(~l) ~>7/
for any ~1, ~2 G [-L, +L], ~1 r ~2
(4.1.7)
and that
there are at most two disjoint open intervals (a, b)
w(~) 6 (rl, or) for ~ ~ (a, b), and
either w(a) - rl, w(b) - ~ or w(a) = c~,
such that
w(b) -~.
(4.1.8)
We note that (4.1.7) and (4.1.8) is invariant under small C 1 perturbations. The subset in
CI ([-L, +L]; IR2)
and (4.1.7) and (4.1.8) are satisfied} (4.1.9)
is open. We define an integral operator
T's x [0, II --+ C 1([-L, L]; R 2)
by
l; ,f;
T(Y, Ix)(~) = y(-L) + z(Y) exp d( + -- f (Y(()) d(
L -~e 6 L
f f
Ix ~ C r/(Y(r)) exp dr d(,
s L L 2S
(4.1.1o)
where z(Y) is given by (4.1.5b). It is clear that a fixed point of T(Y, Ix) is a solution
of (4.1.1).
It is a matter of routine analysis to show that T maps s x [0, 1] continuously into
C 1([-L, L]; R2). Furthermore, we can verify, by taking d/d~ twice on (4.1.10), that T
maps ~ x [0, 1] into a bounded, with bound independent of Ix, subset of C 2([-L, L]; R2).
Thus T is a compact operator from C 1([-L, L]; R 2) x [0, 1] into C 1([-L, L]; R2).
We recall the following fixed point theorem ([57], Theorem IV. 1).
PROPOSITION 4.1.4. Let X be a real normed vector space and [2 a bounded open subset
of X. Let T 9 x [0, 1] --+ X be a compact operator. If
(i) T (x, #) ~ x for x E 0s tx ~ [0, 1], and
(ii) the Leray-Shauderdegree DI(T(., O) - I, ~) 7/=O, where I is the identity operator,
then T (x, 1) -- x has at least one solution in s](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-393-320.jpg)
![396 H. Fan and M. Slemrod
To solve our problem, we take X = C 1([-L, +L]; I1~2). We can see that (ii) is satisfied.
Indeed,
T(Y,O)- Y= Yo- Y, (4.1.11)
where
~:= y(L) - y(-L) f~,
_~2 J_ exp
f_LL exp(w-) d ( L
(-~e 2 ) d( + y(-L).
We note that T(Y, O) = Yo ~ $2 is the solution of (4.1.4) when/z = 0. It is a fixed function,
independent of Y and/z. Then we have DI(T(., 0) - I, S2)= DI(Yo- I, f2) = 1, as
desired.
Now, we preceed to verify (i) of Proposition 4.1.4. We assume, for contradiction, that
there is a fixed point of T (Y,/z),
Y = (u, w)(~) 6 052. (4.1.12)
Then one of the following cases must hold:
Case A. II(u(~), w(~))llcl<t-L,c~;R2) = M + 1.
This case is impossible to occur under the condition (4.1.3c).
Case B. The condition (4.1.7) is violated.
In this case, there are ~1,~2 6 [-L,+L], ~1 < ~2, such that (U(~l), W(~l))=
(U(~2), tO(~2)) and W(~I) ~ r/.
The curve (u(~), w(~)) in (u, w)-plane near ~ = ~1 and ~ = ~2 cannot go across
each other. 1 This is because if otherwise, the curve (u(~), w(~)) in (u, w)-plane plus
a CI([-L, +L]; R 2) perturbation still intersects itself and hence is not in S2. Thus,
(u(~), w(~)) is not in 052 which yields a contradiction.
From Lemma 4.1.1, we know that if (u(~), w(~)) stays inside the region w ~< or, the
curve (u(~), w(~)) cannot intersect itself. Thus, w(~3) > c~ for some ~3 E [-L, +L].
1 Here, we clarify the meaning of "two curves go across each other": For two curves, (Ul, Wl)(~) and
(u2, w2)((), to cross each other in (u, w)-plane, they have to intersect each other first:
(Ul, Wl)(~l) = (U2, W2)((2)
at some points ~1, (2. For convenience, we parameterize the two curves by the length of curve s with s = 0
denoting above point of intersection. If the two curves coincide with each other near s = 0, then the orientation
of the parameterization should be such that
(u 1, Wl)(S) -- (U2, W2)(S) (1)
over [s_, s+] with 0 6 [s_, s+]. In particular, if no such coincidence is present, then s_ = s+ = 0. We further
let the interval [s_, s+] be the largest on which (1) holds. We use the following notations: Tj(s) is the tangential
direction of the j-th curve, j -- 1, 2, k the normal direction of the (u, w)-plane, which is a constant vector in R3.
We say that curves (Ul, Wl)(S) and (u2, w2)(s) go across each other if (1) holds and
(T1(s) • T2(s)) 9k
does not change sign on an open interval containing [s_, s+ ].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-394-320.jpg)
![Dynamicflows with liquid~vaporphase transitions 397
We can further describe the curve (u(~), w(~)) in the (u, w)-plane as follows. There is
an interval [-L, 01] such that w(~) ~<c~and w(01) =ol, and by Lemma 4.1.1, w'(O1) > O.
As ~ increases from 01, (u(~), w(~)) moves into the region ot < w < /3. As long as
w'(~) > 0 and w(~) 6 (c~,/3), the curve (u(~), w(~)) in the (u, w)-plane is convex with
respect to w. Let (01,02) be the largest interval such that w'(~) > 0 and w(~) 6 (or,/~).
(w+, u+
(w_, u_) //
~,, ~=o 1
I
Fig. 5.
W
Then either W(02) -- fl or w(02) E (ly, fl) and w' (02) = 0 holds. For definiteness, we
assume that w(02)= ~ and w~(02) > 0, since the other case is simpler. In view of
Lemma 4.1.1, this interval is followed by another interval [02,03] in which w(~) ~>
and u~(~) > 0 while w(~) has one and only one critical point which is a local maximum
point and w(03) =/3. This shows that ~2 ~ [--L, 03). Then there is the maximum interval
[03, 04) in which w(~) ~ [or,~], w'(~) < 0 and the curve (u(~), w(~)) in the (u, w)-plane
is concave with respect to w. We see that at the right end of the interval, either
w(04) = ~, w'(04) < 0
or
w(04)/> ~, w'(04) = 0
holds. We claim that wf(04) = 0 is impossible because if otherwise the concavity would
make w'(~) > 0 for ~ > 04 and near 04. This would force the curve (u, w)(~) to go
across itself in the region w ~>c~ in order to reach w(L) = w+. This is contradictory to
(u, w)(.) ~ OX2.Thus, w(04) =or and w'(04) < 0 hold. This also shows that if ~2 ~ [03, 04),
then ~1 6 [01,02] and
W' (~1) >/0, W'(~2) ~< 0, w(~) > w(~l) for ~ e (~1, ~2).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-395-320.jpg)
![398 H. Fan and M. Slemrod
='ql
~ I I
,w,u., , i-,,
/
(w,u) /
n a 3
=r/2
w
Fig. 6.
This, however, will lead to a contradiction by integrating (4.1.1b):
f~
2
0 < [w(~) - w(~e)] d~ = e[w'(~e) - w'(~,)] ~<0.
1
Above description shows that the point of self intersection ~2 r [01,04].
Following [03, 04] is the interval [04, 05) in which r/< w(~) ~<or. Let [04, 05) be the
largest of such interval. Then
w(05) = r/, w'(05) < 0 (4.1.13)
holds because if otherwise, w(~) would have a local minimum point in [04,05) and
w(05) = or, see Figure 6. By Lemma 4.1.1, u(~) would decrease over the interval [04, 05).
After ~ = 05, w(~) would enter the w < c~region. Then the curve (u, w)(.) in (u, w)-plane
would have to go across itself in order to connect to (u+, w+), which is prohibited. We
further claim that over the interval (05, L],
w(~) <~.
Indeed, if otherwise, either w(~) > ot for some ~ > 05 or w(~) is less than a and has
multiple extreme points in [05, L]. The case of multiple extreme points are impossible
in view of Lemma 4.1.1. The other case that w(~4) > c~ for some ~4 ~ (05, L] is also
impossible since it and (4.1.13) imply that there are at least three disjoint open intervals
(a, b), bounded away from each other, such that w(a) = ot (or rl) and w(b) = 7/ (or c~).
But this is impossible for a function (u, w)(.) 6 0s This claim implies that the points
of self-intersection satisfies ~2 ~ [04, 05], ~1 E [--L, 01] and w(~2) ~ [r/, or]. Thus, we have
w'(~2) < 0and w'(~l) > 0, and w(~) ~>w(~2) = W(~l) for~ ~ [~1, ~2]. Integrating (4.1.1b)
over [~1, ~2] and using (U(~l), W(~l)) = (u(~2), w(~2)), we obtain
f~2
0 < [w(~) - w(~2)] d~ - e[w'(~2) - w'(~,)] < O,
1](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-396-320.jpg)
![Dynamic flows with liquid~vaporphase transitions 399
which is a contradiction. Thus, Case B cannot happen.
Case C. The condition (4.1.8) is violated. That is, there are more than two disjoint open
intervals (a, b) such that 77< w(~) < ot for ~ 6 (a, b) and w(a) = rl (or c~), w(b) = ot (or rl).
Since (u, w)(~) 6 0S-2, the number of disjoint open connected component intervals (a, b)
with w(a) = rl (or or) and w(b) = ot is four or more. Two of such intervals are (al, bl),
(a2, b2) with a2 = bl and w'(bl) = 0. If w(bl) = rl and hence w(al) = w(b2) -- or, then
= bl is a local minimum point of w(~). According to our discussion of Case B, it is
necessary that bl < 04 and it is impossible that w(b2) = c~. This contradiction shows that
w(bl) 7~ rl9
We claim that the other possibility w(bl ) -- ot cannot happen either. Indeed, if w(bl) = ol
and hence w(al) -- w(b2) = ~ < ot, the point ~ = bl is a local maximum point for w(~).
We see that w(~) ~<o~ for all ~ E [-L, L] since if otherwise, w(~) would have multiple
extreme points in one of the connected component of {~ ~ [-L, L]: w(~) ~< o~} which
is impossible according to Lemma 4.1.1. Then, the number of disjoint open connected
component intervals (a, b) with w(a) = rl (or c~) and w(b) = o~ is just two, not four or
more. This contradiction shows that Case C cannot occur.
Summarizing our analysis for above three cases, we find that if (u(~), w(~)) E 0$2,
then Y = (u(~), w(~)) cannot be a fixed point of T(Y, lz) for # E [0, 1]. Applying
Proposition 4.1.4, we see that T(Y, 1) has a fixed point.
To prove the existence of solutions of (4.1.0), we need to pass to the limit L --+ oo. We
follow Dafermos [20] and extend (u(~), w(~)) as follows
[
(u+, w+),
(u(~'/~) w(~'/~)) = {
' ' ' /(u-,w-),
if~ > L,
if~ <-L.
By the hypothesis (4.1.4), we see that {(u(.; L), w(.; L))} is precompact in C((-c~, cx~);
R2). So, there is a sequence Ln --+ cx~ as n --+ cx~ such that (u(~; Ln), w(~; Ln)) --+
(u(~, cx~), w(~, cx~)) uniformly as n --+ cx~. By integrating (4.1.1a, b) with/z = 1 twice
from ~0, we can prove the limit (u(~, cx~), w(~, c~)) satisfies (4.1.1a, b). It remains to
prove that (u(-+-cx~,~), w(-+-cx~,cx~)) = (u+, w+). To this end, we manipulate (4.1.1a, b)
to obtain
d(exp(~2/2e)y'(~))-1 [f(y(~))' exp(~e2)l
d~ e
or
if0'
exp(~Z/Ze)y'(~) = y'(O) + - V f (y)y'(~)exp -~e d~.
6
(4.1.14)
Applying Gronwall's inequality on (4.1.15), we obtain
I ly'0 Iex"(
~< M exp 2e ' (4.1.15)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-397-320.jpg)
![400 H. Fan and M. Slemrod
where R > 0 depend at most on M, v and e > 0. Inequality (4.1.21) holds for y (~; L) also.
Then
(u(+~, ~), w(+cc, ~)) = (u+, w+)
follows from (4.1.21) easily. It remains to prove that the solution (u(~, cx~),w(~, cxz))
constructed above satisfies (4.1.2) and (4.1.3). Indeed, the same reasoning for Case B
and C implies that (u(~, cx~),w(~, e~)) satisfies (4.1.8) and (4.1.9) also. Since r/6 (if),or)
is chosen arbitrarily, (4.1.2) and (4.1.3) hold for (u(~, c~), w(~, cx~)). D
COROLLARY 4.1.5. Let (u(~), w(~)) be a solution of (4.1.1) or (4.1.0) satisfying (4.1.2),
(4.1.3). Then,
(i) The subset of [-L, +L]
{~ e I-L, +/-,l" w(~) <~o~}
has at most two connected components. Furthermore, each components must have
-L or +L as one of its endpoints.
(ii) The set
consists of at most two connected components.
(iii) The set
{~ e I-L, +L]. w(~) >/r
if nonempty, is an interval.
PROOF. This is proved in our discussion in the proof of Theorem 4.1.3, Case B. D
The assumption (4.1.4) in above theorem can be replaced by a weaker one, as stated in
the following theorem.
THEOREM 4.1.6. The conclusion of Theorem 4.1.3 remains valid if (4.1.4a) is replaced
by
sup ([u(~) I nt- Iw(~)l) <~M1,
where M1 is independent of tx ~ [0, 1] and L > 1.
PROOF. The proof is the same as that of Theorem 1.3 in [77]. IS]
Theorems 4.1.3 and 4.1.6 give the conditions under which (4.1.0) has a connecting orbit
for w+ < ot and u_ < u+. Slemrod [77] proved the following theorem for the case w+ < c~
and u_ < u+:](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-398-320.jpg)
![Dynamicflows with liquid~vaporphase transitions 401
THEOREM 4.1.7. Assume that w+ < ~ and u_ < u+. Then, there is a solution of (4.1.0)
satisfying
w(~) ~<o~, (4.1.16)
if every possible solution of (4.1.1) satisfies
(4.1.17)
for some constant C independent of tt E [0, 1] and L > 1.
4.2. A-priori estimates
In this section, we shall prove the a-priori estimates needed in Theorems 4.1.3 and 4.1.7
as well as some e-independent estimates. Let denotes a solution of (4.1.1) with
the properties (4.1.2) and (4.1.3). For clarity, we shall use (u(~), w(~)) instead of
(ue (~), we (~)) in this section if no confusion is expected.
THEOREM 4.2.1. Suppose w+ < ~ and u_ < u+. Let (ue(~), we(~)) be a solution
of (4.1.1) with the properties (4.1.2) and (4.1.3). Then,
(4.2.1)
where C is, throughout this section, a constant independent of e > 0, # E [0, 1] and
1 < L ~<+cx~.
PROOF. When # = 0, our assertion can be easily verified. Thus, we assume # > 0 in the
rest of the proof. We first prove uE(~) ~> C. Let ~e be a local minimum point of u~(~).
Then either
we(~e) ~ (or,fl), wte(~e)< 0 (4.2.2)
or
!
we(~) E (o~,fl), w~(~) > 0 (4.2.3)
hold.
Case A. (4.2.2)holds.
In this case, by Lemma 4.1.1, w(~e) < fl since if otherwise both ue(~) and we(~)
would have critical points in the set {~ E [-L, +L]: we(~) ~>fl}, which is an interval
by Corollary 4.1.5. Thus, w(~e) <~ot and hence
~ 9 {~ E [--L, +L]" //)e(~) ~ or} -- [-L, 011U [04, 7t-L], (4.2.4)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-399-320.jpg)
![402 H. Fanand M. Slemrod
where 01 ~ 04. If 01 < 04 and ~s e [-L, 01], then w'(~s) < 0 implies that ws(~) also has a
critical point in [-L, 01] which is prohibited by Lemma 4.1.1. Thus, ~s 6 [04, L]. We can
regard the curve (u, w)(~) in the (u, w)-plane as a function u(w). Then we have
dus(~) u'(~)
dws(~) w'(~)
Performing a calculation on (4.1.1), we obtain
d (dus(~)) (dus(~)
s~ dws(~) -- # dws(~)
dus(se
x dws(~)
v/-P'(We(es )))
+v/-p'(w~(~))). (4.2.5a)
This implies that, as ~ increases, dus(es)/dws(es) is decreasing if Idus(~j)/dwe(es)l <~
v/-P'(We(~)) and is increasing if Idue(es)/dwe(es)l >~v/-p'(ws(es)). Thus the "initial"
condition
dus(~)
dws(~) ~=~
=0 (4.2.5b)
leads to that for ~ 6 [04, +L]
dus(~)
dws(~)
~< max (v/-p'(w)) (4.2.6)
w+>>.
w>~
f
and hence
us(~) >~u+ + (or - w_) max (v/-p' (w) ).
we[w+,~]
(4.2.7)
Case B. (4.2.3) holds.
By Corollary 4.1.5, [-L, L] can be divided as
[-L, L] -- [-L, 011 U (01,02) [,-J[02, 03] U (03, 04) U [04, +L], (4.2.8)
where, of course 01 ~<02 ~ 03 ~ 04, and
{~ e [-L, -+-L]: ws(~) ~ o/} -- [-L, 011U [04, L],
{~ e [-L, nt-L]" t/)s(~) e (c~,fl) } -- (01,02) U (03, 04),
{~ E[-L,-+-L]" tOs(~)/> fl} =[02,031.
(4.2.9a)
(4.2.9b)
(4.2.9c)
It is clear that when (4.2.3) holds, ~s 6 (01,02) 5;
f ~ (cf. Figure 5).](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-400-320.jpg)
![Dynamic flows with liquid~vaporphase transitions 403
According to the sign of ~e, we have two cases:
Case B(1).
~e ~>0. (4.2.10)
Since ~e is a local minimum point of u(~), ue dx > 0 for ~ E (~e, ~e + 6) for some 6 > 0.
Then we can define
/71 "-- sup{ ~"> ~e" ue dx > 0 for ~ E (~e, ~')}. (4.2.11)
Since we(~e) ~>ot by (4.2.3), and w~e(~e) > 0, there is a local maximum point /72 of
We(~) with/72 > ~e. We can further require that/72 is the least of such points, i.e.,
/72 "-- sup{ ~"> ~e" w; (() > 0}. (4.2.12)
Then, by Lemmas 4.1.1 and 4.1.2,/71 ~ (~e,/72) and hence (cf. Figure 5)
/71 > /72 > ~e- (4.2.13)
By integrating (4.1.1a) on (~e, ~) where ~ E (~e,/72), we obtain
0 < euedx -- -~'u;(~') d~" + #[p(we(~)) - p(we(~e))].
It follows from (4.2.10) and (4.2.11) that -~ue, dx < 0 for ~ 6 (~e,/71). Thus, in view
of (4.2.3), we have
0 < eu;(~) <~ #[p(we(~e))- p(we(~ee))]
~<#[p(we(se)) - p(c~)] for ~eE (~ee,/72). (4.2.14)
Therefore,
O/ < 1/3e(/72) ~< Wl (4.2.15)
holds, where
WO :-- y, 1131 := V
in Figure 1. Equation (4.2.13) also yields a useful inequality
0 < eule(~) ~<#(p(fl) - p(c~)) (4.2.16)
for ~ E [~e,/711.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-401-320.jpg)
![404 H. Fan and M. Slemrod
Using (4.1.1), we can obtain
d2we -Ix
(~) = ~u~ dx
(dwe(~) 2]
~[l + p'(we(~)) due(~)) "
Hence, if
dwe(~)
due(~) 2 maxwe[wo,tO1](V/[ p' (W)l)
and ~ 6 [~e, ~1], then
d2Wen -lz 1
du 2 (~)~< ~<-- <0.
2eu~dx 2(p(fl) - p(ot))
8n
Thus, as ~ decreases from//2 to ~, dw/du will increases from 0 and eventually
dwe (~) [ _ 1
[ i
du~(~) ~=,73 2max~e[wo,wll(V/lp'(w)l)
for some r/3 6 (//2, ~e)- Let
04 "-- sup{r/3 E (~e,//2): (4.2.19)is satisfied}.
Then,
2 maxwe [wo,W1](%/I P! (W)I) du~(~) ~=.4 du~(~) ~=.2
= f uE(r/4)d2wen
au~(,T2 d-~e2.(~) d(ue (~))
ue (/72)- ue (r/4)
>
or
0 ~ Ue(r/2) -- Ue(r/4)
p(fi) - p(c~)
max~ e[wo,w,](v/Ip ' (w)1)
From (4.2.18), we also see that
dwe (~) />
due(~) 2 maxwe [wo,wl](v/[ P' (w)l)
(4.2.17)
(4.2.18)
(4.2.19)
(4.2.20)
(4.2.21)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-402-320.jpg)
![Dynamicflows with liquid~vaporphase transitions 405
or
du~(~)
dw~(~)
2 max (v/lp'(w)l)
wE[wO,Wl]
for ~ e (~e, 004).Thus,
f
w~(rl4) due (~)
0 ~ Ue(002) -- Ue(~e) = Ue(002) -- Ue(004) § -5-~-_.--7L~dwe
aw~(~) tlw~ L~)
p(fl) - p(~)
~< + 2 max
max~e[~0, Wl](v/I p'(w) l) we[~0,w~]
(v/lp!(w)l)(Wl - too), (4.2.22)
where we used (4.2.15) and 004 E (~e, 001)- Similarly, we can prove that
0 ~ Ue(001) --Ue(002)
p(r - p(c~)
~<
maxwe [w0,w,](v/I p ' (w)l)
§ max (v/lp'(w)l)(wl - wo). (4.2.23)
tOE[tO0,tt)l]
Then we obtain
Ue(~e) ~ Ue(001)-
2(p(fl) - p(ot))
maxw E[u,0,tO1](V/ ]P! (W)l)
-4 max (v/Ip'(w)l)(w,-wo).
we[wo,wl]
(4.2.24)
If ue(001) ~> u+, then, (4.2.24) shows that ue(~) is bounded from below uniformly in
e > 0, # ~ [0, 1] and L > 1. Now, we devote our attention to the case when
b/e(001) < U-k-.
!
Then, 001 < L because ue(L) = u+. By the definition (4.2.11), of 001, Ue(001) -- 0. Then,
by Lemma 4.1.1 and 4.1.2, 001 has to be an extreme point for ue(~). Since ue dx > 0 for
E (~e, 001), /71 is a local maximum point. Lemmas 4.1.2 and 4.1.1 implies that either
Wle(001) > 0 and We(001) ~ (0/, fl) (4.2.25)
or
!
We(001) < 0 and We(001) E (or, fl). (4.2.26)
The case (4.2.25) cannot happen because it implies that 171 E [-L, 01] which violates the
known fact that 001 > ~e E [01,02). Then (4.2.26) infers that there is a local minimum point
05 > 004of ue(~) which satisfies
!
ue(005) -- 0 and we(005)~<c~. (4.2.27)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-403-320.jpg)
![406 H. Fan and M. Slemrod
Then our argument for the Case A applies and gives us
lu+ - ue(r/5)l ~<(or - w+) max (v/-p' (w) ).
w~[w+,a]
(4.2.28)
Using (4.2.28) in (4.2.24), we obtain the desired result
2(p(fl) - p(ot))
//~(~) ~ //e(/~l)-
maxw e[w0,11)
1](g/I pl(W)l)
- 4 max (v/Ip'(w)l)(Wl -- 1/30)
we[wO,Wl]
2(p(fl) - p(a))
/> u~(~5)-
max~e [~o,~, ](v/I p ' (w)l)
-4 max (v/lp'(w)l)(wl-wo)
w~[wo,wl]
i> u+ -- (c~ -- w+) max (V/--p'(w))
wc[w+,~]
2(p(fl) -- p(ot))
maxwe[wO,Wl](v/I P'(w) l)
-4 max (v/lp'(w)l)(Wl-WO),
we[wo,wll
(4.2.29)
which proves that uE(~) is bounded from below uniformly in e > 0, # E [0, 1] and L > 1.
Case B(2) ~e < 0.
The proof is similar to Case B(1). The only difference is that instead of (4.2.11), we
define
r/1 --inf{ff <~," u'(~)> Ofor~ e (~,~E)}
and change the rest of the proof accordingly.
Similarly, we can also prove that ue(~) is bounded from above uniformly in e > 0, #
[0, 1] and L > 1. D
In the remainder of this section, we adopt the following notation:
u* "- sup{ue(~) I~ e I~, e 9 (0, 1)},
u, "- inf{u,(~) I~ e R, e 9 (0, 1)}.
(4.2.30a)
(4.2.30b)
Once we established the a-priori estimates for ue(~), we can proceed to prove the
following results for we(~) by using the similar argument used in [20].
THEOREM 4.2.2. Assume w+ < a and u_ < u+. Let (ue(~), wE(~)) be a solution
of (4.1.1) satisfying (4.1.2) and (4.1.3). Then](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-404-320.jpg)
![Dynamicflows with liquid~vaporphase transitions 407
(i) IIw~(~)llf(t-z,+zl;R2)~ C(e) where C(e) is independent of# e [0, 1], L > 1.
(ii) /f Ip(w)l ~ ec, as Iwl ~ ~, and if# = 1, then IIw~(~)IIc(t-L,+L~;R2) ~ C where
C is independent of L > 1 and e > O.
PROOF. We only prove that w(~) is bounded from above uniformly. The other part of the
proof is similar and is omitted.
(i) Without loss of generality, we assume we(~) has a local maximum point re. We
further assume that
re ~<0. (4.2.31)
The proof for the other case is similar. By Lemmas 4.1.1 and 4.1.2,
!
ue(re) > 0. (4.2.32)
We define
O "-- inf{~ < re" we dx > 0}. (4.2.33)
!
It is clear we(q) ~>0. Integrating (4.1.1b), we obtain
0 >/-ew'e(O) = - ~we dx d~ 4-/z(ue(o) - ue (re)). (4.2.34)
By the definition (4.2.33), we see that ~we dx ~<0 on (7, re) and hence
f Te
0>~ ~wedxd~ >~#(ue(o)-ue(re))>~u,-u*. (4.2.35)
If ~ ~<min(-1, re), then
fo ~ f0 ~
~'w'e(~') d~" <~ - w'e(~') d~" -- we(q) - we(~).
From the definition (4.2.33), we know that either 0 = -L or 0 is a local minimum point
of ue(~), In view of Lemmas 4.1.1 and 4.1.2, we(q) e [min(w_, w+), fl]. Then above
inequality yields
we (~) ~< - ~w'e(~) d~" 4- we (7) ~<u* - u, 4- fl (4.2.36)
for all ~ ~<min(- 1, re). In other words, we(~) is bounded from above uniformly in e > 0,
/z e [0, 1] and L > 1 if ~ ~<min(-1, re). For ~ e (-1, re], we have, from (4.1.1b), that
f re
0 ~> -ew~ dx - -~'w'~(~') d~" +/z(u~(~) - u~(r~)) ~>u, - u*.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-405-320.jpg)
![408 H. Fan and M. Slemrod
This implies that
we(re) ~<we(-1) + Cl (e) <. u* - u, + ~ + Cl (e).
Thus, the statement (i) is proved.
(ii) It remains to consider the case when re 6 (-1,0] and/x = 1. For each e, we can
choose 0 6 (-2,-1) such that ue(O) <<.u* - u,. By integrating (4.1.1a), with/x -- 1, on
[0, re], we obtain
L
%'e
p(wE(rE)) =eu'E(rE)--eu'e(O)+ p(wE(O))-- ~uEdxd~
fo
<~-eulE(O) + p(wE(O)) - ~uE dx d~. (4.2.37)
Every term on the fight-hand side of (4.2.33) is bounded uniformly in e > 0 and L > 1.
Thus, by virtue of the assumption on p in the theorem, WE(re) are bounded from below
uniformly in e > 0 and L > 1. D
THEOREM 4.2.3. Assume w+ < ot and u_ > u+. Let (uE(~), wE(~)) be a possible
solution of (4.1.1) satisfying wE(~) ~ or. Then
(i) [[uE(~)IIc([_L,+L];R2 <~C where C is a constant independent of L > 1, /~ ~ [0, 1]
and e > O.
(ii) [[we(~)[[C([_L,+L];~2 <~C(e) where C(e) is independent of l~ ~ [0, 1], L > 1.
(iii) If [p(w)[--+ 00, as [w[--~ 00, and if l~ = 1, then [[wE(~)[[C([-L,+L];R2 ~<C where
C is independent of L > 1 and e > O.
PROOF. The proof is almost the same as that of Theorem 4.2.2. V1
Combining Theorems 4.1.3, 4.1.4, 4.2.1 and 4.2.2, we obtain the following result.
THEOREM 4.2.4. (i) Assume w+ < ot and u_ < u+. There is a solution (uE(~), wE(~))
of (4.1.0) satisfying
for any ~l, ~2 e (-00, +00), ~l r ~2
and wE(~l) ~> w := rain(w_, w+)
(4.2.38)
and that
there are at most two disjoint open intervals (a, b) such that (4.2.39)
wE(~) ~ (t~, or), and (4.2.39a)
either we (a) = 6v, we (b) = ot
or we (a) = ~, we (b) = 6v.
(4.2.39b)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-406-320.jpg)
![Dynamicflows withliquid~vaporphasetransitions 409
(ii) For the case w+ < ~ and u_ > u+, there is a solution of (4.1.0) satisfying
w~(~) <<.~.
(iii) There is a subsequence {Sn}, Sn --+ O+ as n --+ ~, such that (us, (~), we, (~)) given
in (i) and (ii) converges a.e. to a weak solution (u(~), w(~)) of the Riemannproblem (3.1).
Furthermore, the solutions we constructed have at most two phase boundaries.
PROOF. (i) and (ii) Theorems 4.1.6, 4.2.1, 4.2.2 and 4.2.3 provide the a-priori estimates
needed by Theorems 4.1.3 and 4.1.7 Thus, parts (i), (ii) are established.
(iii) From Corollary 4.1.5, we know that the solutions of (4.1.0) provided in (i),
ue(~) and ws(~) are piecewise monotone. Thus, ({us(~), Ws(~)) given in (i) has total
variation bounded uniformly in e > 0. Then the classical Helly's theorem states that
there is a sequence {Sn}, en --+ 0+ as n --+ oe, such that (ue(~), we(~)) converges almost
everywhere. Apply this limit to the weak form of (3.14), we see that the limit is a weak
solution of (3.1). D
4.3. Solutions constructed by vanishing similarity viscosity are also admissible by
traveling wave criterion
In this section, we shall prove that solutions constructed by vanishing similarity viscosity
is also admissible by traveling wave criterion. Let (us(~), we(~)) denote the solution
of (4.1.0). From last section, we see that (ue(~), ws(~)) are bounded uniformly in e > 0.
Let us denote the upper and lower bounds of ue(~) by u, and u* respectively. Similarly,
the upper and lower bounds of we (~) is denoted by w, and w* respectively.
For simplicity of presentation, we restrict ourselves to the case w_ < ot </3 < w+.
In this case, solutions (ue(~), we(~)) of (4.1.0) have the following shapes: there are two
points ~ = 01 < 02, depending on e, such that
we (~) ~<or, for ~ E (-cx~, 01], (4.3. la)
ot < ws(~) </3 for ~ E (01,02), Ws(01) = Or, We(02) =/3 (4.3.1b)
and
ws(~) t>/3 for ~/> 02. (4.3.1c)
According to Lemmas 4.1.1 and 4.1.2, over each of the intervals (-co, 01) and (02, OO),
there are three possibilities:
(i) The function we (~) has one local extreme point and us (~) is monotone.
(ii) The function us (~) has one local extreme point and we (~) is monotone.
(iii) Both us(~) and ws(~) are monotone.
' d2 /dw 2 > 0 hold. According to
Over the interval (01,02), inequalities ws(~) > 0 and us
shapes (i)-(iii), there are nine different combinations of shapes for (us (~), we (~)). We also
see that over each of the regions w ~<or, c~ < w </3 and w ~>/3, we can consider the curve
(us (~), ws (~)) in (u, w)-plane as the curve of the function Us(w) or We(u), depending on
which of us(~) and ws(~) is monotone.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-407-320.jpg)
![410 H. Fan and M. Slemrod
LEMMA 4.3.1. Let ~ > 0 be some fixed small number so that
w_ <~-3 </~+3 < w+. (4.3.2)
(a) In the region w <. c~-6 (or w ~ fl +6) either dUE(w)/dw or dWE(u)/du is uniformly
bounded in e.
(b) In the region ~ - 6 <. w <. fl + 3, dUE(w)/dw is uniformly bounded.
PROOF. (a) According to the possibilities for shapes of (uE(~), we (~)) in the region w ~<ot
and w/> r, there are three cases:
Case A. There is a critical point for uE(~) in the region w ~<oe (or w ~>13).
We can calculate from (4.1.0) to get
d (dUE(__w))(dUE(w)) e
e ~-~ dw = dw
+ p'(wE(~)). (4.3.3)
At the critical point ~ = re of uE(~), dUE(w)/dwl~=r~ = 0. Then Equation (4.3.3) says that
dUE(w)
dw ~=~
is decreasing as ~ increases from re. But dUE(w)/dw cannot decrease to below
-maxw,<~w<~w, v/lff(w)l since d/d~(dUE(w)/dw) will become positive if dUE(w)/dw
reaches -maxw,<<w<<w, v/lff(w)l. Similarly, as $ decreases from re, dUE(w)/dw will
increase but never reaches maxw, <<w<<.
w, v/lff(w)l because if it does, d/d~(dUE(w)/dw)
will become negative. This shows that
dUE(w)
dw
~< max v/lp'(w)l (4.3.4)
w, <<.
vo<<.
w*
if uE(~) has a critical point re with WE(re) ~<Otor WE(re) ~>ft.
Case B. The function wE(~) has a critical point in the region w ~<ot (or w ~>fl). Then,
uE(~) is necessarily monotone in the region.
If the critical point is in the region w ~<or, then it is the absolute minimum point of
wE(~), according to Lemma 4.1.1. If the critical point re is in the region w ~>r, then it
is the absolute maximum point of wE(~). Thus, WE(re) is in the region w ~<c~ - 3 (or
w>~f+~).
From (4.1.0), we derive that
du )--[l+ du )]" (4.3.5)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-408-320.jpg)
![Dynamic flows with liquid~vaporphase transitions 411
At the critical point ~ --re, we have d~(u)J~=re --0. Similar to our analysis follow-
ing (4.3.3), we can prove that
dWe (u)
du
1
~< max (4.3.6)
~I~,,~-~luI~+~,~*~ v/Ip'(w)l
for w E [w,, ot - 6] U [fl + 6, w*].
Case C. There is no critical point for ue(~) or we(~) in the region w ~<ot (or w >t/3).
For this case, we claim that one of IdUe(w)/dwl and IdWe(u)/dul are bounded
uniformly. To prove this claim, it suffices to prove if one of them is not bounded uniformly,
then the other is. There are the following three possibilities:
Subcase C1. The function dUe(w)/dw is not bounded from below uniformly in e in the
region w ~<or. Let the absolute minimum point of dUe (w)/dw in the closed region w ~<ot
be ~ -- re. Then there is a sequence en, such that
duen] --+ -cx3
dw ~=r~n
as n --+ oc. (4.3.7)
For simplicity, we denote this sequence by e.
In the region w ~<c~, it is necessary that dwe,/d~ > 0 in order to connect to w(ec) =
w+ > /3 > or. Therefore, we have due,/d~ < 0 in the region w ~< c~ due to (4.3.7).
Evaluating Equation (4.3.5) around the point ~ = re, we find that dUe (w)/dw is increasing
at ~ = re, the absolute minimum point of dUe(w)/dw over w_ ~< w ~< or. This implies
we (re) -- w_ and hence re = -oc. Now, applying our reasoning from (4.3.5)-(4.3.6) and
using (4.3.15):
dWe (u)
du
~=~n
--+0 as n-~ cx~,
we can obtain (4.3.6).
Subcase C2. The function dUe(w)/dw is not bounded from above uniformly in e in
the region w ~< or. Let the absolute maximum point of dUe(w)/dw in the closed region
{~: we(~) ~<cr} = [-oc, 01] be ~ = re. Then there is a sequence en, such that
dUe(w)
dw
oc as n ~ oe. (4.3.8)
For simplicity, we denote this sequence by e.
Similar to Subcase C1, when w ~<or, it is necessary that dwe,/d~ > 0. Therefore, we
have duen/d~ > 0 in the region w ~<or. By Equation (4.3.5), we find that dUe(w)/dw is
increasing at ~ as long as
dUe(w)
>i max v/Ip'(w)l.
dw ~=~](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-409-320.jpg)
![412 H. Fan and M. Slemrod
This implies we(re) = or. Recalling that We(O1)--~Ol and We(02)-"fl, we can see that
re = 01 and
dUe (w) /> dUe (w)
dw dw ~=01
for 01 ~ ~ ~ 02 (4.3.9)
due to Lemma 4.1.2. Then we have
f
w(O2) dUe(w)
Uen (02) -- Uen (01) --
,JtO(01) dw
duen] (13 -or) ~ oo
~ d w / > ~ ~=01
as n --+ c~. This violates the uniform boundedness of ue(~), Theorem 4.2.1. This
contradiction shows that Subcase C2 cannot occur.
Subcase C3. The function dUe(w)/dw is not bounded uniformly in e in the region
w>r
The proof for this case is similar to Subcases C 1 and C2.
Combining the Cases A-C, we complete the proof of (a).
(b) Since We(u) is convex in the region ot ~< w ~< 13, the absolute extreme values of
dWe(u)/du over the region ot - 6 ~< w ~< fl + 6 must occur in the region [or - 6, c~] U
[fl, fl + 6]. Our proof for Subcase C2 for (a) shows that dUe (w)/dw is bounded uniformly
from above when w 6 [or - 6, or]. Now we shall prove that dUe(w)/dw is also bounded
uniformly from below when w ~ [or - 6, or]. To this end, we assume its contrary, i.e., there
is a sequence {en} such that
dUe(w)]
dw ~=r~
--~ -cx~ (4.3.10)
as n ~ o0 for some re 6 R with wen(re) 6 [c~- 6, c~]. Equation (4.3.3) implies that
dUe(w)/dw is decreasing as ~ decreases when
dUe(w)
~<- max V/Ip'(w)l.
dw w,<<.w<~w*
Let ~ = ~1 ) --OO be the point such that W(~I) -- W- and w_ ~< we(~) ~ we(re) for
~ (~l, re ]. Then, we have
_ fw(r~) dUe(w)
Ren(7~e)-- Uen(~1) dW(~l) dw
du en I
~dw<~---d-~w ~=~(c~ -~ - w_) ~ -oc
as n --+ cxz. This violates the uniform boundedness of we(~), Theorem 4.2.2. This
contradiction proves that dUe(w)/dw is uniformly bounded from above in the region
[ol - a,~].
The same proof can be used to prove that dUe(w)/dw is uniformly bounded when
w ~ [f,/~ +a]. D](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-410-320.jpg)
![414 H. Fan and M. Slemrod
For convenience, we parameterize the base curve (U(s), W(s)) where s is the length of
the arc joining points (u_, w_) and (U(s), W(s)). In this kind of parameterization, the s
defined by (u(~), w(~)) = (U(s), W(s)) increases when ~ increases.
Now, we study the discontinuities of (u(~), w(~)). Let ~0 be a point of discontinuity of
(u(~), w(~)). We use C~o to denote the portion of the base curve in the (u, w)-plane that
connects points (u(~0-), w(~0-)) and (u(~0+), w(~0+)). We fix (t/, tb) E C~0. Similar
to [20], we define, for n large, ~e, (w; t/, tb) to be the branch of the inverse function of
w = wen(~) for which
(4.3.14)
as n ~ oo. We further define, for n large, ~e,, fien, W'enby the relations
~en := ~En(t~) + e(, (4.3.15)
fi~n(() "-- ue, (~e,), (4.3.16)
w"~n(() "- wen(~en). (4.3.17)
LEMMA 4.3.3. Let ~o be a point of discontinuity of (u(~), w(~)). For (Fte
n((), wAen
(())
defined above, there is a subsequence of {en}, also denoted by {en}, such that
(blen(~'), WAen(~'))--->(t~(~'), W(~'))E C1(I~;I~2) asn --+ oo (4.3.18)
uniformly for ( in a compact subset of N. (fi(( ), ~(( )) satisfies thefollowing initial value
problem"
d~(()
d(
= -~0(t/(()- u(~0-))+ p(@(())- p(w(~o-)), (4.3.19a)
d~(()
d(
(4.3.19b)
(0) -- t/ ~'(0) -- ffJ. (4.3.19c)
Furthermore, (fi((), ~(()) lies on C~o.
PROOF. Clearly, (fien((), w'~n(()) have uniformly bounded total variation since (ue(~),
we(~)) do. Thus, there is a subsequence of {en}, again denoted by {en}, such that
(fie~ ((), w'~. (()) --+ (fi((), ~(g')) as n --+ (4.3.20)
for any ~"6 IR.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-412-320.jpg)
![Dynamicflows with liquid~vaporphase transitions 415
By Lemma 4.3.1, we can choose a small neighborhood V~0 of (u(~0-), w (~0-)) in the
(u, w)-plane such that
due',
s
e E IR such that (uen (~), we',(s
e)) E V~o} (4.3.21a)
or
dwe',
du
E N such that (ue,,(~), we,,(~)) E V~o] (4.3.21b)
is bounded uniformly in n. Since in each of the region w ~<or, ot ~< w ~< fl, w ~> fl, at
least one of ue(~) and we(~) is monotone, in each of above three regions, one of U(s)
and W(s) is monotone. We can further choose V~0 small and (ua, wa) 6 C~0 n V~0 such
that U(s) or W(s) is monotone in V~0. For definiteness, we can assume, without loss of
generality, that (4.3.21a) holds and both we',(~) and W(s) is monotone in V~0.The proof
for the other case is similar.
There is, for n large,
(4.3.22)
such that
3
(4.3.23a)
3
Iw;n<O
n>l -YnnZg (4.3.23b)
From (4.3.7) and
-+
it is easily seen that
~e',(wa) -+ ~0 (4.3.24)
and hence 0e', --+ ~0. Since W(s) and we',(~) are monotone, the limit liminfn-+ec we',(0e',)
lies between w(~0-) and wa. Thus, extracting, if necessary, another subsequence, we
deduce
Wen(Oe',) --+ tO2 as n --+ oo (4.3.25)
for some 11)2 between w(~0-) and wa. Then, by (2.8a), we have that
limn-+o~ue', (we', (Oe',)) - g(w2) -- U2, (4.3.26)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-413-320.jpg)
![416 H. Fan and M. Slemrod
where (u2, 1/)2) 9 V~o. For simplicity, we shall write e instead of en in the rest of this
section. Integrating Equations (1.3) from 0e to re :-- ~e(tb) + e~', we get
d~s(~')
d~"
= -~o[fie(~')- ue(Oe)] + P(We (~')) - p(we(Oe))
f0 ~
+ eue' (Oe) - (~ - ~o)u~e(~) d~, (4.3.27a)
d~e (~')
d~"
= -[a~(~- .~(o~] - ~o[~(c~- w~(O~]
! !
+ ewe(Oe) - (~ - ~0)we(~) d~. (4.3.27b)
By (4.3.23) ew~e(Oe) and eu~e(Oe) approach 0 as e --+ 0 uniformly in ~'. Recalling that
0e --+ ~o, re ~ ~o as n ~ c~, uniformly in ~" for ~"in compact subsets of R, we see that
the last term in (4.3.27a, b) vanish, as n ~ cxz,uniformly in ~"in a compact set. A classical
theorem of the theory of ordinary differential equations implies that (fie(g), we(~')) --+
(fi(~'), ~(~')), as n --+ ~x~,uniformly on compact subsets of R, and that
dfi(~')
d~"
= --~O(t~(~') -- u2) -t- p(w(~')) -- p(w2), (4.3.28a)
d~(~)
m
d~
[a~- .~] - ~0(~- w~), (4.3.28b)
fi(0) = if, ~(0) = if). (4.3.28c)
By letting V~o shrink to (u(~o-), w(~o-)) so as to force (u2, W2) ~ (U(~0--), W(~0--)),
we obtain (4.3.19).
Similar to our proof of Lemma 4.3.2, we can prove that (fi(~'), ~(~')) is on the curve
(U(s), W(s)) for all ~" e R. We note that (fi(0) = ~(0)) = 07, tb) e C~o, and that as ~"
increases (or decreases) from ~"= 0, the point (fi(~'), ~(~')) moves toward the end point
(u(~o+), w(~o+)) ((u(~o-), w(~o-))) along C~o. The point (fi(~'), ~(~')) cannot cross
(u(~o+), w(~o+)) and (u(~o-), w(~o-)) to go outside of C~o. This is because if it did
go out of C~o, there would be a point ~'l 9 R such that
(a(~l~. ~(~l~)- (.(~o-~. w(~o-~) or (.(~o+~. w(~o+~)
Then the Rankine-Hugoniot condition, satisfied by any jump solution of (3.1) with
speed ~o,
-~o(u(~o+)- u(~o-))+ p(w(~o+))- p(w(~o-))= O, (4.3.29a)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-414-320.jpg)
![Dynamic flows with liquid~vapor phase transitions 417
- - [. - - o (4.3.29b)
yields that at (fi((), ~(()) -- (u(~0-), w(~0-)) or (t~((), ~(()) _= (u(~0+), w(~0+)) by
the uniqueness of solutions of initial value problems of systems of ordinary differential
equations. This, however, violates the (fi(0), ~(0)) = (t7,tb) ensured by (4.3.14)-(4.3.17).
This contradiction proves that (/~((), ~(()) ~ C~0 for all ( E R. F-1
COROLLARY 4.3.4. Let (u(~), w(~)) be a weak solution of (3.1) constructed as the limit
of a convergent sequence {Uen(~), Wen(~)} of solutions of (3.14) with the same initial
data (3.1)3. Then, (u(~), w(~)) is also admissible by the traveling wave criterion based
on (3.13), which is the same as that based on (1.2) when A = 1/4.
PROOF. The limit (u(~), w(~)) has bounded total variation. Then points of discontinuity
of (u(~), w(~)) are points of jump discontinuity. Let ~0 be a point of jump discontinuity
of (u(~), w(~)). Lemma 4.3.3 states that (4.3.19) has a solution. We note that the
system (4.3.19)1 and 2 is equivalent to the traveling wave equation (3.13) and the speed
s = ~0. Indeed, the speed of the jump discontinuity of (u(~), w(~)) at x/t = ~ -~o
is ~0. We note that C~0 in last lemma is the portion of the base curve connecting
the points (u(~0-), w(~o-)) and (u(~o+), w(~0+)). As ( increases from 0 to oo, the
point (fi((), ~(()) moves monotonically toward (u(~0+), w(~0+)) along the curve C~o.
In the ( -+ cxz limit, (fi((), ~(()) must approach to an equilibrium point of (4.3.19)
on C~0.Equilibrium points of (4.19) are points (u l, Wl) that satisfies the Rankin-Hugoniot
condition (4.3.27) with (u(~0+), w(~0+)) replaced by (ul, Wl). Similarly, as ( decreases
from 0 to -c~, the point (/~((), ~(()) will move toward (u (~o-), w(~o-)) and approaches
an equilibrium point in the ( -+ cx~ limit. Thus, when there are only finitely many
equilibrium points for each fixed speed ~0, the jump discontinuity (u(~0-), w(~o-)),
(u (~0+), w(~o+)) can be connected together by finitely many traveling waves of the same
speed ~'o. F-l
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![The Cauchyproblem for the Euler equationsfor compressiblefluids 423
1. Introduction
The Cauchy problem for the Euler equations for compressible fluids in d space dimensions
is the initial value problem for the system of d + 2 conservation laws
Otp+V.m=O,
(mm)
8tm+V. +Vp--O,
P
(1.1)
for (x, t) ~ IRd+l , Rd++l "= ]Rd • (0, OO), with initial data
(p, m, E)lt=o : (po, mo, Eo)(x), x ~ ~d, (1.2)
where (P0, m0, E0)(x) is a given vector function of x 6 IRd.
System (1.1) is closed by the constitutive relations
1 ]ml 2
p = p(p, e), E = + pe. (1.3)
2 p
In (1.1) and (1.3), r -- 1/p is the deformation gradient (specific volume for fluids, strain
for solids), v -- (Vl ..... Vd) T is the fluid velocity, with pv = m the momentum vector, p
is the scalar pressure, and E is the total energy, with e the internal energy which is a given
function of (r, p) or (p, p) defined through thermodynamical relations. The notation a | b
denotes the tensor product of the vectors a and b. The other two thermodynamic variables
are the temperature 0 and the entropy S. If (p, S) are chosen as the independent variables,
then the constitutive relations can be written as
(e, p, O) = (e(p, S), p(p, S), O(p, S)), (1.4)
governed by
P
0 dS = de + p dr = de - --T dp.
p~
(1.5)
For a polytropic gas,
R
p = RpO, e = cvO, Y = 1 +--, (1.6)
Cv
and
p = p(p, S) = xp z e s/cv, tr pz_ leS/cv
e -- ~ , (1.7)
y--1](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-421-320.jpg)
![424 G.-Q. Chen and D. Wang
where R > 0 may be taken to be the universal gas constant divided by the effective
molecular weight of the particular gas, Cv > 0 is the specific heat at constant volume,
y > 1 the adiabatic exponent, and x > 0 can be any constant under scaling.
As will be shown in Section 4, no matter how smooth the Cauchy data (1.2) are,
solutions of (1.1) generally develop singularities in a finite time. Hence, system (1.1) is
complemented by the Clausius inequality
Ot(pa(S)) + V. (ma(S)) ~>0 (1.8)
in the sense of distributions for any a(S) ~ C1,at(S) ~ 0, in order to single out physically
relevant discontinuous solutions, called entropy solutions.
The Euler equations for a compressible fluid that flows isentropically take the following
simpler form:
Otp+V.m=O,
Otm+V. (m|
m) (1.9)
+Vp=0,
where the pressure is regarded as a function of density, p = p(p, So), with constant So. For
a polytropic gas,
P(P) = xoP• V>I, (1.10)
where x0 > 0 is any constant under scaling. This system can be derived as follows. It is
well known that, for smooth solutions of (1.1), the entropy S(p, E) is conserved along
fluid particle trajectories, i.e.,
Ot(/9S) + V. (mS) = O. (1.11)
If the entropy is initially a uniform constant and the solution remains smooth, then (1.11)
implies that the energy equation can be eliminated, and the entropy S keeps the same
constant in later time, in comparison with non-smooth solutions (entropy solutions) for
which only S(x, t) ~ min S(x, 0) is generally available (see [297]). Thus, under constant
initial entropy, a smooth solution of (1.1) satisfies the equations in (1.9). Furthermore,
it should be observed that solutions of system (1.9) are also a good approximation to
solutions of system (1.1) even after shocks form, since the entropy increases across a shock
to third-order in wave strength for solutions of (1.1) (cf. [120]), while in (1.9) the entropy
is constant. Moreover, system (1.9) is an excellent model for isothermal fluid flow with
F = 1, and for shallow water flow with y = 2.
In the one-dimensional case, system (1.1) in Eulerian coordinates is
Otp + Oxm =0,
Otm + Ox --7 + p =0,
OtE + OX(p(E + p)) =0,
(1.12)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-422-320.jpg)
![426 G.-Q. Chen and D. Wang
Another main difficulty is nonstrict hyperbolicity or resonance of (1.16), that is, there
exist some 090 9 Sd-1 and u0 9 I~n such that ~i(u0, coo) -- )U(u0, coo) for some i ~ j. In
particular, for the Euler equations, such a degeneracy occurs at the vacuum states or from
the multiplicity of eigenvalues of the system.
The correspondence of (1.8) in the context of hyperbolic conservation laws is the Lax
entropy inequality:
OtO(u) + V. q(u) ~<0 (1.18)
in the sense of distributions for any C2 entropy-entropy flux pair (r/, q):]1~n ~ ]1~ •
I~d, q = (ql ..... qd), satisfying
V2rl(U) ~ O, Vqi(u) : Vr/(u)Vfi(u), i = 1..... d.
Most sections in this paper focus on the Cauchy problem for one-dimensional hyperbolic
systems of n conservation laws
OtU+Oxf(u)=O, ueR n, xeI~, t>O, (1.19)
with Cauchy data:
ult=0 = u0(x). (1.20)
The Euler equations can describe more complicated physical fluid flows by coupling
with other physical equations.
One of the most important examples is the Euler equations for nonequilibrium
thermodynamic fluids. In local thermodynamic equilibrium as we discussed above,
system (1.1) is closed by the constitutive relation (1.3). When the temperature varies over
a wide range, the gas may not be in local thermodynamic equilibrium, and the pressure p
may then be regarded as a function of only a part e of the specific internal energy, while
another part q is governed by a rate equation:
Ot(pq) 4- Vx . (mq) =
Q(p,e) -q
es(p, e)
(1.21)
and
Iml2
p = p(p, e), E = 4- p(e 4- q), (1.22)
2p
where e > 0 is a parameter measuring the relaxation time, which is small in general,
and Q (p, e) and s (p, e) are given functions of (p, e). The equations in (1.1) and (1.21)
with (1.22) define the Euler equations for nonequilibrium fluids, which model the
nonequilibrium thermodynamical processes.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-424-320.jpg)
![The Cauchyproblemfor the Euler equationsfor compressiblefluids 427
Another important example is the inviscid combustion equations that consist of the Euler
equations in (1.1) adjoined with the continuum chemistry equation:
O,(pZ) 4- V . (mZ) -- -ck(O)pZ, 4>(0)- Ke -~176176 (1.23)
where 00 and K are some positive constants, Z denotes the mass fraction of unburnt gas so
that 1 - Z is the mass fraction of burnt gas. Here we assume that there are only two species
present, the unburnt gas and the burnt gas, and the unburnt gas is converted to the burnt
gas through a one-step irreversible exothermic chemical reaction with an Arrhenius kinetic
mechanism. As regards the equations in (1.1), a modification of the internal energy e is the
only change in these equations. The internal energy of the mixture, e(p, S, Z), is defined
within a constant by
e(p, S, Z) -- Zeu(p, S) 4- (1 - Z)eb(p, S),
with eu and eb the internal energies of the unburnt and burnt gas, respectively. For
simplicity, we assume that both of the burnt and unburnt gas are ideal with the same y-law
so that
eu(p, S) = CvO+ q0, eb = cvO,
with q0 > 0 the normalized energy of formation at some reference temperature for the
unburnt gas for an exothermic reaction. Then
e(p, S, Z) = cvO(p, S) 4- qoZ,
p(p, S)
0 (p, S) -- ~ . (1.24)
Rp
Then the equations in (1.1) and (1.23) with (1.24) define the inviscid combustion equations,
which model detonation waves in combustion.
This paper is organized as follows.
In Section 2, we present a local well-posedness theory for smooth solutions and then
in Section 3 a global well-posedness theory for smooth solutions. In Section 4, we
exhibit the formation of singularity in smooth solutions, the main feature of the Cauchy
problem for the Euler equations. In Section 5, we present a local well-posedness theory for
discontinuous entropy solutions.
From Section 6 to Section 10, we discuss global well-posedness theories for discontin-
uous entropy solutions.
In Section 6, we present a global theory for discontinuous entropy solutions of the
Riemann problem, the simplest Cauchy problem with discontinuous initial data. First we
recall two Lax's theorems for the local behavior of wave curves in the phase space and
the existence of global solutions of the Riemann problem, respectively, for general one-
dimensional conservation laws with small Riemann data. Then we discuss the construction
of global Riemann solutions and their behavior for the isothermal, isentropic, and non-
isentropic Euler equations in (1.12)-(1.15)with large Riemann data, respectively.
In Section 7, we focus on the global discontinuous solutions obtained from the Glimm
scheme [130], called Glimm solutions. We first describe the Glimm scheme for hyperbolic](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-425-320.jpg)
![428 G.-Q. Chenand D. Wang
conservation laws and a global well-posedness theory for the Glimm solutions, including
the existence, decay, and Ll-stability of the Glimm solutions. The Glimm scheme is
also applied to the construction of global entropy solutions of the isothermal Euler
equations with large initial data. We also present an alternative method, the wave-front
tracking method, to construct global discontinuous solutions, which can be identified
with a trajectory of the standard Riemann semigroup, and to yield the L 1-stability of the
solutions.
In Section 8, our focus is on general global discontinuous solutions in L~ A B ~oc
satisfying the Lax entropy inequality and without specific reference on the method for
construction of the solutions. We first describe a theory of generalized characteristics
and its direct applications to the decay problem of the discontinuous solutions under the
assumption that the traces of the solutions along any space-like curves are functions of
locally bounded variation. Then we study the uniqueness of Riemann solutions and the
asymptotic stability of entropy solutions in BV for gas dynamics, without additional a
priori information on the solutions besides the natural Lax entropy inequality.
In Section 9, our focus is on the one-dimensional system of the isentropic Euler
equations and its global discontinuous solutions in L~ satisfying only the weak Lax
entropy inequality. We first carefully analyze the system and its entropy-entropy flux pairs.
Then we describe a general compactness framework, with a proof for the case y = 5/3,
for establishing the existence, compactness, and decay of entropy solutions in L~, and
the convergence of finite-difference schemes including the Lax-Friedrichs scheme and the
Godunov scheme. We discuss the stability of rarefaction waves and vacuum states even
in a broader class of discontinuous entropy solutions in Le~. We also record some related
results for the system of elasticity and the non-isentropic Euler equations.
In Section 10, we discuss global discontinuous solutions for the multidimensional
case. We describe a shock capturing difference scheme and its applications to the
multidimensional Euler equations for compressible fluids with geometric structure. Then
we present some classifications and phenomena of solution structures of the two-
dimensional Riemann problem, especially wave interactions and elementary waves, for
the Euler equations and some further results in this direction.
In Section 11, we consider the Euler equations for compressible fluids with source terms.
Our focus is on two of the most important examples: relaxation effect and combustion
effect. Some new phenomena are reviewed.
We remark that, in this paper, we focus only on some recent developments in the
theoretical study of the Cauchy problem for the Euler equations for compressible fluids.
We refer the reader to other papers in these volumes, as well as Glimm and Majda [134],
Godlewski and Raviart [138], LeVeque [189], Lions [201], Perthame [255], Tadmor [296],
Toro [306], and the references cited therein for related topics including various kinetic
formulations and approximate methods for the Cauchy problem for the Euler equations.
2. Local well-posedness for smooth solutions
Consider the three-dimensional Euler equations in (1.1) and (1.7) for polytropic compress-
ible fluids staying away from the vacuum, which are rewritten in terms of the density p 6 R,](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-426-320.jpg)
![The Cauchyproblemfor the Euler equationsfor compressiblefluids 429
the velocity v 6 R 3, and the entropy S 6 11~(taking tc = Cv -- 1 without loss of generality)
in the form:
Ot/9 + V. (pv) = 0,
Ot(pv) + V. (pv | v) + Vp = 0,
OtS + v. VS-O,
(2.1)
with the equation of state: p = p(p, S) -- p• es, y > 1. System (2.1) is a 5 • 5 system
of conservation laws. It can be written in terms of the variables (p, v, S) in the equivalent
form in the region where the solution is smooth:
Otp -+-v. Vp + yp V . v=0,
p(Otv + v. Vv) + V p- 0,
OrS + v. VS =0,
(2.2)
with p = p(p, S) - pl/• e-S~•
The norm of the Sobolev space HS (Ra) is denoted by
12
Ilglls2= E d[Dag dx.
Ioll~<s
For g 6 L c~([0, T]; HS), define
111g IIIs,r = sup IIg<,t>lls,
o~<t~<T
For the Cauchy problem of (2.2) with smooth initial data:
(p, v, S)lt=0 - (p0, v0, S0)(x), (2.3)
the following local existence theorem of smooth solutions holds.
THEOREM 2.1. Assume (Po, vo, So) 6 H s N Lc~(R3) with s > 5/2 and po(x) > 0. Then
there is a finite time T E (0, cx)), depending on the H s and L ~ norms of the initial
data, such that the Cauchy problem (2.2) and (2.3) has a unique bounded smooth
solution (p, v, S) E C 1(R3 x [0, T]), with p(x, t) > 0 for all (x, t) E R 3 x [0, T], and
(p, v, S) 6 C([0, T]; H s) A C 1([0, T]; HS-1).
Consider the Cauchy problem (1.16) and (1.17) for a general hyperbolic system of
conservation laws with the values of u lying in the state space G, an open set in 1t~
n.
The state space G arises because physical quantities such as the density should be positive.
Assume that (1.16) has the following structure of symmetric hyperbolic systems: For all
u ~ G, there is a positive definite symmetric matrix Ao(u) that is smooth in u and satisfies
CO1In ~<A0 (u) <~C0In (2.4)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-427-320.jpg)
![430 G.-Q. Chen and D. Wang
with a constant co uniform for u E G1, for any G1 C G1 C G, such that Ai(u) =
A0(u)Vfi (u) is symmetric, where Vfi (u), i = 1..... d, are the n • n Jacobian matrices
and In is the n • n identity matrix. A consequence of this structure for (1.16) is that the
linearized problem of (1.16) and (1.17) is well-posed (see Majda [223]). The matrix A0(u)
is called the symmetrizing matrix of system (1.16). Multiplying (1.16) by the matrix A0(u)
and denoting A(u) = (A1 (u) ..... Ad(u)) yield the system:
Ao(u)Otu + A(u)Vu = 0. (2.5)
An important observation is that almost all equations of classical physics of the form (1.16)
admit this structure. For example, the equations in (2.2) for polytropic gases are
symmetrized by the 5 • 5 matrix
(yp) 1 0 O)
Ao(p, S) = 0 p(p, S)I3 0 .
0 0 1
Therefore, Theorem 2.1 is a consequence of the following theorem on the local existence
of smooth solutions, with the specific state space G - {(p, v, S)7-" p > 0} C ~5, for the
general symmetric hyperbolic system (1.16).
THEOREM 2.2. Assume that u0" R d --+ G is in H s fq Loo with s > d/2 + 1. Then, for
the Cauchy problem (1.16) and (1.17), there exists a finite time T -- T(llu011s, Ilu011Lo~)
(0, oo) such that there is a unique bounded classical solution u 6 C 1(Rd x [0, T]) with
u(x, t) ~ G for (x, t) ~ Rd x [0, T] and u 6 C([0, T]; H s) 0 CI([0, T]; HS-1).
The proof of this theorem proceeds via a classical iteration scheme. An outline of the
proof of Theorem 2.2 (thus Theorem 2.1) is given as follows.
To prove the existence of the smooth solution of (1.16) and (1.17), it is equivalent
to construct the smooth solution of (2.5) and (1.17) by applying the symmetrizing
matrix A0(u). Choose the standard mollifier j (x) 6 C~ (Rd), supp j (x) ___{x" Ixl ~< 1},
j(x) >~ 0, fRdj(x)dx = 1, and set je(x) = e-dj(x/e). For k- 0, 1, 2..... take e~ =
2-~eo, where eo > 0 is a constant, and define u~ 6 Coo (Rd) by
u~(x) = Jeku0(x) - f•d je~ (X -- y)u0(y) dy.
We construct the solution of (2.5) and (1.17) through the following iteration scheme: Set
u~ t) - u~ and define u~+1(x, t), for k - 0, 1, 2..... inductively as the solution of
the linear equations:
A0(uk)0tu k+l + A(u/~)Vu/~+1 -- O, uk+l It=0 -- u~+1 (x). (2.6)
From the well-known properties of the mollification: Ilu~ - u011s ~ 0, as k --+ ~, and
Ilu~ - u0 II0 <~ C0ek Ilu0 Ill, for some constant Co, it is evident that u~+1 E C ~ (R d X [0, Tk])](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-428-320.jpg)
![The Cauchyproblem for the Euler equationsfor compressiblefluids 431
is well-defined on the time interval [0, Tk]. Here Tk > 0 denotes the largest time where the
estimate Illu~ - U0111~,Tk~< C1 holds for some constant C1 > 0. Then there is a constant
T, > 0 such that Tk >~ T, (T0 = c~) for k -- 0, 1, 2..... which follows from the following
estimates"
k+l
Illu~+l - u~ <- c,, Illu, Ills-l,T, ~<Ce, (2.7)
for all k - 0, 1, 2..... with some constant C2 > 0.
From (2.6), we obtain
A0(uk)0t (u~+l- u ~) + A (u~:)V(u~+1- u~) = E~:, (2.8)
where
Ek = -(A0(u k) -- Ao(uk-1))Otuk- (A(u k) -- A(uk-1))Vu k-
Use the standard energy estimate method for the linearized problem (2.8) to obtain
IIIu~+' - u~l110,~ ~ CeCY ([[u~+' - u~ll0 + zl IIE~ I110,T)
The property of mollification, (2.7), and Taylor's theorem yield
I1-~+' - -~oIio ~<C2-~' ItiE, it10,~ ~ cI IIu~ - u~-' II10,~.
For small T such that C2T exp(CT) < 1, one obtains
OO
El Iluk+l - u l110 T
k=l
which implies that there exists u E C ([0, T]; L 2(•d)) such that
lim IIIu~ - ul110,~ - 0. (2.9)
k--+cx~
From (2.7), we have Illu~llls,T + Illut~llls_l,T ~ C, and u~(x, t) belongs to a bounded set
of G for (x, t) ~ ~d x [0, T]. Then the interpolation inequalities imply that, for any r with
O<.r<s,
1-r/s r/s 1-r/s
Illu~ - u' Ill ~ <~CsII1-~ - -' III0,~ II1-~ --' II ,,~ <~c lllu~ --' III
r, 0, T
(2.10)
From (2.9) and (2.10), limk~ IIluk- ulllr, T --0 for any 0 ~< r < s. Thus, choosing
r > d/2 + 1, Sobolev's lemma implies
.k __~u in C([0, t]; C 1(I~d)). (2.11)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-429-320.jpg)
![432 G.-Q. Chen and D. Wang
From (2.8) and (2.11), one can conclude that u~ ~ u in C([0, T]; C(IRd)), u a cl(]l~ d •
[0, T]), and u(x, t) is the smooth solution of (1.16) and (1.17).
To prove u ~ C([0, T]; H s) A C 1([0, T]; HS-1), it is sufficient to prove u ~ C([0, T];
HS), since it follows from the equations in (2.5) that u ~ C l ([0, T]; Hs-l). The proof can
be further reduced to verifying that u(x, t) is strongly fight-continuous at t = 0, since the
same argument works for the strong fight-continuity at any other t ~ [0, T), and the strong
fight-continuity on [0, T) implies the strong left-continuity on (0, T] because the equations
in (2.5) are reversible in time.
REMARK 2.1. Theorem 2.2 was established by Majda [223] which relies solely on
the elementary linear existence theory for symmetric hyperbolic systems with smooth
coefficients (Courant and Hilbert [77]), as we illustrated above. Moreover, a sharp
continuation principle was also proved there: For u0 ~ H s, with s > d/2 + 1, the interval
[0, T) with T < oe is the maximal interval of the classical H s existence for (1.16) if and
only if either II(Ut, Du)IIz~ ~ oo as t --+ T, or, as t ~ T, u(x, t) escapes every compact
subset K ~ G. The first catastrophe in this principle is associated with the formation
of shock waves in the smooth solutions, and the second is associated with a blow-up
phenomenon.
Kato also gave a proof of Theorem 2.2, in [164], which uses the abstract semigroup
theory of evolution equations to treat appropriate linearized problems. In [165], Kato also
formulated and applied this basic idea in an abstract framework which yields the local
existence of smooth solutions for many interesting equations of mathematical physics. See
Crandall and Souganidis [78] for related discussions.
In [226], Makino, Ukai and Kawashima established the local existence of classical solu-
tions of the Cauchy problem with compactly supported initial data for the multidimensional
Euler equations, with the aid of the theory of quasilinear symmetric hyperbolic systems; in
particular, they introduced a symmetrization which works for initial data having compact
support or vanishing at infinity. There are also discussions on the local existence of smooth
solutions of the three-dimensional Euler equations (2.1) in Chemin [35].
REMARK 2.2. For the one-dimensional Cauchy problem (1.19) and (1.20), it is known
from Friedrichs [122], Lax [175], and Li and Yu [195] that, if u0(x) is in C 1 for all x E R
with finite C 1 norm, then there is a unique C 1 solution u(x, t), for (x, t) 6 IR x [0, T], with
sufficiently small T. As a consequence, the one-dimensional Euler equations in (1.12)-
(1.15) admit a unique local C 1 solution provided that the initial data are in C 1 with finite
C 1 norm and stay away from the vacuum.
3. Global well-posedness for smooth solutions
Consider the Cauchy problem for the one-dimensional isentropic Euler equations of gas
dynamics in (1.14), for x ~ IR and t > 0, with initial data:
(p, re)It=0 = (p0, mo)(x), (3.1)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-430-320.jpg)
![The Cauchyproblemfor the Euler equationsfor compressiblefluids 433
and y-law for pressure:
p(p) = p• ~, > 1. (3.2)
For the case 1 < y ~< 3, which is of physical significance, system (1.14) is genuinely
nonlinear in the sense of Lax [181] in the domain {(x, t)" p(x, t) ~ 0}. For p > 0, consider
the velocity v - m/p and vo(x) - mo(x)/po(x). The eigenvalues of (1.14) are
1 --V--C, ~,2--V+C,
where c = pO, with 0 - ~ 6 (0, 1], is the sound speed. The Riemann invariants of (1.14)
are
Wl -- Wl (p, V) :--- V -+-
pO pO
0 ' W2 -- w2(p, V) := V 0
Set
WlO(X) "-- Wl (po(x), VO(X)), W20(X) "-- W2(Ro(X), VO(X))
as the initial values of the Riemann invariants. With the aid of the method of characteristics
(see Lax [178]), the following global existence theorem of smooth solutions of (1.14)
and (3.1) can be proved.
THEOREM 3.1. Suppose that the initial data (Po, vo)(x), with po(x) > O, are in CI(]R),
with finite C 1 norm and
' (x)>~o,
Wl0 W20(X ) ~>0, (3.3)
for all x ~ IR. Then the Cauchy problem (1.14) and (3.1) has a unique global C 1 solution
(p, v)(x, t), with p(x, t) > Ofor all x E R and t > O.
PROOF. First we show that, if po(x) > 0, no vacuum will develop at any time t > 0 for the
smooth solution. From the first equation of (1.14),
d
dt p = -pOx v, (3.4)
where
d
= 8t + v(x, t)G
dt
denotes the directional derivative along the direction
dx
=v(x,t).
dt
(3.5)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-431-320.jpg)
![434 G.-Q. ChenandD. Wang
For any point (2, D 6 N2 :-- {(x, t)" x 6 N, t 6 N+}, N+ = (0, oe), the integral curve
of (3.5) through (2, D is denoted by x = x(t; 2, D. At t = 0, it passes through the point
(x0(2, D, 0) "- (x(0; 2, D, 0). Along the curve x = x(t; 2, D, the solution of the ordinary
differential equation (3.4) with initial data:
Plt=o -- po(xo(~, i))
(f0 )
P(x, D = po(xo(x, D) exp - OxV(X(t; 2, D, t) dt > O.
To prove the global existence of the C 1 solution (p, v)(x, t), given the local existence
from Remark 2.2, it is sufficient to prove the following uniform a priori estimate: For any
fixed T > 0, if the Cauchy problem (1.14) and (3.1) has a unique C 1 solution (p, v)(x, t)
for x 6 R and t 6 [0, T), then the C l norm of (p, v)(x, t) is bounded on 1R x [0, T].
For a smooth solution (p, v) of system (1.14), one can verify by straightforward
calculations that the derivatives of the Riemann invariants Wl and w2 along the
characteristics are zero:
t__ 0
w 1 , w2 --0, (3.6)
where ~ = Ot + )~20x and ~ = Ot -a
t- ~.10x are the differentiation operators along the
= 0 in (3.6) with respect to the spatial variable
characteristics. Differentiate the equation w 1
x to obtain
02txWl -Jr-~,202xWl @ OWl~,2(OxWl)2 + Ow2~,2OxWlOxt02 =0.
~ - 2COxW2, by setting r = OxW l and noticing
Since 0 = w 2 --- w 2
!
1 +0 1-0 W 2
~,2 = ~,2(Wl, W2) = 2 Wl + 2 w2, OxW2 2c
one has
1-0
1 +Or2 + w~2r =0.
r~ + 2 4c
Set
0-1 0-1
s - ~ In p = ~ ln(wl - w2).
2 20
Then
1-0
= -- W 2 0to 2 S =
Ot02S 4c and s ~ t
1-0
4c
!
~ W 2.](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-432-320.jpg)
![The Cauchy problemfor the Euler equationsfor compressiblefluids 435
Thus
, 1 +Or2
r + ~ +s'r--O.
2
Set
g - eSr - - p(O-1)/2OxWl"
Then
1-0
,1-4-0(0 )20
g -------~ [Wl - w2l g2. (3.7)
Similarly, for h - - p(O-1)/2OxW2, one has
h~ ~ m ~
I-0
1+o(o )20
2 ~[wl - w21 h2. (3.8)
Let x -- x(fl, t) be the forward characteristic passing through any fixed point (/3, 0) at
t = 0, defined by
dx(fl, t)
dt
x(3, o)= 3.
According to (3.6), wl is constant along characteristics, and thus wl(x(fl, t),t)=
wl (13,0) = Wl0(fl) and sup [wl (x, t)[ = sup [wlo(x)[. Similarly, w2 is constant along
the backward characteristics corresponding to the eigenvalue )~l, and sup [wz(x,t)[ --
sup [wz0(x)[. For any given point (x(fl, t), t) on the forward characteristic x = x(fl, t),
there exists a unique c~= c~(fl, t) ~>fl such that wz(x(fl, t), t) = wzo(ot). Therefore, along
the characteristic x = x(fl, t), one has from (3.7) that
Then
where
1-0
1+o(o )20
dt = 2 ~ [w,0(fl)- w2o(ot(fl, t))]
0-1
glt-o - Po(fl)-r- l/.)'10(/~)"
g(x(fl, t), t) 2,
g(x(fl, t), t) = Po(fl) 052 W'lo(fl)
1 +foK(fl, r)dr
K (fl, t) =
1-0
1+o(o )20
2 ~ Iwlo(fl) -- W20(OC(fl, '))l
0-1
po(3)-z- w'lo(3).
(3.9)
(3.10)
(3.11)](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-433-320.jpg)
![436 G.-Q. Chen and D. Wang
From (3.3), K(/~, t) ~>0. Thus, g(x(/~, t), t) is bounded, and
1-0
g(x(~,t),t)
is also bounded. Similarly, OxW2is also bounded from (3.8). As a consequence, the C 1
norms of p = (0(Wl - w2)/2) 1/~ and v = (wl + w2)/2 are bounded on • x [0, T]. The
proof is complete. 7q
REMARK 3.1. In the proof of Theorem 3.1, the second-order derivatives of the Riemann
invariants are formally used. However, the final equality (3.10) does not involve
these second-order derivatives. Some appropriate arguments of approximation or weak
formulation can be used to show that the conclusion is still valid for C 1 solutions.
REMARK 3.2. For the global existence of smooth solutions of general one-dimensional
hyperbolic systems of conservation laws, we refer the reader to Li [194] which contains
some results and discussions on this subject. Also see Lin [197,198] and the references
cited therein for the global existence of Lipschitz continuous solutions for the case that
discontinuous initial data may not stay away from the vacuum. For the three-dimensional
Euler equations for polytropic gases in (2.1), Serre and Grassin in [141,142,273] studied
the existence of global smooth solutions under appropriate assumptions on the initial
data for both isentropic and non-isentropic cases. It was proved in [141] that the three-
dimensional Euler equations for a polytropic gas in (2.1) have global smooth solutions,
provided that the initial entropy So and the initial density P0 are small enough and the initial
velocity v0 forces particles to spread out, which are of similar nature to the condition (3.3).
4. Formation of singularities in smooth solutions
The formation of shock waves is a fundamental physical phenomenon manifested in
solutions of the Euler equations for compressible fluids, which are a prototypical
example of hyperbolic systems of conservation laws. This phenomenon can be explained
by mathematical analysis by showing the finite-time formation of singularities in the
solutions. For nonlinear scalar conservation laws, the development of shock waves can
be explained through the intersection of characteristics; see the discussions in Lax [180,
181] and Majda [223]. For systems in one space dimension, this problem has been
extensively studied by using the method of characteristics developed in Lax [178],
John [161], Liu [206], Klainerman and Majda [170], Dafermos [83], etc. For systems
with multidimensional space variables, the method of characteristics has not been proved
tractable. An efficient method, involving the use of averaged quantities, was developed
in Sideris [282] for hyperbolic systems of conservation laws and was further refined in
Sideris [283] for the three-dimensional Euler equations. Also see Majda [223].](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-434-320.jpg)
![438 G.-Q. Chen and D. Wang
If the smooth solution stays away from the vacuum, i.e., the density p has a positive
lower bound, then one concludes that g(x(fl, t), t) will blow up at a certain finite time
f
if W'lo(fl) < 0. Under the condition (4.2) and if Wlo(fl) <~-e in (4.3), g(x(fl, t), t) will
!
blow up at some finite time which is less than or equal to T. defined in (4.4). If w20(/3) ~<0
or further Wtzo(fl) <<.-e, similar consequence can be obtained from (3.8). This completes
the proof of Theorem 4.1. D
REMARK 4.1. The argument was developed in Lax [178] for 2 x 2 hyperbolic systems
of conservation laws with genuine nonlinearity. The implication of the result is that the
first derivatives of solutions blow up in a finite time, while the solutions stay themselves
bounded and away from the vacuum. This is in agreement](https://image.slidesharecdn.com/handbookofmathematicalfluiddynamics-231020124408-f2a8e2a2/85/Handbook-of-Mathematical-Fluid-Dynamics-pdf-436-320.jpg)
- Handbook of Mathematical Fluid Dynamics.pdf
- 1. Preface The motion of fluids has intrigued scientists since antiquity and we may say that the field of mathematical fluid dynamics originated more than two centuries ago. In 1755 Euler [2] gave a mathematical formulation of the principle of conservation of mass in terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear and angular momentum by PDEs. Material symmetry and frame invariance were used by Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint- Venant [11] and Stokes [12]. In the 19th century no sharp distinction was drawn between mathematicians and physicists as we sometime see in more recent times. The formulation of the equations of fluid motion could be considered as either mathematics or physics. The first work in fluid dynamics that has a "modern" mathematical flavor may have been done by Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous problem. Riemann recognized the mathematical nature of the entropy. This notion led him to his duality method for solving the non-characteristic Cauchy problem for linear hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his contemporaries. What we now call the Cauchy problem for a PDE and the search for its solution did not have the significance that it is accorded nowadays. Even Poincar6 did not raise that kind of question in his Th~orie des tourbillons. For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28. Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much as Riemann, Leray developed new mathematical tools which proved to have independent interest: e.g., weak solutions (that we now call Leray's solutions in this context) and topological degree (a joint work with Schauder [4]). Since the 1930s, the interest that mathematicians devote to fluid dynamics has unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic displays such a variety of models and questions that thousands of scientists, among them many mathematicians, focus their research on fluid dynamics.
- 2. vi Preface Because of the intense activity and the rapid increase of our knowledge, it appeared desirable to set up a landmark. Named "The Handbook of Mathematical Fluid Dynamics", it is a collection of refereed review articles written by some of the very best specialists in their discipline. The authors were also chosen for the high quality of their expository style. We, the editors, are much indebted to our colleagues who enthusiastically accepted this challenge, and who made great efforts to write for a wide audience. We also thank the referees who worked hard to ensure the excellent quality of the articles. Of course, the length of these articles varies considerably since each topic can be narrow or wide. A few of them have the appearance of a small book. Their authors deserve special thanks, for the immense work that they achieved and for their generosity in choosing to publish their work in this Handbook. At the begining of our editorial work, we decided to restrict the contents to mathematical aspects of fluid dynamics, avoiding to a large extent the physical and the numerical aspects. We highly respect these facets of fluid dynamics and we encouraged the authors to describe the physical meaning of their mathematical results and questions. But we considered that the physics and the numerics were extremely well developed in other collections of a similar breadth (see, for instance, several articles in the Handbook of Numerical Analysis, Elsevier, edited by P. Ciarlet and J.-L. Lions). Furthermore, if we had made a wider choice, our editing work would have been an endless task! This has been our only restriction. We have tried to cover many kinds of fluid models, including ones that are rarefied, compressible, incompressible, viscous or inviscid, heat conducting, capillary, perfect or real, coupled with solid mechanics or with electromagnetism. We have also included many kinds of questions: the Cauchy problem, steady flows, boundary value problems, stability issues, turbulence, etc. These lists are by no mean exhaustive. We were only limited in some places by the lack, at present, of mathematical theories. Our first volume is more or less specialized to compressible issues. There might be valid mathematical, historical or physical reasons to explain such a choice, arguing, for instance, for the priority of Riemann's work, or that kinetic models are at the very source of almost all other fluid models under various limiting regimes. The truth is more fortuitous, namely that the authors writing on compressible issues were the most prompt in delivering their articles in final form. The second and third volumes will be primarily devoted to problems arising in incompressible flows. Last, but not least, we thank the Editors at Elsevier, who gave us the opportunity of making available a collection of articles that we hope will be useful to many mathematicians and those beyond the mathematical community. We are also happy to thank Sylvie Benzoni-Gavage for her invaluable assistance. Chicago, Lyon September 2001 Susan Friedlander and Denis Serre susan@math.uic.edu denis.serre @umpa.ens-lyon.fr
- 3. Preface vii References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] A.-L. Cauchy, Bull. Soc. Philomathique (1823), 9-13; Exercices de Math6matiques 2 (1827), 42-56, 108- 111; 4 (1829), 293-319. L. Euler, M6m. Acad. Sci. Berlin 11 (1755), 274-315; 15 (1759), 210-240. J. Leray, J. Math. Pures Appl. 12 (1933), 1-82; 13 (1934), 331-418; Acta Math. 63 (1934), 193-248. J. Leray and J. Schauder, Ann. Sci. Ecole Norm. Sup. (3) 51 (1934), 45-78. L. Lichtenstein, Math. Z. 23 (1925), 89-154; 26 (1926), 196-323; 28 (1928), 387-415; 32 (1930), 608-725. P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vols. 1, 2, Oxford Univ. Press (1998). J. Nash, Bull. Soc. Math. France 90 (1962), 487-497. C.L.M.H. Navier, M6m. Acad. Sci. Inst. France 6 (1822), 375-394. S.D. Poisson, J. Ecole Polytechnique 13 (1831), 1-174. B. Riemann, G6tt. Abh. Math. C1.8 (1860), 43-65. B. de Saint-Venant, C. R. Acad. Sci. Paris 17 (1843). G.G. Stokes, Trans. Cambridge Philos. Soc. 8 (1849), 207-319. W. Wolibner, Math. Z. 37 (1933), 698-726.
- 4. List of Contributors Blokhin, A., Sobolev Institute of Mathematics, Novosibirsk, Russia (Ch. 6) Cercignani, C., Politecnico di Milano, Milano, Italy (Ch. 1) Chen, G.-Q., Northwestern University, Evanston, IL (Ch. 5) Fan, H., Georgetown University, Washington DC (Ch. 4) Feireisl, E., Institute of Mathematics AV (?R, Praha, Czech Republic (Ch. 3) Galdi, G.E, University of Pittsburgh, Pittsburgh, PA (Ch. 7) Slemrod, M., University of Wisconsin-Madison, Madison, WI (Ch. 4) Trakhinin, Yu.,Sobolev Institute of Mathematics, Novosibirsk, Russia (Ch. 6) Villani, C., UMPA, ENS Lyon, Lyon, France (Ch. 2) Wang, D., University of Pittsburgh, Pittsburgh, PA (Ch. 5)
- 5. CHAPTER 1 The Boltzmann Equation and Fluid Dynamics C. Cercignani Dipartimento di Matematica, Politecnico di Milano, Milano, Italy Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. The basic molecular model ......................................... 4 3. The Boltzmann equation ........................................... 5 4. Molecules different from hard spheres ................................... 9 5. Collision invariants .............................................. 10 6. The Boltzmann inequality and the Maxwell distributions ......................... 12 7. The macroscopic balance equations ..................................... 13 8. The H-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 9. Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 10. The linearized collision operator ...................................... 21 11. Boundary conditions ............................................. 22 12. The continuum limit ............................................. 25 13. Free-molecule and nearly free-molecule flows ............................... 33 14. Perturbations of equilibria .......................................... 36 15. Approximate methods for linearized problems ............................... 38 16. Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 17. Polyatomic gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 18. Chemistry and radiation ........................................... 52 19. The DSMC method ............................................. 57 20. Some applications of the DSMC method .................................. 61 21. Concluding remarks ............................................. 63 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME I Edited by S.J. Friedlander and D. Serre 9 2002 Elsevier Science B.V. All rights reserved
- 6. Preface The motion of fluids has intrigued scientists since antiquity and we may say that the field of mathematical fluid dynamics originated more than two centuries ago. In 1755 Euler [2] gave a mathematical formulation of the principle of conservation of mass in terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear and angular momentum by PDEs. Material symmetry and frame invariance were used by Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint- Venant [11] and Stokes [12]. In the 19th century no sharp distinction was drawn between mathematicians and physicists as we sometime see in more recent times. The formulation of the equations of fluid motion could be considered as either mathematics or physics. The first work in fluid dynamics that has a "modern" mathematical flavor may have been done by Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous problem. Riemann recognized the mathematical nature of the entropy. This notion led him to his duality method for solving the non-characteristic Cauchy problem for linear hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his contemporaries. What we now call the Cauchy problem for a PDE and the search for its solution did not have the significance that it is accorded nowadays. Even Poincar6 did not raise that kind of question in his Th~orie des tourbillons. For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28. Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much as Riemann, Leray developed new mathematical tools which proved to have independent interest: e.g., weak solutions (that we now call Leray's solutions in this context) and topological degree (a joint work with Schauder [4]). Since the 1930s, the interest that mathematicians devote to fluid dynamics has unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic displays such a variety of models and questions that thousands of scientists, among them many mathematicians, focus their research on fluid dynamics.
- 7. The Boltzmann equation andfluid dynamics 3 1. Introduction We say that a gas flow is rarefied when the so-called mean free-path of the gas molecules, i.e., the average distance covered by a molecule between to subsequent collisions, is not completely negligible with respect to a typical geometric length (the radius of curvature of the nose of a flying vehicle, the radius of a pipe, etc.). The most remarkable feature of rarefied flows is that the Navier-Stokes equations do not apply. One must then resort to the concepts of kinetic theory of gases and the Navier-Stokes equations must be replaced by the Boltzmann equation [43]. Thus the Boltzmann equation became a practical tool for the aerospace engineers, when they started to remark that flight in the upper atmosphere must face the problem of a decrease in the ambient density with increasing height. This density reduction would alleviate the aerodynamic forces and heat fluxes that a flying vehicle would have to withstand. However, for virtually all missions, the increase of altitude is accompanied by an increase in speed; thus it is not uncommon for spacecraft to experience its peak heating at considerable altitudes, such as, e.g., 70 km. When the density of a gas decreases, there is, of course, a reduction of the number of molecules in a given volume and, what is more important, an increase in the distance between two subsequent collisions of a given molecule, till one may well question the validity of the Euler and Navier-Stokes equations, which are usually introduced on the basis of a continuum model which does not take into account the molecular nature of a gas. It is to be remarked that, as we shall see, the use of those equations can also be based on the kinetic theory of gases, which justifies them as asymptotically useful models when the mean free path is negligible. In the area of environmental problems, the Boltzmann equation is also required. Understanding and controlling the formation, motion, reactions and evolution of particles of varying composition and shapes, ranging from a diameter of the order of 0.001 gm to 50 gm, as well as their space-time distribution under gradients of concentration, pressure, temperature and the action of radiation, has grown in importance, because of the increasing awareness of the local and global problems related to the emission of particles from electric power plants, chemical plants, vehicles as well as of the role played by small particles in the formation of fog and clouds, in the release of radioactivity from nuclear reactor accidents, and in the problems arising from the exhaust streams of aerosol reactors, such as those used to produce optical fibers, catalysts, ceramics, silicon chips and carbon whiskers. One cubic centimeter of atmospheric air at ground level contains approximately 2.5 x 1019 molecules. About a thousand of them may be charged (ions). A typical molecular diameter is 3 x 10-10 m (3 x 10-4 gm) and the average distance between the molecules is about ten times as much. The mean free path is of the order of 10-8 m, or 10-2 l,tm. In addition to molecules and ions one cubic centimeter of air also contains a significant number of particles varying in size, as indicated above. In relatively clean air, the number of these particles can be 105 or more, including pollen, bacteria, dust, and industrial emissions. They can be both beneficial and detrimental, and arise from a number of natural sources as well as from the activities of all living organisms, especially humans. The particles can have complex chemical compositions and shapes, and may even be toxic or radioactive.
- 8. 4 C. Cercignani A suspension of particles in a gas is known as an aerosol. Atmospheric aerosols are of global interest and have important impact on our lives. Aerosols are also of great interest in numerous scientific and engineering applications [175]. A third area of application of rarefied gas dynamics has emerged in the last quarter of the twentieth century. Small size machines, called micromachines, are being designed and built. Their typical sizes range from a few microns to a few millimiters. Rarefied flow phenomena that are more or less laboratory curiosities in machines of more usual size can form the basis of important systems in the micromechanical domain. A further area of interest occurs in the vacuum industry. Although this area existed for a long time, the expense of the early computations with kinetic theory precluded applications of numerical methods. The latter could develop only in the context of the aerospace industry, because the big budgets required till recently were available only there. The basic parameter measuring the degree of rarefaction of a gas is the Knudsen number (Kn), the ratio between the mean free path )~ and another typical length. Of course, one can consider several Knudsen numbers, based on different characteristic lengths, exactly as one does for the Reynolds number. Thus, in the flow past a body, there are two important macroscopic lengths: the local radius of curvature and the thickness of the viscous boundary layer 8, and one can consider Knudsen numbers based on either length. Usually the second one (Kn~ = )~/8), gives the most severe restriction to the use of Navier- Stokes equations in aerospace applications. When Kn is larger than (say) 0.01, the presence of a thin layer near the wall, of thickness of the order )~ (Knudsen layer), influences the viscous profile in a significant way. This and other effects are of interest in both high altitude flight and aerosol science; in particular they are all met by a shuttle when returning to Earth. Another phenomenon of importance is the formation of shock waves, which are not discontinuity surfaces, but thin layers (the thickness is zero only if the Euler model is adopted). When the mean free path increases, one witnesses a thickening of the shock waves, whose thickness is of the order of 6)~. The bow shock in front of a body merges with the viscous boundary layer; that is why this regime is sometimes called the merged layer regime by aerodynamicists. We shall use the other frequently used name of transition regime. When Kn is large (few collisions), phenomena related to gas-surface interaction play an important role. They enter the theory in the form of boundary conditions for the Boltzmann equation. One distinguishes between free-molecule and nearly free-molecule regimes. In the first case the molecular collisions are completely negligible, while in the second they can be treated as a perturbation. 2. The basic molecular model According to kinetic theory, a gas in normal conditions (no chemical reactions, no ionization phenomena, etc.) is formed of elastic molecules rushing hither and thither at high speed, colliding and rebounding according to the laws of elementary mechanics. Monatomic molecules of a gas are frequently assumed to be hard, elastic, and perfectly smooth spheres. One can also consider these molecules to be centers of forces that move
- 9. The Boltzmann equation andfluid dynamics 5 according to the laws of classical mechanics. More complex models are needed to describe polyatomic molecules. The rules generating the dynamics of many spheres are easy to describe: thus, e.g., if no body forces, such as gravity, are assumed to act on the molecules, each of them will move in a straight line unless it happens to strike another molecule or a solid wall. The phenomena associated with this dynamics are not so simple, especially when the number of spheres is large. It turns out that this complication is always present when dealing with a gas, because the number of molecules usually considered is extremely large: there are about 2.7.1019 in a cubic centimeter of a gas at atmospheric pressure and a temperature of 0~ Given the vast number of particles to be considered, it would of course be a hopeless task to attempt to describe the state of the gas by specifying the so-called microscopic state, i.e., the position and velocity of every individual sphere; we must have recourse to statistics. A description of this kind is made possible because in practice all that our typical observations can detect are changes in the macroscopic state of the gas, described by quantities such as density, bulk velocity, temperature, stresses, heat-flow, which are related to some suitable averages of quantities depending on the microscopic state. 3. The Boltzmann equation The exact dynamics of N particles is a useful conceptual tool, but cannot in any way be used in practical calculations because it requires a huge number of real variables (of the order of 102~ The basic tool is the one-particle probability density, or distribution function P(1)(x, ~, t). The latter is a function of seven variables, i.e., the components of the two vectors x and ~ and time t. Let us consider the meaning of p(1) (x, ~, t); it gives the probability density of finding one fixed particle (say, the one labelled by 1) at a certain point (x, ~) of the six-dimensional reduced phase space associated with the position and velocity of that molecule. In order to simplify the treatment, we shall for the moment assume that the molecules are hard spheres, whose center has position x. When the molecules collide, momentum and kinetic energy must be conserved; thus the velocities after the impact, ~'l and ~'2, are related to those before the impact, ~l and ~2, by ~' In (~1 ' 1 =~1--n 9 --~2)] '= In (~1 ' ~2 ~2 "+- n 9 -- ~2)] (3.1) where n is the unit vector along ~1 -- ~t Note that the relative velocity 1" V --" ~1 -- ~2 (3.2) satisfies V'= V - 2n(n. V), (3.3)
- 10. 6 C. Cercignani i.e., undergoes a specular reflection at the impact. This means that if we split V at the point of impact into a normal component Vn, directed along n and a tangential component Vt (in the plane normal to n), then Vn changes sign and Vt remains unchanged in a collision. We can also say that n bisects the directions of V and -W = -(~t 1 - ~i)" Let us remark that, in the absence of collisions, p(1) would remain unchanged along the trajectory of a particle. Accordingly we must evaluate the effects of collisions on the time evolution of p(1). Note that the probability of occurrence of a collision is related to the probability of finding another molecule with a center at exactly one diameter from the center of the first one, whose distribution function is p(1). Thus, generally speaking, in order to write the evolution equation for p(1) we shall need another function, p(2), which gives the probability density of finding, at time t, the first molecule at Xl with velocity ~1 and the second at X2 with velocity ~2; obviously p(2) = p(2) (Xl, x2, ~ 1, ~2, t). Hence p(1) satisfies an equation of the following form: Op(1) Op(l) -Jr-~1" -- G - L. (3.4) Ot OX1 Here L dXl d~l dt gives the expected number of particles with position between Xl and x 1-+-dx 1and velocity between ~1 and ~1-+d~ 1which disappear from these ranges of values because of a collision in the time interval between t and t + dt and G dxl d~ 1dt gives the analogous number of particles entering the same range in the same time interval. The count of these numbers is easy, provided we use the trick of imagining particle 1 as a sphere at rest and endowed with twice the actual diameter 0- and the other particles to be point masses with velocity (~i - ~l) = Vi. In fact, each collision will send particle 1 out of the above range and the number of the collisions of particle 1 will be the number of expected collisions of any other particle with that sphere. Since there are exactly (N - 1) identical point masses and multiple collisions are disregarded, G - (N - 1)g and L -- (N - 1)/, where the lower case letters indicate the contribution of a fixed particle, say particle 2. We shall then compute the effect of the collisions of particle 2 with particle 1. Let x2 be a point of the sphere such that the vector joining the center of the sphere with x2 is an, where n is a unit vector. A cylinder with height [V. n[ dt (where we write just V for V2) and base area dS = 0-2 dn (where dn is the area of a surface element of the unit sphere about n) will contain the particles with velocity ~2 hitting the base dS in the time interval (t, t + dt); its volume is 0-2 dn[V. n[ dt. Thus the number of collisions of particle 2 with particle 1 in the ranges (Xl, Xl + dxl), (~1, ~1 + d~l), (X2, X2 + dx2), (~2, ~2 + d~2), (t,t +dt) occuring at points of dS is p(2)(Xl, x2,/~ 1, ~2, t) dxl d/~1d~2~ dn[V2 9n[ dt. If we want the number of collisions of particle 1 with 2, when the range of the former is fixed but the latter may have any velocity/~ 2 and any position x2 on the sphere (i.e., any n), we integrate over the sphere and all the possible velocities of particle 2 to obtain: 1dxl d/~1dt = dxl d~l dt f R3fu- P(2)(Xl' Xl + 0-n'/~l' ~2' t)lV" nl0-2dnd~2, (3.5)
- 11. TheBoltzmannequationandfluiddynamics 7 where B- is the hemisphere corresponding to V. n < 0 (the particles are moving one toward the other before the collision). Thus we have the following result: L--(N-1)O-ZJR3 ft3- P(Z)(xl'xl+o-n'~jl'~2't)](~2-~l)'nld~zdn" (3.6) The calculation of the gain term G is exactly the same as the one for L, except for the fact that we have to integrate over the hemisphere B +, defined by V. n > 0 (the particles are moving away one from the other after the collision). Thus we have: G--(N-1)O-2 fR3 f13+P(2)(Xl'Xl+o-n'~l'~2't)l(~2-~l)'nld~2dn" (3.7) We can now insert in Equation (3.4) the information that the probability density p(2) is continuous at a collision; in other words, although the velocities of the particles undergo the discontinuous change described by Equations (3.1), we can write" p(2) (x1, ~ 1, x2, ~2, t) -- p(2)(x1, ~1 -- n(n. V), x2, ~2 -+-n(n. V), t) if Ix1 - x21 -- o-. (3.8) For brevity, we write (in agreement with Equations (3.1))" ~fl -- ~1 -- n(n. V), ~2 -- ~2 -+- n(n. V). (3.9) Inserting Equation (3.8) in Equation (3.5) we thus obtain: G--(N- 1)o 2 JR3 ft~+ p(2)(Xl, x, + o-n, ~'1,~2, t)[(~2 - ~l)" n[ d~2 dn (3.10) which is a frequently used form. Sometimes n is changed into -n in order to have the same integration range as in L; the only change (in addition to the change in the range) is in the second argument of p(2), which becomes Xl - o-n. At this point we are ready to understand Boltzmann's argument. N is a very large number and o- (expressed in common units, such as, e.g., centimeters) is very small; to fix the ideas, let us consider a box whose volume is 1 cm 3 at room temperature and atmospheric pressure. Then N ~ 1020 and o- ~ 10-8 cm. Then (N - 1)o-2 ~ No 2 ~ 104 cm2 -- 1 m2 is a sizable quantity, while we can neglect the difference between Xl and Xl -4-o-n. This means that the equation to be written can be rigorously valid only in the so called Boltzmann-Grad limit, when N --+ cxz,o- --+ 0 with No 2 finite.
- 12. 8 C. Cercignani In addition, the collisions between two preselected particles are rather rare events. Thus two spheres that happen to collide can be thought to be two randomly chosen particles and it makes sense to assume that the probability density of finding the first molecule at x l with velocity ~l and the second at x2 with velocity ~2 is the product of the probability density of finding the first molecule at Xl with velocity ~1 times the probability density of finding the second molecule at x2 with velocity ~2. If we accept this we can write (assumption of molecular chaos): P(2)(Xl, ~1, x2, ~2, t ) -- p(1)(Xl,~l,t)p(1)(x2,~2, t) (3.11) for two particles that are about to collide, or, letting a = 0 P(2)(Xl, ~1, Xl + o'n, ~2, t ) = P(1)(Xl,l~l,t)p(1)(Xl,~2, t) for (~2 - ~1)" n < 0. (3.12) Thus we can apply this recipe to the loss term (3.4) but not to the gain term in the form (3.5). It is possible, however, to apply Equation (3.12) (with ~'1,~2' in place of ~1, ~2) to the form (3.8) of the gain term, because the transformation (3.9) maps the hemisphere 13+ onto the hemisphere B-. If we accept all the simplifying assumptions made by Boltzmann, we obtain the following form for the gain and loss terms: L L ' t)](~ -~).nld~2dn , G = Na 2 p(1) (Xl, ~:1,t) p(1) (Xl, ~2' 2 1 3 - (3.13) L--Nff2 fR3 fl~- P(1)(Xl'l~l't)p(1)(Xl'~2't)[(~2-~l)'ll]d~2dll" (3.14) By inserting these expressions in Equation (3.6) we can write the Boltzmann equation in the following form: Op(1) Op(1) + ~. at OXl s 't) = Na 2 [P(1)(Xl,l~'l,t)p(1)(Xl,l~2 ' 3 - -- P(1)(Xl,l;1,t)p(1)(Xl,l~2, t)]](l;2 - ~1)" n] d~2 dn. (3.15) We remark that the expressions for ~'1 and ~2t given in Equations (3.1) are by no means the only possible ones. In fact we might use a different unit vector to, directed as V', instead of n. Then Equations (3.1) is replaced by: , - 1 , 1 ~J2- ~- ~l~Jl- ~J21to, (3.16)
- 13. The Boltzmann equation andfluid dynamics 9 where ~ = 89 (~j1+ ~2) is the velocity of the center of mass. The relative velocity V satisfies v' : ~lVI. (3.17) The recipes (3.13) and (3.14) can be justified at various levels of rigor [36,113,39,47]. We finally mention that we have for simplicity neglected any body force acting on the molecules, such as gravity. It is not hard to take them into account; if the force per unit mass acting on the molecules is denoted by X, then a term X. 0p(1)/0~1 must be added to the left-hand side of Equation (3.8). 4. Molecules different from hard spheres In the previous section we have discussed the Boltzmann equation when the molecules are assumed to be identical hard spheres. There are several possible generalizations of this molecular model, the most obvious being the case of molecules which are identical point masses interacting with a central force, a good general model for monatomic gases. If the range of the force extends to infinity, there is a complication due to the fact that two molecules are always interacting and the analysis in terms of "collisions" is no longer possible. If, however, the gas is sufficiently dilute, we can take into account that the molecular interaction is negligible for distances larger than a certain a (the "molecular diameter") and assume that when two molecules are at a distance smaller than a, then no other molecule is interacting with them and the binary collision analysis considered in the previous section can be applied. The only difference arises in the factor o-21(~2 - ~1)" nl which turns out to be replaced by a function of V = I~2 -- ~ I I and the angle 0 between n and V ([39,35,42]). Thus the Boltzmann equation for monatomic molecules takes on the following form: Op(l) Op(1) + ~. Ot Oxl =NfR3 it3_[V(1)(x1 , ~tl,/) e(1)(x1 , ~j~, t) - e(1)(Xl,~,t)p(1)(Xl,~2, t)]n(o, 1~2 -- ~1) d~2d0 d~, (4.1) where e is the other angle which, together with 0, identifies the unit vector n. The function B(O,V) depends, of course, on the specific law of interaction between the molecules. In the case of hard spheres, of course B(O,1~2 - ~ll) = cos0 sin01~2 - ~ll- (4.2) In spite of the fact that the force is cut at a finite range cr when writing the Boltzmann equation, infinite range forces are frequently used. This has the disadvantage of making the integral in Equation (4.1) rather hard to handle; in fact, one cannot split it into the difference of two terms (the loss and the gain), because each of them would be a divergent
- 14. 10 C. Cercignani integral. This disadvantage is compensated in the case of power law forces, because one can separate the dependence on 0 from the dependence upon V. In fact, one can show [39, 35] that, if the intermolecular force varies as the n-th inverse power of the distance, then B(O, 1/~2-/~11) = ffi(O)1/~2- I~11 (n-5)/(n-1), (4.3) where fl(O) is a non-elementary function of 0 (in the simplest cases it can be expressed by means of elliptic functions). In particular, for n = 5 one has the so-called Maxwell molecules, for which the dependence on V disappears. Sometimes the artifice of cutting the grazing collisions corresponding to small values of l0 - zr/2l is used (angle cutoff). In this case one has both the advantage of being able to split the collision term and of preserving a relation of the form (4.3) for power-law potentials. Since solving of the Boltzmann equation with actual cross sections is complicated, in many numerical simulations use is made of the so-called variable hard sphere model in which the diameter of the spheres is an inverse power law function of the relative speed V (see [43]). Another important case is when we deal with a mixture rather than with a single gas. In this case we have n unknowns, if n is the number of the species, and n Boltzmann equations; in each of them there are n collision terms to describe the collision of a molecule with other molecules of all the possible species [43,39]. If the gas is polyatomic, then the gas molecules have other degrees of freedom in addition to the translation ones. This in principle requires using quantum mechanics, but one can devise useful and accurate models in the classical scheme as well. Frequently the internal energy Ei is the only additional variable that is needed; in which case one can think of the gas as of a mixture of species [43,39], each differing from the other because of the value of Ei. If the latter variable is discrete we obtain a strict analogy with a mixture; otherwise we have a continuum of species. We remark that in both cases, kinetic energy is not preserved by collisions, because internal energy also enters into the balance; this means that a molecule changes its "species" when colliding. This is the simplest example of a "reacting collision", which may be generalized to actual chemical species when chemical reactions occur. The subject of mixture and polyatomic gases will be taken up again in Section 16. 5. Collision invariants Before embarking in a discussion of the properties of the solutions of the Boltzmann equation we remark that the unknown of the latter is not always chosen to be a probability density as we have done so far; it may be multiplied by a suitable factor and transformed into an (expected) number density or an (expected) mass density (in phase space, of course). The only thing that changes is the factor in front of Equations (3.1) which is no longer N. In order to avoid any commitment to a special choice of that factor we replace NB(O, V) by B(O, V) and the unknown P by another letter, f (which is also the most commonly used letter to denote the one-particle distribution function, no matter what its
- 15. The Boltzmann equation and fluid dynamics 11 normalization is). In addition, we replace the current velocity variable ~1 simply by ~ and 2 by ~,. Thus we rewrite Equation (4.1) in the following form: Of Of ~ fB (f'ft*-ff*)B(O V)d~ dOde, o-; +~~x= ~ _ ' * (5.1) where V - [~ - ~, 1.The velocity arguments ~i and ~, in f' and f,~ are of course given by Equations (3.1) (or (3.15)) with the suitable modification. The right-hand side of Equation (5.1) contains a quadratic expression Q(f, f), given by: Q(f' f)= fR3fs2 (f' f',- ff,)B(O, V)dl~,dOde. (5.2) This expression is called the collision integral or, simply, the collision term and the quadratic operator Q goes under the name of collision operator. In this section we study some elementary properties of Q. Actually it turns out that it is more convenient to study the slightly more general bilinear expression associated with Q(f, f), i.e.: 1~ fs (f'g1*+g' f*- fg*-gf*)13(O' V)dl;*dOde" Q(f' g) = -2 3 2 (5.3) It is clear that when g = f, Equation (5.3) reduces to Equation (5.2) and Q(f, g) -- Q(g, f). (5.4) Our first aim is to indicate a basic property of the eightfold integral: fR Q(f, g)r 3 where f, g and ~b are functions such that the indicated integrals exist and the order of integration does not matter. Simple manipulations (see [43,39,35]) give the following result: n O(f, g)~b(~) d/~ ,, , , = _ (f g, + g f, - fg, - gf,) 8 3 3 - x (4)+ 4), - dp'- r V) dl~,dl~dOde. (5.6)
- 16. 12 C. Cercignani This relation expresses a basic property of the collision term, which is frequently used. In particular, when g = f, Equation (5.6) reads R3 Q(f' f)r = (f f, - ff,)(dp + ~b, - ~b' - ~bl,)B(0, V) d~j d~jd0 de. (5.7) 3 3 - * We now observe that the integral in Equation (5.6) is zero independent of the particular functions f and g, if r + r = r + r (5.8) is valid almost everywhere in velocity space. Since the integral appearing in the left-hand side of Equation (5.7) is the rate of change of the average value of the function 4~ due to collisions, the functions satisfying Equation (5.8) are called "collision invariants". It can be shown (see, e.g., [39]) that a continuous function 4~has the property expressed by Equation (5.8) if and only if ~b(~) =a +b.~ +cl~l 2, (5.9) where a and c are constant scalars and b a constant vector. The assumption of continuity can be considerably relaxed [5,40,6]. The functions 7t0 = 1, (Tel, 7t2, ~P3)= ~, 7t4 = I~12 are usually called the elementary collision invariants; they span the five-dimensional subspace of the collision invariants. 6. The Boltzmann inequality and the Maxwell distributions In this section we investigate the existence of positive functions f which give a vanishing collision integral: = (f f, - f f,) 13(0, V) d~,, dOde - O. 3 - (6.1) In order to solve this equation, we prove a preliminary result which plays an important role in the theory of the Boltzmann equation: if f is a nonnegative function such that log f Q(f, f) is integrable and the manipulations of the previous section hold when q~= log f, then the Boltzmann inequality: fR log f Q(f, f) d~j ~<0 (6.2) holds; further, the equality sign applies if, and only if, log f is a collision invariant, or, equivalently: f = exp(a + b. ~j+ cl~j12). (6.3)
- 17. The Boltzmann equation and fluid dynamics 13 To prove Equation (6.2) it is enough to use Equation (4.11) with r = log f: ~3 f Q(f' f) d~ log -l fRfB l~ V)dtjdtj de 3 - ~ * and Equation (6.2) follows thanks to the elementary inequality (6.4) (z - y)log(y/z) <~0 (y, z e R+). (6.5) Equation (6.5) becomes an equality if and only if y = z; thus the equality sign holds in Equation (6.2) if and only if: f' f~, = f f, (6.6) applies almost everywhere. But, taking the logarithms of both sides of Equation (6.6), we find that r = log f satisfies Equation (5.8) and is thus given by Equation (5.9). f = exp(r is then given by Equation (6.3). We remark that in the latter equation c must be negative, since f must be integrable. If we let c = -/3, b = 2fly (where v is another constant vector) Equation (6.3) can be rewritten as follows: f -- A exp(-fi I~ - vl2), (6.7) where A is a positive constant related to a, c, Ibl2 (/3,v, A constitute a new set of con- stants). The function appearing in Equation (3.7) is the so called Maxwell distribution or Maxwellian. Frequently one considers Maxwellians with v = 0 (nondrifting Maxwellians), which can be obtained from drifting Maxwellians by a change of the origin in velocity space. Let us return now to the problem of solving Equation (6.1). Multiplying both sides by log f gives Equation (6.2) with the equality sign. This implies that f is a Maxwellian, by the result which has just been proved. Suppose now that f is a Maxwellian; then f = exp(r where r is a collision invariant and Equation (6.6) holds; Equation (6.1) then also holds. Thus there are functions which satisfy Equation (6.1) and they are all Maxwellians, Equation (6.7). 7. The macroscopic balance equations In this section we compare the microscopic description supplied by kinetic theory with the macroscopic description supplied by continuum gas dynamics. For definiteness, in this section f will be assumed to be an expected mass density in phase space. In order to obtain a density, p = p (x, t), in ordinary space, we must integrate f with respect to ~: P = s f d/~. (7.1)
- 18. 14 C. Cercignani The bulk velocity v of the gas (e.g., the velocity of a wind), is the average of the molecular velocities ~ at a certain point x and time instant t; since f is proportional to the probability for a molecule to have a given velocity, v is given by fR31~f dl~ v = fR 3f d/~ (7.2) (the denominator is required even if f is taken to be a probability density in phase space, because we are considering a conditional probability, referring to the position x). Equation (7.2) can also be written as follows: pv -- ~3/~ f d/~ (7.3) or, using components: pvi -- fR3 ~if d/~ (i = 1, 2, 3). (7.4) The bulk velocity v is what we can directly perceive of the molecular motion by means of macroscopic observations; it is zero for a gas in equilibrium in a box at rest. Each molecule has its own velocity ~ which can be decomposed into the sum of v and another velocity c =/~ - v (7.5) called the random or peculiar velocity; c is clearly due to the deviations of ~ from v. It is also clear that the average of c is zero. The quantity PVi which appears in Equation (7.4) is the i-th component of the mass flow or, alternatively, of the momentum density of the gas. Other quantities of similar nature are: the momentum flow mij -- fR3 ~i~j f d~ (i, j -- 1, 2, 3); (7.6) the energy density per unit volume: 1URI~J12f d/~; (7.7) W~ 3 the energy flow: 1 fR ~il/~12fd~ ri -- -2 3 (i, j = 1, 2, 3). (7.8)
- 19. The Boltzmann equation and fluid dynamics 15 Equation (7.6) shows that the momentum flow is described by the components of a symmetric tensor of second order. The defining integral can be re-expressed in terms of c and v. We have [43,39,35]: mij =- pvi l)j -+-Pij, (7.9) where: pij -- fR3 CiCj f d~j (i, j -- 1, 2, 3) (7.10) plays the role of the stress tensor (because the microscopic momentum flow associated with it is equivalent to forces distributed on the boundary of any region of gas, according to the macroscopic description). Similarly one has [43,39,35]: 1 w -- plvl 2 + pe, (7.11) where e is the internal energy per unit mass (associated with random motions) defined by: 1 s le12fd~j. (7.12) pe-- -~ 3 and: (') k ,, ri -- pvi _v_2 -+-e + vj Pij -+-qi j=l (i = 1, 2, 3), (7.13) where qi are the components of the heat-flow vector: if qi -- -~ ci ICl2f d~j. (7.14) The decomposition in Equation (7.13) shows that the microscopic energy flow is a sum of a macroscopic flow of energy (both kinetic and internal), of the work (per unit area und unit time) done by stresses, and of the heat-flow. In order to complete the connection, as a simple mathematical consequence of the Boltzmann equation, one can derive five differential relations satisfied by the macroscopic quantities introduced above; these relations describe the balance of mass, momentum and energy and have the same form as in continuum mechanics. To this end let us consider the Boltzmann equation Of Of +/~. - Q (f, f) (7 15) at "
- 20. 16 C. Cercignani If we multiply both sides by one of the elementary collision invariants ~Pc~ (a = 0, 1, 2, 3, 4), defined in Section 4, and integrate with respect to ~, we have, thanks to Equation (1.15) with g = f and 4~= ~Pc~" fR TtO~(l~)Q(f,f) dl~= O, 3 (7.16) and hence, if it is permitted to change the order by which we differentiate with respect to t and integrate with respect to ~" 3 Ot ~/af d~ + E ~xi ~i~a f d~ -- 0 i=1 (a -- 1, 2, 3, 4). (7.17) If we take successively ot -- 0, 1, 2, 3, 4 and use the definitions introduced above, we obtain 3 Op i~1 O a---t -~" . ~X i ( p v i ) = O, (7.18) 3 0 0 Ot (pvj) -+-Z -~xi (pvivj -+-Pij) - 0 i=1 (j = 1, 2, 3), (7.19) ) Ot -2plvle + pe -[- ~ pl)i Ivl 2 + e + UjPij -+"qi = O. i=1 j=l (7.20) The considerations of this section apply to all the solutions of the Boltzmann equation. The definitions, however, can be applied to any positive function for which they make sense. In particular if we take f to be a Maxwellian in the form (5.7), we find that the constant vector v appearing there is actually the bulk velocity as defined in Equation (7.2) while fl and A are related to the internal energy e and the density p in the following way: fl = 3/(4e), A- p(4yre/3) -3/2. (7.21) Furthermore the stress tensor turns out to be diagonal (Pij -- (2pe)6ij, where ~ij is the so-called "Kronecker delta" (= 1 if i -- j;= 0 if i 5~ j)), while the heat-flow vector is zero. We end this section with the definition of pressure p in terms of f; p is nothing else than 1/3 of the spur or trace (i.e., the sum of the three diagonal terms) of pij and is thus given by: 1 f. [elZfd,~" (7.22) P=3 3
- 21. The Boltzmann equation andfluid dynamics 17 If we compare this with the definition of the specific internal energy e, given in Equation (3.11), we obtain the relation: 2 p = -=pe. (7.23) .5 This relation also suggests the definition of temperature, according to kinetic theory, T = (2e)/R, where R is the gas constant equal to the universal Boltzmann constant k divided by the molecular mass m. Thus: 1 s lel2fd/~" (7.24) T = 3,oR 3 8. The H-theorem Let us consider a further application of the properties of the collision term Q(f, f) of the Boltzmann equation: Of of + 1~.-2- = Q(f, f). (8.1) Ot ox If we multiply both sides of this equation by log f and integrate with respect to ~j, we obtain: 07-[ 0 Ot + -~x "J = S, (8.2) where 7-[= fR3 f log f ds e, (8.3) J = fR3 f~f log f d/~, (8.4) S = fR3 log f Q(f, f) d~. (8.5) Equation (8.2) differs from the balance equations considered in the previous section because the right-hand side, generally speaking, does not vanish. We know, however, that the Boltzmann inequality, Equation (5.2), implies: S~<0 and S=0 iff fisaMaxwellian. (8.6) Because of this inequality, Equation (8.2) plays an important role in the theory of the Boltzmann equation. We illustrate the role of Equation (8.6) in the case of space-
- 22. 18 C. Cercignani homogeneous solutions. In this case the various quantities do not depend on x and Equation (8.2) reduces to 07-t = S ~<0. (8.7) Ot This implies the so-called H-theorem (for the space homogeneous case): 7-/is a decreasing quantity, unless f is a Maxwellian (in which case the time derivative of 7-/ is zero). Remember now that in this case the densities p, pv and pe are constant in time; we can thus build a Maxwellian M which has, at any time, the same p, v and e as any solution f corresponding to given initial data. Since 7-/decreases unless f is a Maxwellian (i.e., f = M), it is tempting to conclude that f tends to M when t ~ cx~. This conclusion is, however, unwarranted from a purely mathematical viewpoint, without a more detailed consideration of the source term S in Equation (8.7), for which [47] should be consulted. If the state of the gas is not space-homogeneous, the situation becomes more complicated. In this case it is convenient to introduce the quantity H- fs2 7-/dx, (8.8) where s is the space domain occupied by the gas (assumed here to be time-independent). Then Equation (8.2) implies dt ~< J. n &r, (8.9) ~2 where n is the inward normal and dcr the surface element on OS2. Clearly, several situations may arise (see [43] and [47] for a detailed discussion). It should be clear that H has the properties of entropy (except for the sign); this identification is strengthened when we evaluate H in an equilibrium state (see [43,39, 35]) because it turns out to coincide with the expression of a perfect gas according to equilibrium thermodynamics, apart from a factor -R. A further check of this identification is given by an inequality satisfied by the right-hand side of Equation (8.9) when the gas is able to exchange energy with a solid wall bounding S2 (see Section 11 and [43,39,47]). 9. Model equations When trying to solve the Boltzmann equation for practical problems, one of the major shortcomings is the complicated structure of the collision term, Equation (4.2). When one is not interested in fine details, it is possible to obtain reasonable results by replacing the collision integral by a so-called collision model, a simpler expression J (f) which retains only the qualitative and average properties of the collision term Q(f, f). The equation for the distribution function is then called a kinetic model or a model equation.
- 23. The Boltzmann equation andfluid dynamics 19 The most widely known collision model is usually called the Bhatnagar, Gross and Krook (BGK) model, although Welander proposed it independently at about the same time as the above mentioned authors [14,173]. It reads as follows: ,l(f) -- v[~(l~) - f (l~)], (9.1) where the collision frequency v is independent of ~ (but depends on the density p and the temperature T) and q~ denotes the local Maxwellian, i.e., the (unique) Maxwellian having the same density, bulk velocity and temperature as f: -- p(2yr RT) -3/2exp[- I/~- vI2/(2RT)]. (9.2) Here p, v, T are chosen is such a way that for any collision invariant ~pwe have ~3 r d~ -- JR3 r d~. (9.3) It is easily checked that, thanks to Equation (9.3): (a) f and q~ have the same density, bulk velocity and temperature; (b) J (f) satisfies conservation of mass, momentum and energy; i.e., for any collision invariant: s ~(/~)J(f) d~ -- O; (9.4) (c) J (f) satisfies the Boltzmann inequality fRlog f J(f) d/~ ~<0 3 (9.5) the equality sign holding if and only if, f is a Maxwellian. It should be remarked that the nonlinearity of the BGK collision model, Equation (9.1), is much worse than the nonlinearity in Q(f, f); in fact the latter is simply quadratic in f, while the former contains f in both the numerator and denominator of an exponential, because v and T are functionals of f, defined by Equations (6.2) and (6.27). The main advantage in the use of the BGK model is that for any given problem one can deduce integral equations for p, v, T, which can be solved with moderate effort on a computer. Another advantage of the BGK model is offered by its linearized form (see [43, 113,35]). The BGK model has the same basic properties as the Boltzmann collision integral, but has some shortcomings. Some of them can be avoided by suitable modifications, at the expense, however, of the simplicity of the model. A first modification can be introduced in order to allow the collision frequency v to depend on the molecular velocity, more precisely on the magnitude of the random velocity c (defined by Equation (6.5)), while requiting that Equation (9.4) still holds. All the basic properties, including Equation (9.5), are retained, but the density, velocity and temperature appearing in q~ are not the local ones of the gas,
- 24. 20 C. Cercignani but some fictitious local parameters related to five functionals of f different from p, v, T; this follows from the fact that Equation (9.3) must now be replaced by fR3 v(Icl)~r(~J)cP(~J)d~J- fR3 v(Icl)!#(~j)f(~j) d~j. (9.6) A different kind of correction to the BGK model is obtained when a complete agreement with the compressible Navier-Stokes equations is required for large values of the collision frequency. In fact the BGK model has only one parameter (at a fixed space point and time instant), i.e., the collision frequency v; the latter can be adjusted to give a correct value for either the viscosity/z or the heat conductivity x, but not for both. This is shown by the fact that the Prandtl number Pr = lZ/CpX (where Cp is the specific heat at constant pressure) turns out [39,35] to be unity for the BGK model, while it is about to 2/3 for a monatomic gas (according to both experimental data and the Boltzmann equation). In order to have a correct value for the Prandtl number, one is led [87,62] to replacing the local Maxwellian in Equation (9.1) by rP(I~) = p(rc)-3/2(detA)I/e exp(-(~j - v). [A(/~- v)]), (9.7) where A is the inverse of the matrix A-1 = (2RT/Pr)I- 2(1 - Pr)p/(p Pr), (9.8) where Iis the identity and p the stress matrix. If we let Pr -- 1, we recover the BGK model. Only recently [3] this model (called ellipsoidal statistical (ES) model) has been shown to possess the property expressed by Equation (9.5). Hence the H-theorem holds for the ES model. Other models with different choices of q5 have been proposed [151,35] but they are not so interesting, except for linearized problems (see [43,39,35]). Another model is the integro-differential model proposed by Lebowitz, Frisch and Helfand [114], which is similar to the Fokker-Planck equation used in the theory of Brownian motion. This model reads as follows: 3 [02f J (f)= D ~ -UfT~2-~-~ - o,)y]], RT O~k (9.9) where D is a function of the local density p and the local temperature T. If we take D proportional to the pressure p = pRT, Equation (9.9) has the same kind of nonlinearity (i.e., quadratic) as the true Boltzmann equation. The idea of kinetic models can be naturally extended to mixtures and polyatomic gases [151,127,81,43].
- 25. TheBoltzmannequationandfluiddynamics 21 10. The linearized collision operator On several occasions we shall meet the so-called linearized collision operator, related to the bilinear operator defined in Equation (5.3) by Lh = 2M -1Q(Mh, M), (10.1) where M is a Maxwellian distribution, usually with zero bulk velocity. When we want to emphasize the fact that we linearize with respect to a given Maxwellian, we write LM instead of just L. A more explicit expression of Lh reads as follows Lh--f~ fB M*(h'+h~*-h*-h)B(O'V)dl~*dn' 3 + (10.2) where we have taken into account that M'M', = MM,. Because of Equation (4.10) (with Mh in place of f, M in place of g and g in place of q~),we have the identity: f~t3 MgLh dl~ = lf, f, (h'+h', . . . . h h,)(f+f, g g,) 4 3 3 + x 13(0, V) d/~, d/~dn. (10.3) This relation expresses a basic property of the linearized collision term. In order to make it clear, let us introduce a bilinear expression, the scalar product in the Hilbert space of square summable functions of ~ endowed with a scalar product weighted with M: (g, h) -- f~3 ~hM d~j, (10.4) where the bar denotes complex conjugation. Then Equation (1.7) (with ~, in place of g) gives (thanks to the symmetry of the expression in the right-hand side of Equation (10.3) with respect to the interchange g r h): (g, Lh) -- (Lg, h). (10.5) Further: (h, Lh) <~0 (10.6) and the equality sign holds if and only if hf + hi, - h - h, = 0, (10.7) i.e., if and only if h is a collision invariant.
- 26. 22 C. Cercignani Equations (10.5) and (10.6) indicate that the operator L is symmetric and non-positive in the aforementioned Hilbert space. 11. Boundary conditions The Boltzmann equation must be accompanied by boundary conditions, which describe the interaction of the gas molecules with the solid walls. It is to this interaction that one can trace the origin of the drag and lift exerted by the gas on the body and the heat transfer between the gas and the solid boundary. The study of gas-surface interaction may be regarded as a bridge between the kinetic theory of gases and solid state physics and is an area of research by itself. The difficulties of a theoretical investigation are due, mainly, to our lack of knowledge of the structure of surface layers of solid bodies and hence of the effective interaction potential of the gas molecules with the wall. When a molecule impinges upon a surface, it is adsorbed and may form chemical bonds, dissociate, become ionized or displace surface molecules. Its interaction with the solid surface depends on the surface finish, the cleanliness of the surface, its temperature, etc. It may also vary with time because of outgassing from the surface. Preliminary heating of a surface also promotes purification of the surface through emission of adsorbed molecules. In general, adsorbed layers may be present; in this case, the interaction of a given molecule with the surface may also depend on the distribution of molecules impinging on a surface element. For a more detailed discussion the reader should consult [39,110] and [41]. In general, a molecule striking a surface with a velocity ~ reemerges from it with a velocity ~ which is strictly determined only if the path of the molecule within the wall can be computed exactly. This computation is very hard, because it depends upon a great number of details, such as the locations and velocities of all the molecules of the wall and an accurate knowledge of the interaction potential. Hence it is more convenient to think in terms of a probability density R(~ ~--+ ~; x, t; r) that a molecule striking the surface with velocity between ~ and ~j~+ d~~at the point x and time t will re-emerge at practically the same point with velocity between ~ and ~ + d~ after a time interval r (adsorption or sitting time). If R is known, then we can easily write down the boundary condition for the distribution function f (x, ~, t). To simplify the discussion, the surface will be assumed to be at rest. A simple argument ([43,39,35]) then gives: f(x,/~, t)I/~ .n[ = dr R(~j' --+ ~; x, t; r)f(x, ~', t - r)l~'' n[ d~' '.n<O (x ~ 0S'2,~. n > 0). (11.1) The kernel R can be assumed to be independent of f under suitable conditions which we shall not detail here [39,110,41]. If, in addition, the effective adsorption time is small compared to any characteristic time of interest in the evolution of f, we can let r = 0 in
- 27. The Boltzmann equation andfluid dynamics 23 the argument of f appearing in the right-hand side of Equation (3.4); in this case the latter becomes: f (x, ~j, t)I~j" nl = f~'.n<0e(~t ~/~; x, t)f(x,/~', t)l~'" nl d~j' (x 6 Y2, ~ 9n > 0), (11.2) where fO G R(~j' --+ ~j; x,t)= drR(~j' --+ ~j; x,t; r). (11.3) Equation (11.2) is, in particular, valid for steady problems. Although the idea of a scattering kernel had appeared before, it is only at the end of 1960's that a systematic study of the properties of this kernel appears in the scientific literature [35,110,41]. In particular, the following properties were pointed out [36,35,110, 41,34,108,48,109,37]: (1) Non-negativeness, i.e., R cannot take negative values: R(~j' --+ ~j; x, t; r) ~>0 (11.4) and, as a consequence: R(se' -+/~; x, t) ~>0. (11.5) (2) Normalization, if permanent adsorption is excluded; i.e., R, as a probability density for the totality of events, must integrate to unity: R(/~' --+ ~; x,t" r)dse -- 1 dr '.n~>0 (11.6) and, as a consequence: f~ R (~' ~ se; x, t) d~ -- 1. (11.7) '-n>/O (3) Reciprocity; this is a subtler property that follows from the circumstance that the microscopic dynamics is time reversible and the wall is assumed to be in a local equilibrium state, not significantly disturbed by the impinging molecule. It reads as follows: les'.nIMw(~')R(es' ~ se;x, t; r)= I~ .nlMw(~)R(-~ --~ -se'; x, t; r) (11.8) and, as a consequence: les'. nlMw(fj')R(es' ~/~; x,t) = I/~"nlMw(es)R(-es -+ -/~'; x,t). (11.9)
- 28. 24 C.Cercignani Here Mw is a (non-drifting) Maxwellian distribution having the temperature of the wall, which is uniquely identified apart from a factor. We remark that the reciprocity and the normalization relations imply another property: (3') Preservation of equilibrium, i.e., the Maxwellian Mw must satisfy the boundary condition (11.1): Mw(~)l~. nl = dr '.n<O R(~j' --+/~;x, t; r)Mw(/~')l~j'' nl d~j' (11.10) equivalent to: Mw(~)l~" nl = f~'.n<O R(8' ~ ~; x, t)Mw (/~')l/~'. nl dS'. (11.11) In order to obtain Equation (11.10) it is sufficient to integrate Equation (11.8) with respect to ~' and r, taking into account Equation (11.6) (with -~ and -~' in place of ~' and ~, respectively). We remark that one frequently assumes Equation (11.10) (or (11.11)), without mentioning Equation (11.8) (or (11.9)); although this is enough for many purposes, reciprocity is very important when constructing mathematical models, because it places a strong restriction on the possible choices. A detailed discussion of the physical conditions under which reciprocity holds has been given by B~irwinkel and Schippers [10]. The basic information on gas-surface interaction, which should be in principle obtained from a detailed calculation based on a physical model, is summarized in a scattering kernel. The further reduction to a small set of accommodation coefficients can be advocated for practical purposes, provided this concept is firmly related to the scattering kernel (see [43, 39,47] for further details). In view of the difficulty of computing the kernel R(~j' --~ ~) from a physical model of the wall, frequently one constructs a mathematical model in the form of a kernel R (~' ~ ~j) which satisfies the basic physical requirements expressed by Equations (11.5), (11.7), (11.9) and is not otherwise restricted except by the condition of not being too complicated. One of the simplest kernels is R(~j' ~ ~j) = ctMw(~)1~. nl + (1 - ct)3(~ - ~' + 2n(~' 9~)). (11.12) This is the kernel corresponding to Maxwell's model [121], according to which a fraction (1 -ct) of molecules undergoes a specular reflection, while the remaining fraction c~ is diffused with the Maxwellian distribution of the wall Mw. This is the only model for the scattering kernel that appeared in the literature before the late 1960's. We refer to original papers and standard treatises for details on more recent models [110,41,34,108,48,109,37, 10,121,135,91,100,112,99,68,174,38,54,53,22,43,39], among which the most popular in recent years has been the so-called Cercignani-Lampis (CL) model. It is remarkable that, for any scattering kernel satisfying the three properties of normalization, positivity and preservation of equilibrium, a simple inequality holds. The
- 29. The Boltzmann equation andfluid dynamics 25 latter was stated by Darrozbs and Guiraud [70] who also sketched a proof. More details were given later [37,39]. It reads as follows fin--- fR~'3 n flog f d~<~-(2RTw)-lfR ~ .3 nl~ 12fd~j (x ~ 0S'2). (11.13) Equality holds if and only if f coincides with Mw (the wall Maxwellian) on 0S-2 (unless the kernel in Equation (11.2) is a delta function). We remark that if the gas does not slip upon the wall, the right-hand side of Equation (11.13) equals -qn/(RTw) where qn is the heat-flow along the normal, according to its definition given in Section 5. If the gas slips on the wall, then one must add the power of the stresses Pn "v to qn. In any case, however, the right-hand side equals qn (w), where q(nw)is the heat-flow in the solid at the interface, because the normal energy flow must be continuous through the wall and stresses have vanishing power in the solid, because the latter is at rest. If we identify the function H introduced in Section 8 with -rl/R (where r/is the entropy of the gas), the inequality in Equation (11.13) is exactly what one would expect from the Second Principle of thermodynamics. 12. The continuum limit In this section we investigate the connection between the Navier-Stokes equations and the Boltzmann equation by using a method, which originated with Hilbert [85] and Enskog [74]. The discussion is made complicated by the various possible scalings. For example, if we denote by (~,,7) the microscopic space and time variables (those entering in the Boltzmann equation) and by (x, t) the macroscopic variables (those entering in the fluid dynamical description), we can study scalings of the following kind = s-ix, (12.1) 7 - s -~ t, (12.2) where ot is an exponent between 1 and 2. For ot -- 1, this is called the compressible scaling. If c~ > 1, we are looking at larger "microscopic" times. We now investigate the limiting behavior of solutions of the Boltzmann equation in this limit. Notice first that the compressible Euler equations, 8tp + div(pv) = O, Ot(pvi) + div pvvi + -~peei - 0, 1 1 5 Ot[p(e + -~lvl2)] + div[pv(-~lvl2 + -~e)] - 0, (12.3)
- 30. 26 C. Cercignani are invariant with respect to the scaling t -+ e-it, x -+ e-1 x. Here, ei denotes the unit vector in the i-th direction and p is related to p and e -- 3RT/2 by the state equation for a perfect gas. To investigate how these equations change under the scalings (12.1)-(12.2), let v (x, t)- p (x, t) = p( -lx, Te(x,t)--T(e-lx, y=c~- 1, (12.4) where (p, v, T) solve the compressible Euler equations (12.3). We easily obtain Otpe + div(peve) = O, (12.5) 1 62(1_c~ ) Otve + (ve" Ox)ve = pe OxPe" ' (12.6) 2Te(0x. ve)=0 OtTe + (ve . Ox)Te + -~ (12.7) The scaling of the bulk velocity field ve in (12.4) is done in a dimensionally consistent way. We expect that the continuum limit of the Boltzmann equation under the scaling (12.1)- (12.2) will be given by the asymptotic behavior of (pe, ve, Te), satisfying (12.3), in the limit e --+ 0. We will now investigate this limit. To this end, let r/= e2(~ and expand Vr/ ~ ve __ VO -+- ?/Vl -+-//2V2 @ "'', pO = pe _ PO + ~lPl + r/2p2 -+- 99", T o =--T e= To + oT1 + 02T1 +'". If we collect the terms of order 0-1 in (12.4), we have Oxpo -0 (12.8) and the terms of order 77 ~ give OtPo + div(povo) = O, Otvo + (vo-0x)Vo = -Oxp______j_l, (12.9) Po 2 OtTo + (vo. 0x)To + ~ Todiv vo -- 0.
- 31. The Boltzmann equation andfluid dynamics 27 From (12.8) and the perfect gas law p0 - p0T0, which we assume to hold at zeroth order, it follows that poTo is constant as a function of the space variables. The first and third equations in (12.9) now imply 2 Ot(poTo)+ div(p0T0v0) -- - ~-p0T0divv0. (12.10) .5 As poTois only a function of t, say A(t), Equation (12.10) implies that 3AI divv0--5 (t)/A(t). (12.11) Under suitable assumptions, Equation (12.11) implies that A1(t) = 0 and then divv0 = 0. This is the case, for example, if we are in a box with nonporous walls, because then the normal component of v0 is 0 and we can use the divergence theorem. A similar argument applies to the case of a box with periodicity boundary conditions; or if we are in the entire space 9~3 and the difference between v0 and a constant vector decays fast enough at infinity. Assuming that we have conditions which imply that div v0 = 0, we easily get from the continuity equation that P0 will be independent of x if the initial value is, and the same for To. But with this knowledge Equations (12.9) then actually entail that P0 and To are constant. Therefore, if the initial conditions are "well prepared" in the sense that v~(x, 0) is a divergence-free vector field and p~ (x, 0), Te(x, 0) are constant, we expect as a first-order approximation for p~, v~, T ~ the solution of the equations div v0 - 0, Otvo + (vo. Ox)VO- - ~ OxPl (12.12) p0 which are the incompressible Euler equations. This limit, known as the "low velocity limit", is well known at the macroscopic level. We refer to Majda's book [120] for references and a detailed discussion. The variable r/-1 enters into the theory as the square of the speed of sound. If this parameter is large compared to typical speeds of the fluid, then the incompressible model is well suited to describe the time evolution, provided that the initial velocity field is divergence-free and the initial density and temperature are constant. The incompressible fluid limit was met in the last section in connection with the Stokes paradox. In fact, it seems that this limit and the derivation of the steady incompressible Navier-Stokes equations from the Boltzmann equation were first considered in connection with the flow past a body at small values of the Mach number [32]. We remark that in the steady case div v0 - 0 and P0 and To turn out to be constant without using the boundary conditions. Let us examine the kinetic picture as described by the Boltzmann equation. The above discussion suggests that if ot e (1,2), then in the scaling (12.1)-(12.2) the solutions of the Boltzmann equation will converge to a local Maxwellian distribution whose parameters satisfy the incompressible Euler equations. This assertion can actually be proved rigorously [71,9].
- 32. 28 C. Cercignani For ct = 2 something special happens. Of course, the incompressible Euler equations are invariant under the scaling (12.1)-(12.3); however, for c~= 2 the incompressible Navier- Stokes equations, Otv + (v. 0x)V= -Oxr + v A v, 0x. v = 0 (12.13) (where r = p/p and v = Iz/p is the kinematic viscosity) are also invariant under the same scaling. It is therefore of great interest to understand whether the Boltzmann dynamics "chooses" in this limit the Euler or the Navier-Stokes evolution. We can expect that the answer is Navier-Stokes. In other words, considering larger times than those typical for Euler dynamics (e-zt instead of e-c~t, c~ < 2), dissipation becomes nonnegligible. We can see an illustration of this behavior in the following example: consider two parallel layers of fluid moving with velocities v and v + 6v. Suppose that we want to decide whether there is any momentum transfer between these layers (which is expected for the Navier-Stokes equations, but not for the Euler equations). The momentum transfer can in principle be affected by the trend to thermalization typical of the Boltzmann collision term, but a scaling argument shows that it is proportional to e~-Z6v, with the consequence that it remains only relevant for c~= 2. These considerations can be put on a rigorous basis, and, of course, the viscosity coefficient can be computed in terms of kinetic expressions (see [43,39,35]). The incompressible Navier-Stokes equations can be derived from the Boltzmann equation if the time interval is such that smooth solutions of the continuum equations exist. The tool which yields this result is a truncated Hilbert expansion and one gets local convergence for the general situation [71] or global and uniform convergence if the data are small in a suitable sense [9]. Let us consider now the cases in which compressibility is not negligible. If we scale space and time in the same way, the scaled Boltzmann equation becomes: Otf e + I~. Oxfe = _1Q(fe, fe). (12.14) E We will use the abbreviation Dt f := Otf + ~ 9Oxf. Of course, we expect that eDtf e -+ 0 as e --+ O, (12.15) and if fe __+ fo, (12.16) the limit f0 must satisfy Q(fO, f0) = o. (12.17)
- 33. TheBoltzmannequationandfluiddynamics 29 This implies, as we know, that f0 is a local Maxwellian distribution: f~ p(x,t) ( I~-v(x,t)l 2) (2zrRT(x, t))3/2 exp - 2RT(x, t) " (12.18) The fields (p, x, T), which characterize the behavior of the local Maxwellian distribution M in space and time, are expected to evolve according to continuum equations which we are going to derive. First, let us emphasize again that these fields are varying slowly on the space-time scales which are typical for the gas described in terms of the Boltzmann equation. From the conservation laws (I. 12.36) f g/uQ(f, f)d~j-O (c~=O ..... 4), (12.19) we readily obtain, as we know, f ~Po~(Ot f + I~. Oxf) d/~ --0. (12.20) This is a system of equations for the moments of f which is in general not closed. However, if we assume f = M and use the identities (for M they are identities; for a general f they are definitions given in (I.12.3-7), where e = 3RT) -- f M ds e, (12.21)1 , v--fMseds e, (12.21)2 3 1 if 2M , w = -~pRT + ~plv] 2- ~ s e ds e (12.21)3 we readily obtain from (12.20) that Otp + div(pv) = O, (12.22) Ot(pvi) + div(fM~i) --O, (12.23) (3 1)l(f ) Ot -~peZ+ ~p[vl 2 + ~div M~jI~I 2 --0. (12.24) These equations are nothing but Equations (12.19)-(12.21), specialized to the case of a Maxwellian distribution. This is of crucial importance if we want to write
- 34. 30 C. Cercignani Equations (12.23) and (12.24) in closed form. To do so we have to express f M~i and f M~ I~12in terms of the fields (p, v, T). To this end, we use the elementary identities: f M(~j - vj)(~i - vi)d~ 16ijpRT, f M(f~ - v) I(/~- v)l 2d/~ = O, (12.25) which transform Equation (12.23) into Ot(pvi ) + div(pvvi ) = --Oxi p, (12.26) with p = p R T. (12.27) Equation (12.27) is the perfect gas law. Obviously, the quantity p defined by Equa- tion (12.27) has the meaning of a pressure. Recalling that the internal energy e is related to temperature T by 3 e = -~RT, (12.28) using this and Equation (12.25) we transform Equation (12.24) into: ( 1 )) ( ( 1 )) Ot p(e + ~lvl 2 + div pv e + ~lvl 2 - -div(pv). (12.29) The set of Equations (12.22), (12.26) and (12.29) express conservation equations for mass, momentum and energy respectively and can be rewritten as the Euler equations (12.3). For smooth functions, an equivalent way of writing the Euler equations in terms of the field (p, v, T) is Otp + div(pv) = 0, 1 Otv + (v. Ox)V+ -Oxp = O, (12.30) P 2 OtT + (v. ax)T + =Tax. v =0. .5 However, in this form we lose the conservation form as given in (12.3), in which the time derivative of a field equals the negative divergence of a current which is a nonlinear function of this field. Before going on, some comments on our limits are in order, because one might suspect an inconsistency in the passage from a rarefied to a dense gas. Recall that the Boltzmann equation holds in the Boltzmann-Grad limit (Nor 2 = O(1)). In the continuum limit, we
- 35. The Boltzmann equation and fluid dynamics 31 have to take Nff 2 -- 1/e ~ oo. This, at first glance, seems contradictory, but there is really no problem. The Boltzmann equation holds for a perfect gas, i.e., for a gas such that the density parameter 6 - N~r3/V, where V is the volume containing N molecules, tends to zero. The parameter 1 Nff 2 -- -- N1/3S 2/3 (12.31) Kn V 2/3 may tend to zero, to cxz or remain finite in this limit. These are the three cases which occur if we scale N as 3-m (m ~>0), for m < 2, m > 2 and m = 2 respectively. In the first case the gas is in free-molecular flow and we can simply neglect the collision term (Knudsen gas), in the second we are in the continuum regime which we are treating here, and we cannot simply "omit" the "small" term, i.e., the left-hand side of the Boltzmann equation, because the limit is singular. In the third case the two sides of the Boltzmann equation are equally important (Boltzmann gas) and this is the case dealt with before for solutions close to an absolute Maxwellian distribution. In spite of the fact that we face a singular perturbation problem, Hilbert [85] proposed an expansion in powers of e. In this way, however, we obtain a Maxwellian distribution at the lowest order, with parameters satisfying the Euler equations and corrections to this solution which are obtained by solving inhomogeneous linearized Euler equations [85,43, 39,35]. In order to avoid this and to investigate the relationship between the Boltzmann equation and the compressible Navier-Stokes equations, Enskog introduced an expansion, usually called the Chapman-Enskog expansion [74,39,35]. The idea behind this expansion is that the functional dependence of f upon the local density, bulk velocity and internal energy can be expanded into a power series. Although there are many formal similarities with the Hilbert expansion, the procedure is rather different. As remarked by the author [35,39], the Chapman-Enskog expansion seems to introduce spurious solutions, especially if one looks for steady states. This is essentially due to the fact that one really considers infinitely many time scales (of orders e, e2..... e n .... ). The author [35,39] introduced only two time scales (of orders e and e2) and was able to recover the compressible Navier-Stokes equations. In order to explain the idea, we remark that the Navier-Stokes equations describe two kinds of processes, convection and diffusion, which act on two different time scales. If we consider only the first scale we obtain the compressible Euler equations; if we insist on the second one we can obtain the Navier-Stokes equations only at the price of losing compressibility. If we want both compressibility and diffusion, we have to keep both scales at the same time and think of f as f(x, g;, t) -- f (ex, ~,, et, e2t). (12.32) This enables us to introduce two different time variables tl - et, t2 - e2t and a new space variable Xl - ex such that f - f(xl, ~, tl, t2). The fluid dynamical variables are functions
- 36. 32 c. Cercignani of xl, tl, t2, and for both f and the fluid dynamical variables the time derivative is given by O Of 2 Of m = e~ + e ~. (12.33) Ot Otl Ot2 In particular, the Boltzmann equation can be rewritten as Of 62 Of + + e!~"Oxf = Q(f, f) Otl -~2 If we expand f formally in a power series in e, we find that at the lowest order f is a Maxwellian distribution. The compatibility conditions at the first order give that the time derivatives of the fluid dynamic variables with respect to tl is determined by the Euler equations, but the derivatives with respect to t2 are determined only at the next level and are given by the terms of the compressible Navier-Stokes equations describing the effects of viscosity and heat conductivity. The two contributions are, of course, to be added as specified by (12.33) in order to obtain the full time derivative and thus write the compressible Navier-Stokes equations. It is not among the aims of this article to describe the techniques applied to and the results obtained from the computations of the transport coefficients, such as the viscosity and heat conduction coefficients, for given molecular interaction. For this we refer to standard treatises [64,86,75]. The results discussed in this section show that there is a qualitative agreement between the Boltzmann equation and the Navier-Stokes equations for sufficiently low values of the Knudsen number. There are however flows where this agreement does not occur. They have been especially studied by Sone [154]. New effects arise because the no-slip and no temperature jump boundary condition do not hold. In addition to the thermal creep induced along a boundary with a nonuniform temperature, discovered by Maxwell, two new kinds of flow are induced over boundaries kept at uniform temperatures. They are related to the presences of thermal stresses in the gas. The first effect [154,153,138] is present even for small Mach numbers and small temperature differences and follows from the fact that there are stresses related to the second derivatives of the temperature (see Section 15). Although these stresses do not change the Navier-Stokes equations, they change the boundary conditions; the gas slips on the wall, and thus a movement occurs even if the wall is at rest. This effect is particularly important in small systems, such as micromachines, since the temperature differences are small but may have relatively large second derivatives; it is usually called the thermal stress slip flow [154,153,138]. The second effect is nonlinear [101,155] and occurs when two isothermal surfaces do not have constant distance (thus in any situation with large temperature gradients, in the absence of particular symmetries). In fact, if we assume that in the Hilbert expansion the velocity vanishes at the lowest order, i.e., the speed is of the order of the Knudsen number, the terms of second order in the temperature show up in the momentum equation.
- 37. The Boltzmann equation andfluid dynamics 33 These terms are associated with thermal stresses and are of the same importance as those containing the pressure and the viscous stresses. A solution in which the gas does not move can be obtained if and only if: grad T A grad(I grad TIe) = 0. (12.34) Since Igrad T I measures the distance between two nearby isothermal lines, if this quantity has a gradient in the direction orthogonal to grad T, the distance between two neighboring isothermal lines varies and we must expect that the gas moves. These effects may occur even for sufficiently large values of the Knudsen number; they cannot be described, however, in terms of the local temperature field. They rather depend by the configuration of the system. They should not be confused with flows due to the presence of a temperature gradient along the wall, such as the transpiration flow [139] and the thermophoresis of aerosol particles [162]. Numerical examples of simulations of this kind of flow are discussed in [43]. 13. Free-molecule and nearly free-molecule flows After discussing the behavior of a gas in the continuum limit, in this section we consider the opposite case in which the small parameter is the Knudsen number (or the inverse of the mean free path). By analogy with what we did in the previous section, we might be tempted to use a series expansion of the form (12.2), albeit with a different meaning of the expansion parameter. This, however, does not work in general, for a reason to be presently explained. The factor multiplying the gradient of f in Equation (5.1) takes all possible values and hence also values of order e; thus we should expect troubles from the molecules travelling with low speeds, because then certain terms in the left-hand side can become smaller than the fight- hand side, in spite of the small factor e. This is confirmed by actual calculations, especially for steady problems. Let us now consider the limiting case when the collisions can be completely neglected. This, by itself, does not pose many problems. The Boltzmann equation (in the absence of a body force) reduces to the simple form Dtf = atf + I~ . Oxf =0. (13.1) Since the molecular collisions are negligible, the gas-surface interaction discussed in Section 11 plays a major role. This situation is typical for artificial satellites, since the mean free path is 50 meters at 200 kilometers of altitude. The general solution of Equation (13.1) is in terms of an arbitrary function of two vectors g(., .): f (x, l~, t) = g(x- I~t, ~,). (13.2) In the steady case, Equation (13.1) reduces to I~. Oxf = 0, (13.3)
- 38. 34 C. Cercignani and the general solution becomes: f (x, ~, t) = g(x A l~, l~). (13.4) Frequently it is easier to work with the property that f is constant along the molecular trajectories than with the explicit solutions given by Equations (13.3)-(13.4). The easiest problem to deal with is the flow past a convex body. In this case, in fact, the molecules arriving at the surface of the wall have an assigned distribution function f~, usually a Maxwellian distribution with the density p~, bulk velocity v~, and temperature Too, prevailing far away from the body, and the distribution function of the molecules leaving the surface is given by the boundary conditions. The distribution function at any other point P, if needed, is simply obtained by the following rule: if the straight line through P having the direction of ~ intersects the body at a point Q and ~j points from Q towards P, then the distribution function at P is the same as that at Q; otherwise it equals f~. Interest is usually confined to the total momentum and energy exchanged between the molecules and the body, which, in turn, easily yield the drag and lift exerted by the gas on the body and the heat transfer between the body and the gas. In practice, the temperature of a body is determined by a balance of all forms of heat transfer at the body surface. For an artificial satellite, a considerable part of heat is lost by radiation and this process must be duly taken into account in the balance. The results take a particularly simple form in the case of a large Mach number since we can let the latter go to infinity in the various formulas. One must, however, be careful, because the speed is multiplied by sin 0 in many terms and thus the aforementioned limit is not uniform in 0. Thus the limiting formulas can be used, if and only if, the area where S sin 0 ~< 1 is small. The standard treatment is based on the definition of accommodation coefficients, but calculations based on other models are available [39,50,49]. The case of nonconvex boundaries is, of course, more complicated and one must solve an integral equation to obtain the distribution function at the boundary. If one assumes diffuse reflection according to a Maxwellian, the integral equation simplifies in a considerable way, because just the mass flow at the boundary must be computed [39]. In particular the latter equation can be used to study free-molecular flows in pipes of arbitrary cross section with a typical diameter much smaller than the mean free path (capillaries). If the cross section is circular the equation becomes particularly simple and is known as Clausing's equation [39]. The perturbation of free-molecular flows is not trivial for steady problems because of the abovementioned non-uniformity in the inverse Knudsen number. If one tries a na'fve iteration, the singularity arising in the first iterate may cancel when integrating to obtain moments (cancellation is easier, the higher is the dimensionality of the problem, because a first-order pole is milder, if the dimension is higher). The singularity is always present and, although it may be mild, it can build up a worse singularity when computing subsequent steps. The difficulties are enhanced in unbounded domains where the subsequent terms diverge at space infinity. The reason for the latter fact is that the ratio between the mean free path )~ and the distance d of any given point from the body is a local Knudsen
- 39. The Boltzmann equation andfluid dynamics 35 number which tends to zero when d tends to infinity; hence collisions certainly arise in an unbounded domain and tend to dominate at large distances. On this basis we are led to expect that a continuum behavior takes place at infinity, even when the typical lengths characterizing the size of the body are much smaller than the mean free path; this is confirmed by the discussion of the Stokes paradox for the steady linearized Boltzmann equation (see [43,39,35]). Both difficulties are removed by the so-called collision iteration: the loss term is partly considered to be unknown in the iteration, thus building an exponential term which controls the singularity. The presence of the latter is still felt through the presence of logarithmic terms in the (inverse) Knudsen number. In higher dimensions this is multiplied by a power of (Kn)-1 which typically equals the number of space dimensions relevant for the problem under consideration in a bounded domain. In particular the dependence upon coordinates will show the same singularity (we can think of local Knudsen numbers based on the distance from the nearest wall); as a consequence first derivatives will diverge at the boundary in one dimension and the same will occur for second, or third derivatives, in two, or three, space dimensions, respectively. In an external domain we have, in addition to the low speed effects, the effect of particles coming from infinity, which actually dominates. In particular in one dimension (half-space problems) the terms coming from iterations are of the same order as the lowest order terms; actually for a half-space problem there is hardly a Knudsen number (the local one is an exception). In two dimensions the corrections in the moments are of order Kn-1 log Kn. In three dimensions a correction of order Kn-2 log Kn is preceded by a correction of order Kn -1 . Care must be exercised when applying the aforementioned results to a concrete numerical evaluation, as mentioned above. In fact, for large but not extremely large Knudsen numbers (say 10 ~< Kn ~< 100)logKn is a relatively small number, although log Kn ---> oQ for Kn ---> oo. Hence terms of order log Kn/Kn, though mathematically dominating over terms of order 1/Kn are of the same order as the latter for practical purposes. As consequence, the two kinds of terms must be computed together if numerical accuracy is desired for the aforementioned range of Knudsen numbers. Related to this remark is the fact that any factor appearing in front of Kn in the argument of the logarithm is meaningless unless the term of order Kn -I is also computed. This is particularly important when the factor under consideration depends upon a parameter which can take very large (or very small) values (typically a speed ratio). Thus Hamel and Cooper [70,85] have shown that the first iterate of the integral iteration is incapable of describing the correct dependence upon the speed ratio and have applied the method of matched asymptotic expansions [81] to regions near a body and far from a body. In particular, for the hypersonic flow of a gas of hard spheres past a two-dimensional strip, they find for the drag coefficient elog e] CD = CDf.m. 1 + 2zr ' (13.5) where the inverse Knudsen number e is based on the mean free path )~ = 7r3/2cr2nooS w (or is the molecular diameter and S~, = S~(Tw/Too), whereas n~ and S~ are the number density at infinity).
- 40. 36 C. Cercignani If we consider infinite-range intermolecular potentials, then we have fractional powers rather than logarithms. All the considerations of this section have the important consequence that approximate methods of solution which are not able to allow for a nonanalytic behavior for Kn ~ c~ produce poor results for large Knudsen numbers. 14. Perturbations of equilibria The first steady solutions other than Maxwellian to be investigated were perturbations of the latter. The method of perturbation of equilibria is different from the Hilbert method because the small parameter is not contained in the Boltzmann equation but in auxiliary conditions, such as boundary or initial conditions. The advantage of the method is that we can investigate problems in the transition regime, provided differences in temperature and speed are moderate. Let us try to find a solution of our problem for the Boltzmann equation in the form oo f =Zenfn, n=O (14.1) where at variance with previous expansions e is a parameter which does not appear in the Boltzmann equation. In addition f0 is assumed from the start to be a Maxwellian distribution. By inserting this formal series into Equation (5.1) and matching the various orders in e, we obtain equations which one can hope to solve recursively: 0tfl + ~j" 0xfl = 2Q(fl, f0), (14.2)1 j-1 Otf j + 1~. Oxfj = 2Q(f j, fo) + Z Q(J~' f j-i), i=1 (14.2)j where, as in Section 4, Q(f, g) denotes the symmetrized collision operator and the sum is empty for j = 1. Although in principle one can solve the subsequent equations by recursion, in practice one solves only the first equation, which is called the linearized Boltzmann equation. This equation can be rewritten as follows: Oth + Ij 9Oxh -- LMh, (14.3) where LM denotes the linearized collision operator about the Maxwellian M, i.e., LMh = 2Q(M, Mh)/M, h = fl/M (see Section 10). We shall assume, as is usually done with
- 41. The Boltzmann equation andfluid dynamics 37 little loss of generality, that the bulk velocity in the Maxwellian is zero and we shall denote the unperturbed density and temperature by P0 and To. Although the equation is now linear, and hence all the weapons of linear analysis are available, it is far from easy to solve for a given boundary value problem, such as Couette flow. Yet it is possible to gain an insight on the behavior of the general solution of Equation (14.3) (see [43,39,35]). This insight gives the following picture for a slab problem, provided the plates are sufficiently far apart (several mean free paths). There are two Knudsen layers near the boundaries, where the behavior of the solution is strongly dependent on the boundary conditions, and a central core (a few mean free paths away form the plates), where the solution of the Navier-Stokes equations holds (with a slight reminiscence of the boundary conditions). If the plates are close in terms of the mean free path, then this picture does not apply because the core and the kinetic layers merge. One can give evidence for the above statements just in the case of the linearized Boltzmann equation, but there is a strong evidence that this qualitative picture applies to nonlinear flows as well, with a major exception. In general, compressible flows develop shock waves at large speeds and these do not appear in the linearized description. As already remarked, these shocks are not surfaces of discontinuity as for an ideal fluid, governed by the Euler equations, but layers of rapid change of the solution (on the scale of the mean free path). One can obtain solutions for flows containing shocks from the Navier-Stokes equations, but, since they change significantly on the scale of the mean free path, they are inaccurate. Other regions where this picture is inaccurate are the zones of high rarefaction, where nearly free-molecular conditions may prevail, even if the rest of the flow is reasonably described in terms of Navier-Stokes equations, Knudsen layers and shock layers. The theory of Knudsen layers can be essentially described by the linearized Boltzmann equation. The main result concerns the boundary conditions for the Navier-Stokes equations. They turn out to be different from those of no-slip and no temperature jump. In fact, the velocity slip turns out to be proportional to the normal gradient of tangential velocity and the temperature jump to the normal gradient of temperature. When one can use the Navier-Stokes equations but must use the slip and temperature-jump boundary conditions, one talks of the slip regime; this typically occurs for Knudsen numbers between 10-1 and 10-2. Subtler phenomena may occur if the solutions depend on more than one space coordinate. The most important change with respect to traditional continuum mechanics is the presence of the term with the second derivatives of temperature in the expression of the stress deviator and of the term with the second derivatives of bulk velocity in the expression of the heat flow. These terms were already known to Maxwell [121]. In recent times, their importance has been stressed by Kogan et al. [101] and by Sone et al. [155] (as already mentioned in Section 12). Even in fully three-dimensional problems the solution of the linearized Boltzmann equation reduces to the sum of two terms, one of which, h8, is important just in the Knudsen layers and the other, hA, is important far from the boundaries. The latter has a stress deviator and a heat flow with constitutive equations different from those of Navier- Stokes and Fourier. In spite of this, the bulk velocity, pressure, and temperature satisfy the Navier-Stokes equations when steady problems are considered. In fact, when we take
- 42. 38 C. Cercignani the divergence of the heat flow vector a term proportional to the Laplacian of v vanishes, thanks to the continuity equation, and thus just a term proportional to the temperature gradient survives; then, taking the divergence of the stress, a term grad(AT) vanishes, because of the energy equation. Yet, the new terms in the constitutive relations may produce physical effects in the presence of boundary conditions different from those of no-slip and no temperature jump. In fact, we must expect the velocity slip to be proportional to the shear stress and the temperature jump to the heat flow. 15. Approximate methods for linearized problems Linearization combined with the use of models lends itself to the use of analytical methods, which turn out to be particularly useful for a preliminary analysis of certain problems. Closed form solutions are not so frequent and are practically restricted to the case of half- space problems [43,39,35]. The latter, in turn, are useful to investigate Knudsen layers and compute the slip and temperature jump coefficients. The use of BGK or similar models permits reducing the solution of Boltzmann's integro- differential equation in phase-space to solving integral equation in ordinary space. This is obtained because in the BGK model the distribution function f occurs only in two ways: explicitly in a linear, simple way and implicitly through a few moments (appearing in the local Maxwellian and the collision frequency). Then one can express f in terms of these moments by integrating a linear, simple partial differential equation; then, using the definitions of these moments and the expression of f one can obtain integral equations for the same moments [43,39,35]. These equations can be solved numerically in a much easier way than the Boltzmann equation. This is particularly true in the linearized case. The integral equation approach lends itself to a variational solution. The main idea of this method (for linearized problems) is the following. Suppose that we must solve the equation: 12h=S, (15.1) where h is the unknown,/2 a linear operator and S a source term. Assume that we can form a bilinear expression ((g, h)) such that ((s h)) = ((g, Eh)),for any pair {g, h} in the set where we look for a solution. Then the expression (functional): J(/~) -- ((h, C/~)) - 2((S,/Tt)) (15.2) has the property that if set/t - h + r/, then the terms of first degree in ~ disappear and J(h) reduces to J(h) -+-((r/,/2r/)) if and only if h is a solution of Equation (15.1). In other words if r/is regarded as small (an error), the functional in Equation (15.2) becomes small of second order in the neighborhood of h, if and only if h is a solution of Equation (15.1). Then we say that the solutions of the latter equation satisfy a variational principle, or make the functional in Equation (15.2) stationary. Thus a way to look for solutions of Equation (15.1) is to look for solutions which make the functional in Equation (15.2) stationary (variational method).
- 43. The Boltzmann equation andfluid dynamics 39 The method is particularly useful if we know that ((0,/20)) is non-negative (or non- positive) because we can then characterize the solutions of Equation (15.1) as maxima or minima of the functional (15.2). But, even if this is not the case, the property is useful. First of all, it gives a non-arbitrary recipe to select among approximations to the solution in a given class. Second, if we find that the functional J is related to some physical quantity, we can compute this quantity with high accuracy, even if we have a poor approximation to h. If the error 0 is of order 10%, then J will be in fact computed with an error of the order of 1%, because the deviation of J (/t) from J (h) is of order 02, as we have seen. The integral formulation of the BGK model lends itself to the application of the variational method [58]. Thus in the case of Couette the functional is related to the stress component p12 which is constant and gives the drag exerted by the gas on each plate. Thus this quantity can be computed with high accuracy [58,43]. This method can be generalized to other problems and to the more complicated mod- els [39]. It can also be used to obtain accurate finite ordinate schemes, by approximating the unknowns by trial functions which are piecewise constant [44]. In the case of the steady linearized Boltzmann equation, Equation (14.3), a similar method can be used. Let us indicate by Dh the differential part appearing in the left-hand side (Dh = ~ 90xh for steady problems) and assume that there is a source term as well (an example of a source occurs in linearized Poiseuille flow, see [43,39,35]) and write our equation in the form: Dh-Lh--S. (15.3) If we try the simplest possible bilinear expression f0L g(x, s e)h(x,/j) dx d,~ (15.4) ((g,h)) -- 3 and we use it with Eh = Dh - Lh we cannot reproduce the symmetry property ((Eg, h)) = ((g, Eh)). It works for Lh but not for Dh. There is however a trick [33] which leads to the desired result. Let us introduce the parity operator in velocity space, P, such that P[h(~j)] -- h(-~). Then we can think of replacing Equation (15.3) by PDh- PLh = PS (15.5) because this is completely equivalent to the original equation. In addition, because of the central symmetry of the molecular interaction PLh = LPh and the fact that we had no problems with L is not destroyed by the fact that we use P. On the other hand we have by a partial integration: ((g, PDh)) --((PDg, h)) + ((g+, Ph-)) B -((Pg-,h+)) B. (15.6)
- 44. 40 C. Cercignani Here g+ denote the restrictions of a function defined on the boundary to positive, respectively negative, values of ~ 9n, where n is the unit vector normal to the boundary. In addition, we have put ((g+'h+))B=f~ f+ S2 ~.n>0 Is e. nlg(x, se)h(x, s e) ds e da. (15.7) In the one-dimensional case, the integration over the boundary OS2 reduces to the sum of the boundary terms at x -- 0 and x = L. Clearly the last two terms in Equation (15.6) do not fit in our description. We have two ways out of the difficulty. We first recall a property of the boundary conditions, discussed in Section 11. The boundary conditions must be linearized about the Maxwellian distribution M and this gives them the following form (see below): h+ = ho + Kh-. (15.8) Because of reciprocity (Equation (11.9)), we have ((Pg-, Kh-))B = ((Kg-, Ph-)) B. (15.9) Hence, if we assume that both g and h satisfy the boundary conditions, we have ((g+, Ph-)) B -- ((Pg-' h+)) B = ((ho, Ph-)) B -- ((Pg-' ho)) B + ((Kg-, Ph-)) B -- ((Pg-' Kh-)) B = ((ho, Ph-))B -- ((Pg-,hO))B. (15.10) We remark that we can modify the solution of the problem by adding a combination of the collision invariants with constant coefficients. This does not modify the Boltzmann equation but can be used to modify the boundary conditions. Usually it is possible to dispose of the constant coefficients to make ((h0, Ph-))8 = 0 (and, at the same time, of course, ((h0, P g-))B = 0). We assume that this is the case. Using this relation and Equation (15.9), Equation (15.10) reduces to ((g+, Ph-)) B - ((Pg-,h+))8 -0 (15.11) and the variational principle holds with the operator s = PDh - PLh and the source PS. This variational principle is correct but not so useful, because it can be used only with approximations which exactly satisfy the boundary conditions and it can be complicated to construct these approximations. Thus we follow another procedure by incorporating the boundary conditions in the functional. It is enough to consider J([t) = (([t, PDh - PLh)) - 2((PS, h)) +((Ph-,h+-Kh--2hO))B. (15.12)
- 45. The Boltzmann equation andfluid dynamics 41 In fact, if we let h = h + 0, we find that the terms linear in 0 disappear from J and the variational principle holds. In agreement with what we said before, it is interesting to look at the value attained by J when h = h. Equation (15.12) becomes J(h)--((PS, h)) - ((Ph-,hO))B. (15.13) This result acquires its full meaning only when we examine the expressions for h0 and S. In general S = 0 (an important case in which this is not true, is linearized Poiseuille flow). If we let S -- 0, then we must look at the expression of h0. The boundary source has a special form because it arises from the linearization, about a Maxwellian distribution M, of a boundary condition of the form: f+=Kwf-, (15.14) where Kw is an operator which has several properties, including Mw+ = KwMw-. (15.15) Now, if let f = M(1 + h) in Equation (15.14), we have: KwM-h- KwM- h+ = + - 1. (15.16) M+ M+ This relation is exact. We can now proceed to neglecting terms of order higher than first in the perturbation parameters. We can replace in Kw the temperature and velocity of the wall by those of the Maxwellian M (i.e., To and 0) and obtain a slightly different operator K0. Thus we obtain the operator K, which we used before, by letting KoM_h-/M+ = Kh_. Concerning the source, we have, using Equation (15.15): KwM- KwM- KwMw- h0= -1= - M+ M+ Mw+ Since Mw and M differ by terms of first order, we can replace Kw by K0 because their difference is also of first order and would produce a term of second order in the expression of h0. KoM- KoMw- KoMw- h0 = = 1- ~ . (15.17) M+ Mw+ Mw+ Now, if we neglect terms of higher than first order in the speed of the wall and the temperature difference Tw - To, we have Mw = M (1 + 7t), where (recalling that Mw is determined up to the density that we can choose to be the same as in M) 7t can be explicitly computed to give:
- 46. 42 C. Cercignani and, finally, neglecting again terms of order higher than first: h0 = ap+ - K ~p_. (15.19) Then Equation (15.13) with S = 0 gives J(h) = -((Ph-, ~P+))B+ ((Ph-, K~_)) B = -((Ph-, g/+))B + ((Kh-, Pqt-))B = -((Ph-, g/+))B + ((h+ -- ~+ + K~-, P~-))B = -((Ph-, lp+))B + ((h+-, P~_))B +((KTt---O+,P~-)) B. (15.20) The last term is a known quantity, whereas the first and the second can be combined to give unknown quantities of physical importance. In fact, if we take into account the expression of gr (Equation (15.18)), we obtain: -((Ph-, ~+))B + ((h+-' P~-))B Vii) To ~j" nh(x, ~j)M d~jda f( 4- 2RTo 2 To ~j. nh(x, ~j)M d~jda l(f f RTo Pn "Vwdo" + q(n) Tw - To da~. (15.21) } To Here Pn is the normal stress vector and qn the normal component of the heat flow. The fact that the mass flow vanishes at the wall has been taken into account. Because of the linearity of the problem, it is possible and convenient, without loss of generality, to consider separately the two cases vw = 0 and Tw = To. Then the two terms in the expression above occur in two different problems. For some typical problems Vw vanishes on just one part of the boundary, whereas it is a constant on the remaining part of the latter; then the factor multiplying this constant is the drag on the corresponding part of the boundary. Similarly one can consider the case in which the factor in front of qn vanishes on just one part of the boundary, whereas it is a constant on the remaining part of the latter, and relate the value of the functional to the heat transfer. The two variational principles which have been discussed are related to each other [39]. The integral equation approach and the variational method have been applied with great success to many linearized problems [43,39,35]. Among the most interesting results we mention the calculation of the minimum in the flow rate for Poiseuille flow [30,61,31, 44], first experimentally discovered by Knudsen [98], and the calculation of the drag on a sphere (at low Mach numbers) where the results agree very well [59] with the semi- empirical formula derived by Millikan from his experimental data [126].
- 47. The Boltzmann equation andfluid dynamics 43 16. Mixtures As is well known, air at room pressure and temperature is a mixture, its main components being two diatomic gases, nitrogen and oxygen. This immediately calls for an immediate change in our basic equation, Equation (5.1), which is only suitable for a single monatomic gas. A remarkable feature of aerodynamics at the molecular level is that the evolution equations change in a significant way, when dealing with polyatomic rather than monatomic gases. This is not the case when the gas is treated as a continuum, where, at least at room conditions, only a few changes in the equations occur, the most remarkable being the change of the ratio of specific heats y. If we consider a mixture of monatomic gases, the differences between the various species occur in the values of the masses and in the law of interaction between molecules of different species; in the simplest case, when the molecules are pictured as hard spheres, the second difference is represented by unequal values of the molecular diameters. In the mathematical treatment, a first difference will be in the fact that we shall need n distribution functions f/ (i = 1, 2..... n) if there are n species. The notation becomes complicated, but there is no new idea, except, of course, for the fact that we must derive a system of n coupled Boltzmann equations for the n distribution functions. The arguments are exactly the same, with obvious changes, and the result is a f/ a f/ a f/ -~- ~" if- Xi" at ~x a~ -- ~ fR3 ft~ (fi'f[* - f~f~*)Bik(n" V' lVl)dl;*dn' k=l + (16.1) where ]~ik is computed from the interaction law between the i-th and k-th species, while in the k-th term in the left-hand side, V = ~ - ~j. is the relative velocity of the molecule of the i-th species (whose evolution we are following) with respect to a molecule of the k-th species (against which the former is colliding). The arguments ~f and ~jf.are computed, as before, from the laws of conservation of mass and energy in a collision with the following result: ~j, ~j _ 2/zik n[(~j - ~j.). n], mi 2lZik ~', -- ~j, + n[(~ - ~,). n], mk (16.2) where lZik : mimk/(mi -+-m~) is the reduced mass [39]. To prepare some material for the description of polyatomic gases and chemical reactions, we remark that Equation (16.1) can be rewritten as follows [39]: oj5 ~ ~. oj5 oj5 ~)---7- - -~X -p- xi " O~
- 48. 44 C. CercQnani n -zf. 3 R 3 k=l x xR 3 (fi'f/~, - fi fk,)Wik(l~,/j, 1/~ t,/j~,) d/~, d/~'d/~,. (16.3) Now ~j~,~,, ~, ~, are independent variables (i.e. they are not related by the conservation laws) and Wik(~, ~,[~'~,) ~- Sik(n" V, [Y[)8(mi~, + mk~, -- mi~' -- mk~t,) x 8(mi[~,[ 2 + mk[~,[ 2 -- mi [~[ 2 -- mk[~t,[2), (16.4) where n = (/~ -/~t)/[/~ - ~J'l and Si~(n. V, IVl)= 13ik(n. V, IVl) (mi q- mk)3mimk. (16.5) 2n. V Conservation of momentum and energy is now taken care of by the delta functions appearing in Equation (16.4) [39]. With a slight modification, Equation (16.3) can be extended to the case of a mixture in which a collision can transform the two colliding molecules of species j, 1 into two molecules of different species k, i (a very particular kind of chemical reaction). In this case the relations between the velocities before and after the encounter are different from the ones used so far, but we may still write a set of equations for the n species: aj~ aj~ afi O---t--[- ~" -~X -'[-X i" 0 3 R 3 k,l,j--1 • • (f/fj, - fi fk,)Wi~ (~, l~,l~z, l~Z,)d~, d~Zdl~,, (16.6) where Wi~ gives the probability density that a transition from velocities ~', ~jt,to velocities ~, ~, takes place in a collision which transforms two molecules of species l, j, respectively, into two molecules of species i, k, respectively. It is clear how the previous model is included into the new one when the species change does not occur. The idea of kinetic models analogous to the BGK model can be naturally extended to mixtures and polyatomic gases [151,127,81 ]. A typical collision term of the BGK type will read n Ji(fr) -" Jij(fr) - Z 1)ij[(I)ij(~) - fi(~)], j--1 j=l (16.7) where Vij are the collision frequencies and ~ij is a Maxwellian distribution to be determined by suitable conditions that generalize Equation (9.4). There are some important changes concerning the collision invariants and the definition of macroscopic functions in the case of mixtures. First of all, the collision invariants in
- 49. The Boltzmann equation andfluid dynamics 45 the case of n species have n components and are defined as follows: ~ri (i = 1..... n) is a collision invariant if and only if R ~iQid~=O l (16.8) where Qi denotes the right-hand side of Equation (16.1). There are n + 4 rather than 5 linearly independent collision invariants. There are 3 invariants related to momentum conservation, 7t(n+~)i = mi~ (or-- 1,2, 3), and one related to energy conservation, ~(n+4)i --mil~12; the remaining n invariants are related to the conservation of the number of particles of each species lPij --" ~ij (i, j = 1,...,/7). This, of course, applies when there are no chemical reactions. Concerning the macroscopic quantities and their relation to the moments of the distribution functions, we remark that in the case of mixtures it is more convenient to think that the distribution function is normalized as a number density (this has been already taken into account when giving the expression of the collision invariants). Then the number densities of the single species are given by: n(i)= f (i- 1..... n) (16.9) and the mass density p(i) is given by mi n(i) . The number and mass densities for the mixture are given by: F/ n -- ~ n (i), i=1 (16.10) /7 p = ~ p(i). i=1 (16.11) It is convenient to define the bulk velocities of the single species and the bulk velocity of the mixture as follows: v(i) = fR 3~ fi dl~ (i = 1,..., n), f R3 fi dl~ (16.12) ?/ pv = ~ p(i)v(i). i=1 (16.13) It is usual to define the peculiar velocity c=~ -v. (16.14)
- 50. 46 C. Cercignani The stress tensor for the i-th species is given by: p(i) fR jk -- mi CjCk fi dl~ 3 (i = 1..... n; j,k= 1,2,3); (16.15) and the stress tensor for the mixture is the sum of the various stresses: ~_(i) Pjk -- Pjk i=1 (j, k = 1, 2, 3). (16.16) It is to be remarked that, though these definitions are the most common and natural, they are not used by all the authors. One might, e.g., define a peculiar velocity for each species and use it to define the partial stresses. Then it is no longer true that the stress tensor for the mixture is the sum of the partial stresses. Similarly the thermal energy per unit mass (associated with random motions) is defined for each species by: n(i)e(i)- 2 3 Icl2fi d~ (i = 1..... n) (16.17) and for the mixture by: pe = ~p(i)e(i) i=1 (16.18) A similar procedure can be applied to the heat flow. The pressure is, as usual, 1/3 of the trace of the stress matrix and is related to the temperature by p -- nkB T. Please remark that there is not a constant R such that p = p R T. 17. Polyatomic gases A possible picture of a molecule of a polyatomic gas, suggested by quantum mechanics, is as follows [172]. The molecule is a mechanical system, which differs from a point mass by having a sequence of internal states, which can be identified by a label, assuming integral values. In the simplest cases these states differ from each other because the molecule has, besides kinetic energy, an internal energy taking different values Ei in each of the different states. A collision between two molecules, besides changing the velocities, can also change the internal states of the molecules and, as a consequence, the internal energy enters in the energy balance. From the viewpoint of writing evolution equations for the statistical behavior of the system, it is convenient to think of a single polyatomic gas as a mixture of different monatomic gases. Each of these gases is formed by the molecules corresponding to a given internal energy, and a collision changing the internal state of at least one molecule is considered as a reactive collision of the kind considered above,
- 51. The Boltzmann equation andfluid dynamics 47 Wi/~(~J,/~,I~J',~J',) giving the probability density of a collision transforming two molecules with internal states l, j respectively, and velocities ~t ~jt respectively, into molecules with internal states i, k, respectively, and velocities ~j, ~,, respectively. This model is amply sufficient to discuss aerodynamic applications. We want to mention, however, that it requires nondegenerate levels of internal energy, if there are, e.g., strong magnetic fields which can act on the internal variables such as (typically) the spin of the molecules. In that case, if the molecule has spin s, the distribution function f becomes a square matrix of order 2s + 1 and the kinetic equation reflects the fact that matrices in general do not commute and, as remarked by Waldmann [170,171] and Snider [152], the collision term contains not simply the cross-section but the scattering amplitude, which may not commute with f. It is appropriate now to enquire why we started talking about quantum rather than classical mechanics. The main reason is not related to practice, but rather to history. Classical models of polyatomic molecules are regarded with suspicion since 1887 when Lorentz found a mistake in the proof of the H theorem of Boltzmann [23] for general polyatomic molecules. The question arises from the fact that when one proves the H theorem for a monatomic gas one usually does not explicitly underline (because it is irrelevant in that case) that the velocities ~jt and ~t, are not the velocities into which a collision transforms the velocities ~j and ~,, but the velocities which are transformed by a collision into the latter ones; this is conceptually very important, but the lack of a detailed discussion does not lead to any inconvenience because the expressions for ~t, and ~t are invariant with respect to a change of sign of the unit vector n, which permits an equivalence between velocity pairs that are carried into the pair ~,, ~ and those which originate from the latter pair, as a consequence of a collision. The remarkable circumstance which we have just recalled is related to the particular symmetry of a collision described by a central force, which allows to associate to a collision [~j,, ~] -+ [~jt,,~t] another collision, the so- called "inverse collision" [~t,, ~jt] __+ [~,, ~], which differs from the former just because of the transformation of the unit vector n into -n. When polyatomic molecules are dealt with, the states before and after a collision require more than just the velocities of the mass centers to be described (the angular velocity, e.g., if the molecule is pictured as a solid body). Let us symbolically denote by [A, B] the state of the pair of molecules. Then there is no guarantee that one can correlate an "inverse collision" [At, B t] ~ [A, B], differing from the previous one just because of the change of n into -n with the collision [A, B] ~ [At, Bt]. Now in the original proof of the H theorem for polyatomic molecules proposed by Boltzmann [23], the assumption was implicitly made that there is always such collision. Lorentz remarked [119,24,26] that this is not true in general. Boltzmann recognized his blunder and proposed another proof based on the so-called "closed cycles of collisions" [119,24,26,166]; the initial state [A,B] is reached not through a single collision but through a sequence of collisions. This proof, although called unobjectionable by Lorentz and Boltzmann [26], never satisfied anybody [167,171 ]. For a while the matter was forgotten till a quantum mechanical proof showed that the required property followed from the unitarity of the S matrix [171]. A satisfactory proof of the inequality required to prove the H theorem for a purely classical, but completely general, model was given only in 1981 [51].
- 52. 48 c. Cercignani For aerodynamic applications all these aspects are not so relevant and, in fact, the main problem is to find a sufficiently handy model for practical calculations. Lordi and Mates [118] studied the "two centers of repulsion" model and found that a rather complicated numerical solution was required for a given set of impact parameters. The lack of closed form expressions makes the model impractical for applications, where the numerical solution describing the collision should be repeated many millions of times. Curtiss and Muckenfuss [132] developed the collision mechanics of the so-called sphero- cylinder model, consisting of a smooth elastic cylinder with two hemispherical ends. Whether or not two molecules collide depends on more than one parameter; in addition, there are several "chattering" collisions in a single collision event. The loaded-sphere model had been already developed by Jeans [94] in 1904 and was subsequently developed by Dahler and Sather [69] and Sandier and Dahler [146]. Although it is spherical in geometry, the molecules rotate about the center of mass, which does not coincide with the center of the sphere, with the consequence that it has essentially the same disadvantages as the sphero-cylinder model. The only exact model which is amenable to explicit calculations is the perfectly rough sphere model, first suggested by Bryan [28] in 1894. The name is due to the fact that the relative velocity at the point of contact of the two molecules is reversed by the collision. This model has some obvious disadvantages. First, a glancing collision may result in a large deflection; second, all collisions can produce a large interchange of rotational and translational energy, with the consequence that the relaxation time for rotational energy is unrealistically short; third, the number of internal degrees of freedom is three, rather than two, which makes the model inappropriate for a description of the main components of air, which are diatomic gases. One can disregard the first disadvantage and put a remedy to the second by assuming that a fraction of collisions follow the smooth-sphere rather that rough-sphere dynamics; but there is obviously no escape from the third difficulty. In practical calculations, one has learned, since long time, that one must compromise between the faithful adherence to a microscopic model and the computational time required to solve a concrete problem. This was true in the early days of rarefied gas dynamics (and may still be true nowadays when one tries to find approximate closed form solutions or spare computer time) even for monatomic gases, as we discussed in Section 9 (in connection with the BGK model) and shall discuss later (in connection with Direct Simulation Monte Carlo). As a matter of fact, when trying to solve the Boltzmann equation, one of the major shortcomings is the complicated structure of the collision term; if to this problem, present even in the simplest case, one adds the complication of the presence of internal degrees of freedom, any practical problem becomes intractable, unless one is ready to accept the aforementioned compromise. Fortunately, when one is not interested in fine details, it is possible to obtain reasonable results by replacing the collision integral by a phenomenological collision model, e.g., a simpler expression which retains only the qualitative and average properties of the true collision term. As computers become more and more powerful, the amount of phenomenological simplification diminishes and the calculations may more closely mimic the microscopic models. For polyatomic gases, the basic new fact with respect to the monatomic ones is that the total energy is redistributed between translational and internal degrees of freedom at each collision. Those collisions for which this redistribution is negligible are called elastic,
- 53. TheBoltzmann equationandfluid dynamics 49 while the others are called inelastic. The simplest approach would be to calculate the effect of collisions as a linear combination of totally elastic and completely inelastic collisions, the second contribution being described by a model analogous to the BGK model which was described in Section 9. There are, of course, problems related to the molecule spins and their alignment; they are particularly important when we put the molecules in a magnetic field and peculiar phenomena, which go under the general name of Senftleben-Beenakker effects, arise. There is an entire book devoted to this topic [122] and we shall not deal with these problems. Let us now consider in more detail the case of a continuous internal-energy variable. In this case, it is convenient to take the unit vector n in the center-of-mass system and use the internal energy Ei and Ei, of the colliding molecules. As usual, the values before a collision will be denoted by a prime. Equation (1.2) is replaced by / fo Q(f, f)= d~, Ei 2 dEi, (E~)~ dE~ 3 2 f E-E~ t t t x (E;,) n~__.._~2 dEi,(f f~_ ff,) Jo • n 9n~; E i E(~,~ Ei, Ei,). (17.1) Here E -- m[tj[2/4 + Ei -k- Ei, is the total energy in the center-of-mass system which is conserved in a collision. The kernel B satisfies the reciprocity relation IVIB(E'n n' ' E( , 9 ;E i, ~,~ Ei, Ei,) 9 rE! ) --[Vt[/3(E; n. n~ Ei, Ei, ~ E i t, (17.2) Here we follow a paper by Kug~er [111] and look for a one-parameter family of models, assuming that the scattering is isotropic in the center-of-mass system. The second crucial assumption will be that the redistribution of energy among the various degrees of freedom only depends upon the ratios of the various energies to the total energy E, Ei -- Ei/E, etc. This assumption is valid for collisions of rigid elastic bodies and can be considered as a good approximation for steep repulsive potentials9 It is then possible to write 13 in the following form: IVIB(E n n' ' ' ; 9 ;E i,Ei,~ Ei, Ei,) iv, i2 IVl O'tot(E) 0(el , - - 4roE n ' ei* ~ Ei , ei,; "g). (17.3) The denominator on the fight takes care of normalization. Then the function 0 satisfies the following relations: 1 n-2 f01-6 n-2 ~ ! S--~--de E, 2 ds,0(s , ei, ~ ei, ei,; "g)-~- 1, (17.4)
- 54. 50 C. Cercignani ' ')0(e~, ' .r) (1 -- 6 i -- 6i, 6i, ~ 8i, 6i,, ! ! .15). = (1 - si - si,)O(si, si, ~ s i, si,, (17.5) The dependence of O'tot on E makes it possible to adjust the model to the correct dependence of the viscosity on temperature. The parameter 15is chosen in such a way as to represent the degree of inelasticity of the collisions, r - 0 corresponds to elastic collisions" ! o - ~ 0(8~, Ei, ~ Si, Ei,, 0) = 6 n-2 n-2 l 2 6, 2 = 6(8i -- e~)~(E~ -- 8i, ). (17.6) 15-- cx~ corresponds to maximally inelastic collisions: n22 --~-~-~F(n + 2) ' 9cx~)= 6 6, (1 -- 8i -- 8,i) 0(8~, 8i, ~ 8i, 8i,, (F(n)) 2 9 (17.7) A mixture of the two extreme cases gives the model first proposed by Borgnakke and Larsen [27] in 1975" O(E , Ei, ~ 8i, Ei,, 15) ! 90) + 1 - e-r O(s 8i, ~ 6i, 6i,, = e-r 0(6 , 6i, --+ 6i, 6i,, , 9 (17.8) Ku~6er [111 ] notices an analogy between this model and Maxwell's model for gas-surface interaction, as discussed Section 11, and introduces another model, called the theta model, which would correspond to the Cercignani-Lampis model in this analogy. The Larsen-Borgnakke model has become a customary tool in numerical simulations of polyatomic gases. It can be also applied to the vibrational modes through either a classical procedure that assigns a continuously distributed vibrational energy to each molecule, or through a quantum approach that assigns discrete vibrational levels to each molecule. It would be out of place here to discuss this point in more detail, for which we refer to the book of Bird [20]. We also refrain from discussing the interesting recent developments [12, 13] of an old idea of Boltzmann [25,42] to interpret, in the frame of classical statistical mechanics, the circumstance that at low temperatures the internal degrees of freedom appear to be frozen, as due to the extremely long relaxation times for the energy transfer process. The Larsen-Borgnakke model suffers from the limitation that it considers all the collisions as a mixture of the elastic or completely inelastic collisions, disregarding the possibility of a partially inelastic one. In order to construct a more general model we use a procedure which was first used to deal with models for boundary conditions [43,39,35]. We l start from a sensible approximate kernel 00(s~, 6il ~ 6i, C/l), which is chosen on the basis of intuition, but does not satisfy the basic properties (17.4) and (17.5). At this point we add to it some other terms, which ensure that the three fundamental properties are satisfied, as follows: ! 0(61 , 6il ~ 6i, 6il) ! =- 00(6 I, Eil ~ 8i, Eil)
- 55. The Boltzmann equation and fluid dynamics 51 + F(n -+-2)(1 -- ei- eil)(1 -- U(ei eil))(1 -- H(e~ e~l))/I, (/-,(n))2 ' , fo,fo,i = n-2 n-2 ! 2 dei de/l, O0(e~, eil --'+ ei, eil) ei 2 eil (17.9) I--1- F' (n + 2) (/-'(n)) 2 L1L 1-6i n-2 n-2 2 dei deil (17.10) • (1 -- ei -- eil) H(ei, eil) e i 2 ei 1 . As in the aforementioned case, Equation (17.9) may be interpreted as the linear ! combination of two normalized kernels 01(...) and 02(...), while H(e~,eil ) is a sort of accommodation coefficient depending on the energies of the impinging molecules I (H(e I, ell) must lay in the interval [0, 1]); we write ! ! 0(...) -- H(e~, 8il)01 (...) --}- (1 - H(E~, 8il))02(...), l 0o(...) -- 01 (...)H(e;, C/l); F(n + 2) (1 - ei - eil)(1 -- H(ei eil))/I. 02(. 9.) -- (/_,(n))2 (17.11) The expression of 02(...) is suggested by the requirement that the product (1- t1))02(.. ") must satisfy reciprocity, the definition of I is chosen in order to get the normalization of 02(...). l Cercignani and Lampis [52] proposed the following kernel 00(eI, ei] ~ ei, eil;a, b), containing two parameters a and b. ! 00(e~, eil ~ ei, eil" a, b) -- ba ,,-2 n-2 ,,-2 ,,-2 (1 - ei - eil) 1/2 4 e~-- T !--T ei 4 eil eil ! ! 1/2 7r (l--c/--eil ) • exp[-a (ei - e!) 2 - a (eil -- e;1)2]. (17.12) In order to avoid a singularity in the expression of the kernel (17.12) and therefore also in t t I _ mCfr 2/4 > r, where r is a little positive that of H(e~, eil), it is assumed that E- E i - Eil ! parameter: then 00(...) and H(e~, eil) are limited. This approximate kernel 00(...) for ! a --+ cx~tends to bOel(e~, eil, ei, eil), so that the full kernel given by Equation (10) tends to the Larsen-Borgnakke model. For finite values of a, 00(...) describes a collision in which l ! (and similarly for ell, eil ), but may be different, according to ei is not exactly equal to e i a Gaussian distribution. By fitting the theoretical expression for viscosity with some experimental data (from standard handbooks), the aforementioned authors [52] have obtained the values of the parameters in the case of N2 and 02.
- 56. 52 C Cercignani The aforementioned authors, in a joint paper with J. Struckmeyer [56,57,55], found that a good fitting of some experimental data given in [99] can be obtained. However, they were unable to obtain a unique determination of the parameters a and b. Some examples of possible choices of a and b have been given in [52], in the case of N2 and 02. In those examples, low values of a, for instance, a = 1, a - 0.1, and for each of them a value of b close to its maximum value, were chosen, but it is also possible to choose much higher values of a. In order to obtain more information about the values of the parameters a and b, an obvious way would be to try to fit a second transport coefficient. Unfortunately, the experimental data for fly versus temperature are very scanty [141,73]. The bulk viscosity of gases can only be measured by the attenuation and dispersion of an ultrasonic, acoustic signal. Moreover, it is a difficult method subject to experimental error [73]. Because of this, it is not possible to draw conclusions on the range of applicability of the model. In the application of the kernel to the calculation of transport coefficients, everything does work also without introducing the cut-off. The situation may be different in other problems, for instance in the application of DSMC. Therefore a similar model that does not present singularities and does not require a cut-off was introduced [56], based on the following kernel: ! 00(8~, 8il ---> 8i, 8il; a, b) 2ba n-2 n-2 n-2 n-2 (1 -- 8i -- 8il) 4 t---T-- !-----T 8i 4 8il Ei 8il t t re (1 -- 8i -- eil) -1- (1 - 8 i -- 8il ) X exp[--a(8i 8') 2 t 2] -- --a(8il--8il) . (17.13) Using this kemel, the authors repeated the calculation of heat conductivity, which follows the same procedure as before. The results given in [52] about heat conductivity versus temperature are identical to those calculated with the new kemel. We end this review of models for polyatomic gases by remarking that a model generalizing the ES model (see Section 9) to polyatomic gases has been discussed by Andries et al. [3], who have also shown that the H-theorem holds for this model for polyatomic gases as well. Conceming the boundary conditions for the distribution function for polyatomic gases, we remark that there is not much material published on this subject, perhaps because the theory is not so different from that holding in the monatomic case. Concerning specific models, one should mention an extension of the CL model to polyatomic molecules proposed by Lord [117] as a generalization of the Cercignani-Lampis model [48]. 18. Chemistry and radiation Chemical reactions are important in high altitude flight because of the high temperatures which develop near a vehicle flying at hypersonic speed (i.e., at Mach numbers larger than 5). Up to 2000 K, the composition of air can be considered to be the same as at standard conditions. Beyond this temperature, N2 and 02 begin to react and form NO. At
- 57. The Boltzmann equation andfluid dynamics 53 2500 K diatomic oxygen begins to dissociate and form atomic oxygen O, till 02 completely disappears at about 4000 K. Nitrogen begins to dissociate at a slightly higher temperature (about 4250 K). NO disappears at about 5000 K. Ionization phenomena start at about 8500 K. As we implicitly remarked when we wrote Equation (16.6), the kinetic theory of gases is an ideal tool to deal with chemical reactions of a particular kind, i.e. bimolecular reactions, which can be written schematically as A + B +-~ C + D, (18.1) where A, B, C and D represent different molecular species. We already used the term "molecule", as usual in kinetic theory, to mean also atom (a monatomic molecule); in this section we shall further enlarge the meaning of this term to include ions, electrons and photons as well, when we have to deal with ionization reactions and interaction with radiation. As long as the reaction takes place in a single step with the presence of no other species than the reactants, it is a well-known circumstance that the change of concentration of a given species (A, say) in a space-homogeneous mixture can be written as follows: dnA dt -- kb(T)ncnD - k f(T)nAnB. (18.2) We remark that, in chemistry, one uses the molar density in place of the number density used here, the two being obviously related through Avogadro's number. The rate coefficients kf and kb for the forward and backward (or reverse) reactions, respectively, are functions of temperature and are usually written by a semiempirical argument, which generalizes the Arrhenius formula, in the form: ( Ea) k(T) = AT ~ exp -~-~- , (18.3) where A and r/ (= 0 in the Arrhenius equation) are constants, and Ea is the so-called activation energy of the reaction. It is clear that these equations, though having a flavor of kinetic theory, are essentially macroscopic and can be assumed to hold when the distribution function is essentially Maxwellian. In fact, the above reaction theory can be obtained by assuming that the distribution functions are Maxwellians, whereas the role of internal degrees of freedom may be ignored and the reaction cross-section vanishes if the translational energy Et in the center-of-mass system is less than Ea and equals a constant aR if the energy is larger than Ea. A more accurate theory is obtained [18-20] by assuming that the ratio of the reaction cross-section to the total cross-section is zero when the total collision energy Ec (equal to the sum of Et and the total internal energy of the two colliding molecules Ei) is less than Ea and proportional to the product of a power of Ec - Ea and a power of Ec. The exponents and the proportionality factor are essentially dictated by the number of internal degrees of freedom, the exponent of the temperature in the diffusion coefficient of species A in species B and
- 58. 54 C. Cercignani the empirical exponent r/appearing in Equation (18.3). This theory provides a microscopic reaction model that can reproduce the conventional rate Equations (18.2)-(18.3) in the continuum limit. The model is, however, as in the case of gas-surface interaction and models for polyatomic gases, largely based on phenomenological considerations and mathematical tractability. The ideal microscopic model would consist of complete tabulations of the differential cross-sections as functions of the energy states and n. Some microscopic data, coming from extensive quantum-mechanical computations, supported by experiments, are available, but, unfortunately, not very much is known for reactions of engineering interest. When comparisons can be made, the reaction cross-section provided by the phenomenological model is of the correct order of magnitude. This provides some reasons of optimism about the validity of the results obtained with these models for the highly non-equilibrium rarefied gas flows. Termolecular reactions provide some difficulty to kinetic theory, because the Boltzmann equation essentially describes the effect of binary collisions. They are, however, of essential importance in high-temperature air, where the reverse (or backward) reaction of a dissociation one is a recombination reaction, which is necessarily termolecular, as we shall presently explain. A typical dissociation-recombination reaction can be represented as AB+X+-~A+B+X, (18.4) where A B, A, B and X represent the dissociating molecule, the two molecules produced by the dissociation and a third molecule (of any species), respectively. The latter molecule, in the forward reaction, collides with A B and causes its dissociation. This process is described by a binary collision and is an endothermic reaction, requiring a certain amount of energy, the dissociation energy Eo. The recombination process is an exothermic reaction and it might seem that one could dispense with the "third body" X and consider it as a bimolecular reaction AB +-- A + B. (18.5) However, one can easily see that the energy balance for this event cannot be satisfied in the presence of energy release. In fact if two molecules form an isolated system and are assumed to interact with a potential energy which is attractive at large distances and repulsive at short distances, they can come close enough to orbit one about the other but the repulsive part will eventually separate them. In fact, they cannot form a stable molecule; this is seen by writing the energy equation in the center-of-mass system. The final kinetic energy in this reference system should be zero for the molecule whereas it was positive when the two molecules approached each other and the potential energy is negative. A third molecule X is required to describe the recombination process. In order to keep the binary collision analysis, appropriate for a rarefied gas, we must think of the recombination process as a sequence of two binary collisions. The first of these forms an (unstable) orbiting pair P, that is stabilized by a second collision of this pair with X, as long as this collision occurs within a sufficiently small elapsed time. One can then extend the previous theory based on a binary collision analysis. If the activation energy is assumed to be zero, then the main change is that the cross section acquires a factor proportional to the number density of the species X.
- 59. The Boltzmann equation and fluid dynamics 55 This simple theory is based on a molecular interaction that is attractive at large distance and repulsive at short distances. One can assume a highly simplified scheme by taking a hard sphere core with diameter cr and a scattering of the square-well type at some larger distance cr,. The potential energy is, say, -Q (Q > 0) between cr and cr,. Then if m is the mass of a molecule A, for a distance r > or,, the trajectory of one molecule with respect to the other before the interaction begins, will be a straight line r cos 0 = b (in polar coordinates), where b is the impact parameter. The condition b < or, must be satisfied if the molecules actually interact. Then, assuming for simplicity that A = B (as in the case of recombination of oxigen) the conservation of energy and angular momentum give (d!) 2 1 4Q 1 -~ r 2 m g 2 : b --5 (or < r < a,), (18.6) where V is, as usual, the relative speed. We easily verify that the trajectory has a comer point at a distance r = or, and the molecule we are following is deflected toward the other by an angle Oo cos-l(b) ,[b ( 4Q )-,/2] - -- -cos- -- 1+ V2 . (18.7) a, a, m The orbiting pair P has in this case a very simple motion: in fact the molecules A approach each other and then have a hard sphere collision. After that they tend to separate again; and the "molecule" P will disappear and two molecules A emerge again, unless a third molecule X collides with P, which is stabilized into a A2. The unstable pair P is endowed with an internal energy mv2 EB---~ + Q. (18.8) This energy is stored to be, possibly, converted into kinetic energy (and hence, from a macroscopic viewpoint, heat of reaction) through the process (18.4). Even for this simple scheme we must write three Boltzmann equations: one for the species A, one for the species A2 and one for the unstable species P; even if A is monatomic A2 and P are diatomic and hence have an internal energy. The species X used in the above argument can be any of the three aforementioned species. The species A loses particles when colliding with A (formation of P) and P gains in the same process, but loses in most collisions undergone by a P molecule. This model can be slightly complicated by assuming that there is a potential barrier between or, and cr** (or** > or,). If the potential energy is Ea > 0 for these distances between the A molecules, then the formation can occur if, and only if, the relative speed V is larger than 4Ea/m. Then Ea plays the role of the activation energy. The resulting system of Boltzmann equations reads as follows: -- + ~. -- -2 at Ox 3 + f f, BrAAd/~ dn + (f F, - f g,)BAA2 , 3 +
- 60. 56 C.Cercignani + fR fB (ftf~--ff*)BeAAd~j*dn 3 + + fB (ft f1* - f f *)13AA2d~* dn' 3 + (18.9)~ OF Ot + ~. OF Ox y.f f.f .. d~ 2 ' ' d~j dn + f g,13Bades, = F g, BBA2 , 3 + 3 + -fl~3ft3+Fg, Bt~A2dl~,dn + (F f~ - Ff,)]~AA2 d~j, dn 3 + + fB (FIFI* - FF*)I3A2A2 d~* dn' 3 + (18.9)2 Og Ot ag fR f13 ft t r fR f13 9 = f*]~aad~, dn - gf,13Ba d~, dn ~XX 3 + 3 + -fR3ft3+gF*131~a2df~,an, (18.9)3 where f, F, and g denote the distribution functions for species A, A2, and B, whereas the superscripts r and e are used to discriminate between reactive and elastic collisions when necessary. For simplicity, we have omitted indicating the internal energies of particles A2 and B. The factors 2 take into account the fact that 2 particles of a species disappear or appear at the same time. Models to deal with a chemically reactive gas, akin to the BGK one have been discussed by Yoshizawa [178]. Ionization reactions involve the electronic states and it is unlikely that a purely classical theory will be successful in describing them, because of the selection rules. Yet, one can use the phenomenological approach to provide at least an upper bound for the reaction rates9 As mentioned above, one can, in principle, think of interaction with radiation as if it were a reaction involving photons as "molecules". Here spontaneous emission should also be taken into account. It becomes harder to develop phenomenological models, because one should consider as many species as there are excited levels for each molecule9 Photons can be described by means of the so-called radiative transfer (or radiation transport) equation, which looks like a Boltzmann equation. The analogy is, however, in a sense, artificial, because the number of photons is not conserved. Of 3f ~ + c~o. at ax
- 61. The Boltzmann equation andfluid dynamics 57 -- fR3 K(x, k' ~ k)f(x, k') dk' - v(x, k)f(x, k) + s(x, k). (18.10) Here K is the scattering probability from a wavevector k' to a wavevector k. The direction of k is given by o~and its magnitude is the radiation frequency multiplied by the speed of light c, not to be confused with v(x, k), the total frequency of scattering and absorption events. The term s(x, k) describes the volume radiation source, due to photon emission. If inelastic scattering effects, like fluorescence and stimulated emission, are neglected, then there is no interaction between photons of different frequency and we can replace the arguments by k' and k by the corresponding unit vectors oJ and oY. The emission term can be expressed as a product of the Planck distribution by the volume emission coefficient elkl,T. Boundary conditions for radiation can be described in a way similar to that used for gas-surface interaction. Of course, emission and absorption of radiation occur, along with reflection [149,96]. 19. The DSMC method Kinetic models and perturbation methods are very useful in obtaining approximate solutions and forming qualitative ideas on the solutions of practical problems, but in general they are not sufficient to provide detailed and precise answers for practical problems. Various numerical procedures exist which either attempt to solve for f by conventional techniques of numerical analysis or efficiently by-pass the formalism of the integrodifferential equation and simulate the physical situation that the equation describes (Monte Carlo simulation). Only recently proofs have been given that these partly deterministic, partly stochastic games provide solutions that converge (in a suitable sense) to solutions of the Boltzmann equation. Numerical solutions of the Boltzmann equation based on finite difference methods meet with severe computational requirements due to the large number of independent variables. The only method that has been used for space-inhomogeneous problems in more than one space dimension is the technique of Hicks, Yen and Nordsiek [136,177], which is based on a Monte Carlo quadrature method to evaluate the collision integral. This method was further developed by Aristov and Tcheremissine [4,164] and has been applied with some success to a few two-dimensional flows [165,45]. An additional difficulty for traditional numerical methods is the fact that chemically reacting and thermally radiating flows (and even simpler flows of polyatomic gases) are hard to describe with theoretical models having the same degree of accurateness as the Boltzmann equation for monatomic nonreacting and nonradiating gases. These considerations paved the way to the development of simulation schemes, which started with the work of Bird on the so-called Direct Simulation Monte Carlo (DSMC) method [17] and have become a powerful tool for practical calculations. There appear to be very few limitations to the complexity of the flow fields that this approach can deal with. Chemically reacting and ionized flows can be and have been analysed by these methods. A problem which arises in the applications of the DSMC method is the choice of a model for the molecular collisions. The issue is to have simple computing rules by discarding what
- 62. 58 C. Cercignani is physically insignificant. In the case of monatomic gases, the relation of the deflection angle to the impact parameter and the relative speed appears to be the most important piece of physics. It turns out, however, that the scattering law has little effect and that the observable effects are strongly correlated with the cross-section change with relative speed. This realization led to the idea of the variable hard-sphere (VHS) model [19] which combines the scattering simplicity of the hard-sphere model with a variable cross-section based on a molecular diameter proportional to some power co - 1/2 of the relative speed (co being the power of absolute temperature ruling the change of the viscosity coefficient). This does not produce problems in a single gas, because the heat conductivity varies with approximately the same law as the viscosity coefficient, but problems arise for mixtures. If one wants the correct diffusion coefficient, another modification is needed. Koura and Matsumoto [107] developed the variable soft sphere (VSS) model, which introduces an additional power law parameter and gives the necessary flexibility for mixtures. More complicated models can be devised when the attractive part of the intermolecular force is taken into account [84]. A simpler method has also been proposed [97]. Before discussing the DSMC method and some of its applications in some detail, we remark that, although the DSMC has no rivals for practical computations, some other methods may turn out to be of interest in the future if much more powerful computers will be available. Thus, e.g., discrete velocity models have been an intensely studied subject for many years, before becoming a systematic method of approximating the Boltzmann equation. The DSMC method, the molecular collisions are considered on a probabilistic rather than a deterministic basis. The main aim is to calculate practical flows through the use of the collision mechanics of model molecules. In fact, the real gas is modeled by some hundred thousands or millions of simulated molecules on a computer. For each of them the space coordinates and velocity components (as well as the variables describing the internal state, if we deal with polyatomic molecules) are stored in the memory and are modified with time as the molecules are simultaneously followed through representative collisions and boundary interactions in the simulated region of space. In most applications, the number of simulated molecules is extremely small in comparison with the number of molecules that would be present in a real gas flow. Thus, in the simulation, each model molecule is representing the appropriate number of real molecules, The calculation is unsteady and the steady solutions are obtained as asymptotic limits of unsteady solutions. The flow field is subdivided into cells, which are taken to be small enough for the solution to be approximately constant through the cell. The time variable is advanced in discrete steps of size At, small with respect to the mean free time, i.e., the time between two subsequent collisions of a molecule. This permits a separation of the inertial motion of the molecules from the collision process: one first moves the molecules according to collision- free dynamics and subsequently the velocities are modified according to the collisions occurring in each cell. The rate of occurrence of collisions is given by (hard spheres): (19.1) r- ~ rij, i,j<i
- 63. The Boltzmann equation andfluid dynamics 59 P ft3 n-(~i-~j)dn, rij m Nm + (19.2) where N is the number of molecules in the sample and ~i is the velocity of the i-th molecule (i = 1..... N). Each time a collision occurs, the velocities of a collision pair are modified (and, as a consequence, r also varies). Let Tk be the length of the time interval between the (k - 1)-th and the k-th collision in the time interval [0, At] (t = 0 is the time of the 0-th collision by definition). The time intervals Tk are chosen in this way: after the (k - 1)-th collision, one samples a pair (i, j) on the basis of the probability distribution pij --rij/r (with a fixed velocity for each pair); then one takes Tk = 2N/[(N- 1)rij]. Then one samples a direction 6oof the post-collisional velocity ~i -- ~ j with the probability + distribution p(6o) - rij/r (fixed i, j and I~i - ~Jj l) and replaces ~Ji and ~j by ~j+ and ~j, given by 1 ~? -- ~(~i -Jr-~j -~- 6o[~i -- ~j[), (19.3) 1 (19.4) The operation is repeated until Y~'~kTk exceeds AT. Some variations of Bird's method have appeared, due to Koura [104], Belotserkovskii and Yanitskii [11], Deshpande [72]. They differ from Bird's method because of the procedure used to sample the time interval between two subsequent collisions. In particular Koura [104] introduced the so-called "null collision technique", which uses, for models different from hard spheres, the maximum of the cross-section, when estimating the possible collisions. Then for values of the cross-section less than the maximum, some collisions produce a null effect. A different method was proposed by Nanbu [133]. One does not subdivide At and works with the probability Pi that the i-th molecule collides in [0, At]. One has N Pi -- At Z rij. j-1 (19.5) One starts with rij evaluated at t -- 0 and samples a random number coin [0, 1]. According to whether co is smaller or larger than Pi, the i-th molecule collides or does not collide in [0, At]. If a collision occurs, one samples a collision partner of the i-th molecule from the probability distribution p)i) ._ rij (19.6) Y~L l rik " Then one samples a direction 6o of the post-collisional velocity ~+- ~+ with the + probability distribution p(6o) - rij/r (for fixed values of i, j and I/~+ - ~j I) and replaces /~i by/~+ given by Equation (19.3). Then the procedure is repeated for all the values of i.
- 64. 60 C. Cercignani The method of Nanbu was criticized by Koura [103], who asserted that momentum and energy are not conserved by collisions, because one does not change the velocity of the j-th molecule, when one changes the velocity of the i-th molecule. The criticism is not well- founded, however, because, as Nanbu pointed out [134], the Boltzmann equation requires the overall conservation of momentum and energy of the system at each space point, not the conservation of the same quantities at each single simulated event. What is new in Nanbu's method is precisely the circumstance that it does not try to simulate the N-body dynamics, but rather the description of the system described by the Boltzmann equation. Nanbu's method is now well understood, from both a physical and mathematical standpoint and has been rigorously proven to yield approximations to solutions of the Boltzmann equation, provided the number of test molecules is sufficiently large. The relevant theorem reads as follows [7,8]: THEOREM. If the Boltzmann equation with initial data fo has a smooth, nonnegative solution f (x,/~) ~ L l, then the solution f of Nanbu's method converges weakly in L 1 to f, in the sense that, for any test function q~(x,~j) 6 L~176 f~ f dx d~ ~ f~f dx d~ (19.7) as N --+ ~, Ax, At --+ O. Subsequently, a similar result was proved for Bird's method [169,142]. The fluctuations inevitably occurring in a DSMC calculation can cause problems if the number of molecules is not large enough to have a representative sample in each area of the flow domain. They can, however, have a physical significance and contain information on those occurring in a real gas. The comparisons between the fluctuations in DSMC and the predictions of fluctuating hydrodynamic theory have been reviewed by Garcia [76] and appear to be consistent with the fluctuations in a real gas. The computing task of a simulation method varies with the molecular model. For models other than Maxwell's it is proportional to N for Bird's method, while it is proportional to N 2 for Nanbu's method. Babovsky [7] found a procedure to reduce the computing task of Nanbu's method and make it proportional to N. His modification is based on the idea of subdividing the interval [0, 1] into N equal subintervals. If the random number colies in the j-th segment, one calculates only eij -~- rij At and there is no collision if co < (j/n) - Pij, there is a collision with the j-th molecule if co >~(j/n) - Pij. There is also a condition that eij must satisfy, i.e., eij < 1/N, but this is usually automatically verified, given the size of At. Application of Nanbu's method in the form modified by Babovsky shows that the the computing task is not only theoretically but also practically comparable to that of Bird's method [80]. This modification eventually evolved into what is called the "Finite-Pointset" method. We remark that one may take advantage of flow symmetries in physical space, but all collisions are calculated as three-dimensional events. As for the boundary conditions, Maxwell's model of diffuse reflection (see Section 11) is adequate for many problems. There are many cases. The CL model [48] has been
- 65. The Boltzmann equation andfluid dynamics 61 adapted and extended by Lord [117] for application in DSMC studies. The resulting CLL model [117,150] has been shown to provide a realistic boundary condition with incomplete accommodation [176]. More complicated models would be required to describe chemical reactions which can occur at the surface for high impact energies. 20. Some applications of the DSMC method The first significant application of DSMC method dealt with the structure of a normal shock wave [121], but only a few years later Bird was able to calculate shock profiles [15] that allowed meaningful comparisons with the experimental results then available [16] and with subsequent experiments [147,2]. This long time span is understandable: the method is very demanding of computer resources. In 1964, even with the fastest computers, the restriction on the number of molecules which could be used was such that large random fluctuations had to be expected in the results, and it was difficult to arrive at definite conclusions. Thus the number of simulated molecules and the sample sizes in the computations that could be performed in those years were extremely small in comparison with those that have been routinely employed by an increasing number of workers. The problem of the shock wave structure has continued to be an important test case. Later studies have included comparisons of measured and computed velocity distribution functions within strong shock waves in helium [140]. Early DSMC studies were also devoted to the problem of hypersonic leading edge. This arises in connection with the flow of a gas past a very sharp plate, parallel to the oncoming stream. When the Reynolds number Re = p~ V~L/#~, based on the plate length is very large, the picture, familiar from continuum mechanics, of a potential flow plus a viscous boundary layer is valid everywhere except near the leading and the trailing edge. Estimates obtained already in the late 1960s by Stewartson [161] and Messiter [124] showed that the Knudsen number at the trailing edge is of order Ma~ Re-3/4, where Ma~ is the upstream Mach number. As a consequence, kinetic theory is not needed (for large values of Re) at the trailing edge. For the leading edge, the Knudsen number is of order Ma~; hence in supersonic, or, even more, hypersonic flow (Ma~ ~>5), the flow in the region about the leading edge must be considered as a typical problem in kinetic theory. In particular, the viscous boundary layer and the outer flow are no longer distinct from each other, although [123,82,95] a shock-like structure may still be identified. It is in this connection that the name of merged-layer regime, mentioned in Section 1, arose. There are several methods based on simplified continuum models, represented by the papers of Oguchi [137], Shorenstein and Probstein [148], Chow [66,67], Rudman and Rubin [145], Cheng et al. [65], and Kot and Turcotte [102], which usefully predict surface and other gross properties in this regime. The good agreement between these approaches and experiment gave new evidence for the the importance of the Navier-Stokes equations. Nevertheless, if we go sufficiently close to the leading edge, the Navier-Stokes equations must be given up in favor of the Boltzmann equation. Huang and coworkers [90,88,89] carried out extensive computations based on discrete ordinate methods for the BGK model and were able to show the process of building the flow picture assumed in the simplified continuum models mentioned above.
- 66. 62 c. Cercignani The first DSMC is due to Vogenitz et al. [168] and exhibits a flow structure qualitatively different from the predictions of earlier studies. Their results are supported by the experiments of Metcalf et al. [125]. Validation studies of the DSMC method were also conducted at the Imperial College [83]. Hypersonic flows past blunt bodies were also the object of many simulations, most of the calculations being those made for the Shuttle Orbiter re-entry, for which useful comparisons with measured data were possible [128]. This comparison was concerned with the windward centerline heating and employed an axially symmetric equivalent body. Later comparisons [143] with Shuttle data were for the aerodynamic characteristics of the full three-dimensional shape. Another interesting problem which has been simulated by Ivanov and his coworkers is the reflection on a plane wall of an oblique shock wave generated by a wedge [92,93]. Three-dimensional DSMC calculations have also been made for the flow past a delta wing [29]. The results compare well with wind-tunnel measurements [116] of the flow field under the same conditions. Other important problems are related to separated flows, especially wake flows and flows involving viscous boundary layer separation and reattachment. The first calculations referred to the two-dimensional flow over a sharp flat plate followed by an angled ramp [129]. The results were in a reasonably good agreement with wind tunnel studies, which is not truly two-dimensional because of inevitable sidewall effects. Similar experiments were therefore performed [63] for the corresponding axially symmetric flow, less subject to the aforementioned non-uniformity. The DSMC calculations for these cases [130] show excellent agreement with experimental results. In particular, separation and reattachment of a viscous boundary layer in the laminar regime are correctly predicted. The most remarkable wake flow simulation was for a 70~ spherically blunted cone model that had been tested in several wind tunnels [1,115]. The results of the calculations [131] of the lee side flow that contains the vortex are in good agreement with the experiments and with Computational Fluid Dynamics (CFD) studies of the flow based on the Navier-Stokes equations. In the case of polyatomic gases one has several cross-sections, such as elastic, rotational, vibrational, and also reactive, if chemical reactions occur. Koura [105] has extended his null collision technique [104] to these cases and improved it later [106]. He applied this method to simulate the hypersonic rarefied nitrogen flow past a circular cylinder [106], with particular attention to the simulation of the vibrational relaxation of the gas; he also investigated the effect of changing the number of molecules in each (adaptive) cell and the truncation in the molecular levels. The Direct Simulation Monte Carlo method is not only a practical tool for engineers, but also a good method for probing into uncovered areas of the theory of the Boltzmann equation, such as stability of the solutions of this equation and the possible transition to turbulence [156,60,77,78,157,159,158,21,144,160,20]. We finally remark that the Direct Simulation Monte Carlo method has been used even to uncover the analytical nature of a singularity in a limiting solution of the Boltzmann equation, the structure of an infinitely strong shock wave. The latter arises when the temperature upstream of the shock is taken to be zero; then the solution of the Boltzmann equation is the sum of a delta function term and a more regular distribution. The latter was
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- 74. CHAPTER 2 A Review of Mathematical Topics in Collisional Kinetic Theory Crdric Villani UMPA, ENS Lyon, 46 all#e d'Italie, F-69364 Lyon Cedex 07, France E-mail: cvillani@umpa.ens-lyon.fr Contents Introduction ..................................................... 73 2A. General Presentation ............................................. 75 1. Models for collisions in kinetic theory .................................. 77 2. Mathematical problems in collisional kinetic theory .......................... 95 3. Taxonomy ................................................. 118 4. Basic surgery tools for the Boltzmann operator ............................. 124 5. Mathematical theories for the Cauchy problem ............................. 130 2B. Cauchy Problem ............................................... 141 1. Use of velocity-averaging lemmas .................................... 143 2. Moment estimates ............................................. 147 3. The Grad's cut-off toolbox ........................................ 153 4. The singularity-hunter's toolbox ..................................... 165 5. The Landau approximation ........................................ 180 6. Lower bounds ............................................... 185 2C. H Theorem and Trend to Equilibrium .................................... 189 1. A gallery of entropy-dissipating kinetic models ............................. 191 2. Nonconstructive methods ......................................... 200 3. Entropy dissipation methods ....................................... 203 4. Entropy dissipation functionals of Boltzmann and Landau ....................... 208 5. Trend to equilibrium, spatially homogeneous Boltzmann and Landau ................. 224 6. Gradient flows ............................................... 228 7. Trend to equilibrium, spatially inhomogeneous systems ........................ 235 2D. Maxwell Collisions .............................................. 245 1. Wild sums ................................................. 248 2. Contracting probability metrics ..................................... 249 3. Information theory ............................................ 254 4. Conclusions ................................................ 258 HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME I Edited by S.J. Friedlander and D. Serre 9 2002 Elsevier Science B.V. All rights reserved 71
- 75. 72 C. Villani 2E. Open Problems and New Trends ....................................... 263 1. Open problems in classical collisional kinetic theory .......................... 265 2. Granular media .............................................. 272 3. Quantum kinetic theory .......................................... 279 Bibliographical notes ................................................ 286 Acknowledgements ................................................ 287 References ..................................................... 288
- 76. A review of mathematical topics in collisional kinetic theory 73 Introduction The goal of this review paper is to provide the reader with a concise introduction to the mathematical theory of collision processes in (dilute) gases and plasmas, viewed as a branch of kinetic theory. The study of collisional kinetic equations is only part of the huge field of nonequilibrium statistical physics. Among other things, it is famous for historical reasons, since it is in this setting that Boltzmann proved his celebrated theorem about entropy. As of this date, the mathematical theory of collisional kinetic equations cannot be considered to be in a mature state, but it has undergone spectacular progress in the last decades, and still more is to be expected. I have made the following choices for presentation: (1) The emphasis is definitely on the mathematics rather than on the physics, the modelling or the numerical simulation. About these topics the survey by Carlo Cercignani will say much more. On the other hand, I shall always be concerned with the physical relevance of mathematical results. (2) Most of the presentation is limited to a small number of widely known, mathemat- ically famous models which can be considered as archetypes- mainly, variants of the Boltzmann equation. This is not only for the sake of mathematics: also in modelling do these equations play a major role. (3) Two important interface fields are hardly discussed: one is the transition from particle systems to kinetic equations, and the other one is the transition from kinetic equations to hydrodynamics. For both problematics I shall only give basic considerations and adequate references. (4) Not all mathematical theories of kinetic equations (there are many of them!) are "equally" represented: for instance, fully nonlinear theories occupy much more space than perturbative approaches, and the Boltzmann equation without cut-off is discussed in about the same detail than the Boltzmann equation with cut-off (although the literature devoted to the latter case is considerably more extended). This partly reflects the respective vivacity of the various branches, but also, unavoidably, personal tastes and areas of competence. I apologize for this! (5) I have sought to give more importance to mathematical methods and ideas, than to results. This is why I have chosen a "transversal" presentation: for each problem, corresponding tools and ideas are first explained, then the various results obtained by their use are carefully described in their respective framework. As a typical example, and unlike most textbooks, this review does not treat spatially homogeneous and spatially inhomogeneous theories separately, but insists on tools which apply to both frameworks. (6) At first I have tried to give extensive lists of references, but soon realized that it was too ambitious .... The plan of the survey is as follows. First, a presentation chapter discusses models for collisional kinetic theory and introduces the reader to the various mathematical problems which arise in their study. A central position is given to the Boltzmann equation and its variants. Chapter 2B bears on the Cauchy problem for the Boltzmann equation and variants. The main questions here are propagation of regularity and singularities, regularization effects,
- 77. 74 C. Villani decay and strict positivity of solutions. The influence of the Boltzmann collision kernel (satisfying Grad's angular cut-off or not) is discussed with care. Chapter 2C considers the trend to equilibrium, insisting on constructive approaches. Boltzmann's H theorem and entropy dissipation methods have a central role here. The shorter, but important Chapter 2D studies in detail the case of so-called Maxwell collision kernels, and several links between the theory of the Boltzmann equation and information theory. The ideas in this chapter crucially lie behind some of the most notable results in Chapter 2C, even though, strictly speaking, these two chapters are to a large extent independent. Finally, Chapter 2E discusses selected open problems and promising new trends in the field. Apart from the numerous references quoted in the text, the reader may find useful the short bibliographical notes which are included before the bibliography, to help orientate through the huge literature on the subject. Let me add one final word about conventions: it is quite customary in kinetic theory Oust as in the field of hyperbolic systems of conservation laws) to use the vocable "entropy" for Boltzmann's H functional; however the latter should rather be considered as the negative of an entropy, or as a "quantity of information". In the present review I have followed the custom of calling H an entropy, however I now regret this choice and recommend to call it an information (or just the H functional); accordingly the "entropy dissipation functional" should rather be called "entropy production functional" or "dissipation of information" (which is both closer to physical intuition and maybe more appealing).
- 78. CHAPTER 2A General Presentation Contents 1. Models for collisions in kinetic theory .................................... 77 1.1. Distribution function ........................................... 77 1.2. Transport operator ............................................ 78 1.3. Boltzmann's collision operator ..................................... 79 1.4. Collision kernels ............................................. 82 1.5. Boundary conditions ........................................... 84 1.6. Variants of the Boltzmann equation ................................... 86 1.7. Collisions in plasma physics ....................................... 89 1.8. Physical validity of the Boltzmann equation .............................. 94 2. Mathematical problems in collisional kinetic theory ............................ 95 2.1. Mathematical validity of the Boltzmann equation ........................... 95 2.2. The Cauchy problem ........................................... 100 2.3. Maxwell's weak formulation, and conservation laws ......................... 101 2.4. Boltzmann's H theorem and irreversibility ............................... 1134 2.5. Long-time behavior ........................................... 109 2.6. Hydrodynamic limits ........................................... 111 2.7. The Landau approximation ....................................... 114 2.8. Numerical simulations .......................................... 114 2.9. Miscellaneous .............................................. 115 3. Taxonomy ................................................... 118 3.1. Kinetic and angular collision kernel ................................... 118 3.2. The kinetic collision kernel ....................................... 119 3.3. The angular collision kernel ....................................... 120 3.4. The cross-section for momentum transfer ............................... 120 3.5. The asymptotics of grazing collisions .................................. 121 3.6. What do we care about collision kernels? ................................ 123 4. Basic surgery tools for the Boltzmann operator ............................... 124 4.1. Symmetrization of the collision kernel ................................. 124 4.2. Symmetric and asymmetric point of view ............................... 125 4.3. Differentiation of the collision operator ................................. 125 4.4. Joint convexity ef the entropy dissipation ................................ 126 4.5. Pre-postcollisional change of variables ................................. 126 4.6. Alternative representations ........................................ 127 4.7. Monotonicity ............................................... 128 4.8. Bobylev's identities ........................................... 128 4.9. Application of Fourier transform to spectral schemes ......................... 129 5. Mathematical theories for the Cauchy problem ............................... 130 5.1. What minimal functional space? .................................... 130 5.2. The spatially homogeneous theory ................................... 133 5.3. Maxwellian molecules .......................................... 134 5.4. Perturbation theory ............................................ 134 75
- 79. 76 C. Villani 5.5. Theories in the small ........................................... 136 5.6. The theory of renormalized solutions .................................. 137 5.7. Monodimensional problems ....................................... 139
- 80. A review of mathematical topics in collisional kinetic theory 77 The goal of this chapter is to introduce, and make a preliminary discussion of, the mathematical models and problems which will be studied in more detail thereafter. The first section addresses only physical issues, starting from scratch. We begin with an introduction to kinetic theory, then to basic models for collisions. Then in Section 2, we start describing the mathematical problems which arise in collisional kinetic theory, restricting the discussion to the ones that seem to us most fundamental. Particular emphasis is laid on the Boltzmann equation. Each paragraph contains at least one major problem which has not been solved satisfactorily. Next, a specific section is devoted to the classification of collision kernels in the Boltzmann collision operator. The variety of collision kernels reflects the variety of possible interactions. Collision kernels have a lot of influence on qualitative properties of the Boltzmann equation, as we explain. In the last two sections, we first present some basic general tools and considerations about the Boltzmann operator, then give an overview of existing mathematical theories for collisional kinetic theory. 1. Models for collisions in kinetic theory 1.1. Distributionfunction The object of kinetic theory is the modelling of a gas (or plasma, or any system made up of a large number of particles) by a distribution function in the particle phase space. This phase space includes macroscopic variables, i.e., the position in physical space, but also microscopic variables, which describe the "state" of the particles. In the present survey, we shall restrict ourselves, most of the time, to systems made of a single species of particles (no mixtures), and which obey the laws of classical mechanics (non-relativistic, non- quantum). Thus the microscopic variables will be nothing but the velocity components. Extra microscopic variables should be added if one would want to take into account non- translational degrees of freedom of the particles: internal energy, spin variables, etc. Assume that the gas is contained in a (bounded or unbounded) domain X C I~u (N = 3 in applications) and observed on a time interval [0, T], or [0, +ec). Then, under the above simplifying assumptions, the corresponding kinetic model is a nonnegative function f(t, x, v), defined on [0, T] x X x I~N. Here the space Ii~u -- ~N is the space of possible velocities, and should be thought of as the tangent space to X. For any fixed time t, the quantity f(t, x, v)dx dv stands for the density of particles in the volume element dx dv centered at (x, v). Therefore, the minimal assumption that one can make on f is that for all t/> 0, f(t, ", ") E L~oc(X; L 1(It{N)); or at least that f (t,., .) is a bounded measure on K x R N, for any compact set K C X. This assumption means that a bounded domain in physical space contains only a finite amount of matter.
- 81. 78 C. Villani Underlying kinetic theory is the modelling assumption that the gas is made of so many particles that it can be treated as a continuum. In fact there are two slightly different ways to consider f: it can be seen as an approximation of the true density of the gas in phase space (on a scale which is much larger than the typical distance between particles), or it can reflect our lack of knowledge of the true positions of particles. Which interpretation is made has no consequence in practice. 1 The kinetic approach goes back as far as Bernoulli and Clausius; in fact it was introduced long before experimental proof of the existence of atoms. The first true bases for kinetic theory were laid down by Maxwell [335,337,336]. One of the main ideas in the model is that all measurable macroscopic quantities ("observables") can be expressed in terms of microscopic averages, in our case integrals of the form f f(t, x, v)qg(v)dr. In particular (in adimensional form), at a given point x and a given time t, one can define the local density p, the local macroscopic velocity u, and the local temperature T, by P = fRN f (t, x, v) dv, pu -- fRu f (t, x, v)v dv, Plu] 2 + NpT -- foNf (t, x, l))1l)12dr. (1) For much more on the subject, we refer to the standard treatises of Chapman and Cowling [154], Landau and Lipschitz [304], Grad [250], Kogan [289], Uhlenbeck and Ford [433], Truesdell and Muncaster [430], Cercignani and co-authors [141,148,149]. 1.2. Transport operator Let us continue to stick to a classical description, and neglect for the moment the interaction between particles. Then, according to Newton's principle, each particle travels at constant velocity, along a straight line, and the density is constant along characteristic lines dx/dt - v, dv/dt = 0. Thus it is easy to compute f at time t in terms of f at time 0: f (t, x, v) = f (O, x- vt, v). 1For instance, assume that the microscopic description of the gas is givenby a cloud of n points x1..... Xn in I~N, with velocities Vl..... Vn in I~N. A microscopic configuration is an element (Xl,Vl..... Xn,Vn) of (R N • I~N)n. The "density" of the gas in this configuration is the empirical measure (l/n)y~nt=l ~(xi,vi) it is a probability measure on RN x RN. In the first interpretation, f (x, v)dx dv is an approximation of the empirical measure.In the secondone,there is a symmetric probability density fn on the space (RN x RN)n of all microscopic configurations,and f is an approximation of the one-particle marginal plfn(xl, Vl) -- f fn(xl, Vl ..... Xn, vn)dx2dv2"'" dxndvn. Thus the first interpretation is purely deterministic, while the second one is probabilistic. It is the second interpretation which was implicitlyused by Boltzmann, and which is needed by Landford's validation theorem, see Section2.1.
- 82. A review of mathematical topics in collisional kinetic theory 79 In other words, f is a weak solution to the equation of free transport, of Ot -- + v. Vx f = 0. (2) The operator v. Vx is the (classical) transport operator. Its mathematical properties are much subtler than it would seem at first sight; we shall discuss this later. Complemented with suitable boundary conditions, Equation (2) is the right equation for describing a gas of noninteracting particles. Many variants are possible; for instance, v should be replaced by V/v/1 + Ivl2 in the relativistic case. Of course, when there is a macroscopic force F (x) acting on particles, then the equation has to be corrected accordingly, since the trajectories of particles are not straight lines any longer. The relevant equation would read 0f + v. Vxf + F(x). gvf=O (3) Ot and is sometimes called the linear Vlasov equation. 1.3. Boltzmann's collision operator We now want to take into account interactions between particles. We shall make several postulates. (1) We assume that particles interact via binary collisions: this is a vague term describing the process in which two particles happen to come very close to each other, so that their respective trajectories are strongly deviated in a very short time. Underlying this hypothesis is an implicit assumption that the gas is dilute enough that the effect of interactions involving more than two particles can be neglected. Typically, if we deal with a three- dimensional gas of n hard spheres of radius r, this would mean nr 3 << 1, nr 2 ~_ 1. (2) Moreover, we assume that these collisions are localized both in space and time, i.e., they are brief events which occur at a given position x and a given time t. This means that the typical duration of a collision is very small compared to the typical time scale of the description, and also quantities such as the impact parameter (see below) are negligible in front of the typical space scale (say, a space scale on which variations due to the transport operator are of order 1). (3) Next, we further assume these collisions to be elastic: momentum and kinetic energy are preserved in a collision process. Let v', v.' stand for the velocities before collision, and v, v. stand for the velocities after collision: thus ! Vf 4- V, -- V 4- V,, ,12 2 12 [v'l2 § Iv, -Ivl + Iv, . (4)
- 83. 80 C. Villani V V~- Fig. 1. A binaryelasticcollision. Since this is a system of N + 1 scalar equations for 2N scalar unknowns, it is natural to expect that its solutions can be defined in terms of N - 1 parameters. Here is a convenient representation of all these solutions, which we shall sometimes call the a-representation: v'-v+v* Iv-v,I - -T- + --T--~, , v+v, Iv-v,I v, = 2 ----T --a" (5) Here the parameter a ~ SN- 1 varies in the N - 1 unit sphere. Figure 1 pictures a collision in the velocity phase space. The deviation angle 0 is the angle between pre- and post- collisional velocities. Very often, particles will be assumed to interact via a given interaction potential qb(r) t should be computed as the result of a (r -- distance between particles); then vt and v, classical scattering problem, knowing v, v, and the impact parameter between the two colliding particles. We recall that the impact parameter is what would be the distance of closest approach if the two particles did not interact. (4) We also assume collisions to be microreversible. This word can be understood in a purely deterministic way: microscopic dynamics are time-reversible; or in a probabilistic way: the probability that velocities (v~, v~,) are changed into (v, v,) in a collision process, l is the same that the probability that (v, v,) are changed into (v~, v,). (5) And finally, we make the Boltzmann chaos assumption: the velocities of two particles which are about to collide are uncorrelated. Roughly speaking, this means that if we randomly pick up two particles at position x, which have not collided yet, then the joint distribution of their velocities will be given by a tensor product (in velocity space) of f with itself. Note that this assumption implies an asymmetry between past and future: indeed, in general if the pre-collisional velocities are uncorrelated, then post-collisional velocities have to be correlated2! 2See,for instance,the discussionin Section2.4.
- 84. A review of mathematical topics in collisional kinetic theory 81 Under these five assumptions, in 1872 Boltzmann (cf. [93]) was able to derive a quadratic collision operator which accurately models the effect of interactions on the distribution function f" of Ot coll (t, x, v) = Q(f, f) (t, x, v) (6) = f~N dr, fSN-1 do" B(v- v,, a)(f~f~, - ff,). (7) Here we have used standard abbreviations: f' = f(t, x, v'), f, = f(t, x, v,), f,~ = f (t, x, v~,). Moreover, the nonnegative function B(z, cr), called the Boltzmann collision kernel, depends only on Izl and on the scalar product (z/lz[, or). Heuristically, it can be seen as a probability measure on all the possible choices of the parameter cr 6 SN-l, as a function of the relative velocity z = v - v,. But truly speaking, this interpretation is in general false, because B is not integrable .... The Boltzmann collision kernel is related to the cross-section F, by the identity B(z, a) -- Izlr (z, a). By abuse of language, B is often called the cross-section. Let us explain (7) a little bit. This operator can formally be split, in a self-evident way, into a gain and a loss term, Q(f, f)= Q+(f, f)- Q-(f, f). The loss term "counts" all collisions in which a given particle of velocity v will encounter another particle, of velocity v,. As a result of such a collision, this particle will in general change its velocity, and this will make less particles with velocity v. On the other hand, t, then the vt particle may each time particles collide with respective velocities v~ and v, acquire v as new velocity after the collision, and this will make more particles with velocity v: this is the meaning of the gain term. It is easy to trace back our modelling assumptions: (1) the quadratic nature of this operator is due to the fact that only binary collisions are taken into account; (2) the fact that the variables t, x appear only as parameters reflects the assumption that collisions are localized in space and time; (3) the assumption of elastic collisions results in the formulas giving v~and v~,; (4) the microreversibility implies the particular structure of the collision kernel B; (5) finally, the appearance of the tensor products 3 f~f/, and ff, is a consequence of the chaos assumption. On the whole, the Boltzmann equation reads of Ot + v. Vxf = Q(f, f), t ~ 0, X E ]1~N, U E ~N., (8) or, when a macroscopic force F (x) is also present, of Ot -- + v . Vx f + F(x) . Vvf = Q(f, f), t >~0, x E]I~N vE]~ N , 3In the sense that ff, (f | f)(v, v,), , t I = f f, = (f | f)(v', v,). (9)
- 85. 82 C.Villani The deepest physical and mathematical properties of the Boltzmann equation are linked to the subtle interaction between the linear transport operator and the nonlinear collision operator. We note that this equation was written in weak formulation by Maxwell as early as 1866: as recalled in Section 2.3 below, Maxwell [335,337] wrote down the equation satisfied by the observables f f(t,x, v)qg(v)dv (see in particular [335, Equation (3)]). However Boltzmann did a considerable job on the interpretation and consequences of this equation, and also made them widely known in his famous treatise [93], which was to have a lot of influence on theoretical physics during several decades. Let us make a few comments about the modelling assumptions. They may look rather crude, however they can in large part be completely justified, at least in certain particular cases. By far the deepest assumption is Boltzmann's chaos hypothesis ("Stosszahlansatz") which is intimately linked to the questions of irreversibility of macroscopic dynamics and the arrow of time. For a discussion of these subtle topics, the reader may consult the review paper by Lebowitz [293], or the enlightening treatise by Kac [284], as well as the textbooks [149,410] for the most technical aspects. We shall say just a few words about the subject in Sections 2.1 and 2.4. 1.4. Collisionkernels Since the collision kernel (or equivalently the cross-section) only depends on iv - v, I and V--V, on (Iv v,l' a), i.e., the cosine of the deviation angle, throughout the whole text we shall abuse notations by writing 8(v - v,, ~)= 8(Iv- v,I, cos0), V--V, > cos0= Iv-v,l'a" (lO) Maxwell [335] has shown how the collision kernel should be computed in terms of the interaction potential r In short, here are his formulas (in three dimensions of space), as they can be found for instance in Cercignani [141], for a repulsive potential. For given impact parameter p ~>0 and relative velocity z ~ R 3, let the deviation angle 0 be fo ~-c~ ds/ s2 0 (p, z) = Jr - 2p V/1 - 7 - 4/-z~P2 ~(s) p/so - Jr - 2 ! dO du where so is the positive root of p2 ~bl(s0) - s--~-4 z[ 2 =0. Then the collision kernel B is implicitly defined by p dp B(Izl, cos0) -- sin0 dO ~--Izl. (11)
- 86. A review of mathematical topics in collisional kinetic theory 83 It can be made explicit in two cases: 9 hard spheres, i.e., particles bounce on each other like billiard balls: in this case B(lv - v.[, cos0) is just proportional to Iv - v.[ (the cross-section is constant); 9 Coulomb interaction, ~b(r) = 1/r (in adimensional variables and in three dimensions of space): then B is given by the famous Rutherford formula, 1 B(lv - v,I, cosO) - Iv - v,I 3 sin4(O/2)" (12) A dimensional factor of (e2/47r eom) should multiply this kernel (e = charge of the particle, e0 = permittivity of the vacuum, m = mass of the particle). Unfortunately, Coulomb interactions cannot be modelled by a Boltzmann collision operator; we shall come back to this soon. In the important 4 model case of inverse-power law potentials, ~b(r)--rS_l, s>2, then the collision kernel cannot be computed explicitly, but one can show that B(Iv - v,I, cos0) -- b(cos0)lv - v,I y, s - (2N - 1) • = . (13) s--1 In particular, in three dimensions of space, g = (s - 5) / (s - 1). As for the function b, it is only implicitly defined, locally smooth, and has a nonintegrable singularity for 0 --+ 0: sin N-2 0 b(cos0) ~ KO -1-v v= (N = 3). (14) s--1 Here we have put the factor sinN-2 0 because it is (up to a constant depending only on the dimension) the Jacobian determinant of spherical coordinates on the sphere S u-1 . The nonintegrable singularity in the "angular collision kernel" b is an effect of the huge amount of grazing collisions, i.e., collisions with a very large impact parameter, so that colliding particles are hardly deviated. This is not a consequence of the assumption of inverse-power forces; in fact a nonintegrable singularity appears as soon as the forces are of infinite range, no matter how fast they decay at infinity. To see this, note that, according to (11), f0 f0 f0 Pmax d__ppdO - Izl p dp - ~ . (15) B (Izl, cos 0) sin 0 dO - Izl P dO 2 By the way, it seems strange to allow infinite-range forces, while we assumed interactions to be localized. This problem has never been discussed very clearly, but in 4Inverse power laws are moderately realistic, but very important in physics and in modelling, because they are simple, often lead to semi-explicit results, and constitute a one-parameter family which can model very different phenomena. Van der Waals interactions typically correspond to s = 7, ion-neutral interactions to s = 5, Manev interactions [88,279] to s = 3, Coulomb interactions to s -- 2.
- 87. 84 C. Villani principle there is no contradiction in assuming the range of the interaction to be infinite at a microscopic scale, and negligible at a macroscopic scale. The fact that the linear Boltzmann equation can be rigorously derived from some particle dynamics with infinite range [179] also supports this point of view. As one sees from formula (13), there is a particular case in which the collision kernel does not depend on the relative velocity, but only on the deviation angle: particles interacting via a inverse (2N - 1)-power force (1/r 5 in three dimensions). Such particles are called Maxwellian molecules. They should be considered as a theoretical model, even if the interaction between a charged ion and a neutral particle in a plasma may be modelled by such a law (see, for instance, [164, Theorem 1, p. 149]). However, Maxwell and Boltzmann used this model a lot,5 because they had noticed that it could lead to many explicit calculations which, so did they believe, were in agreement with physical observations. Also they believed that the choice of molecular interaction was not so important, and that Maxwellian molecules would behave pretty much the same as hard spheres. 6 Since the time of Maxwell and Boltzmann, the need for results or computations has led generations of mathematicians and physicists to work with more or less artificial variants of the collision kernels given by physics. Such a procedure can also be justified by the fact that for many interesting interactions, the collision kernel is not explicit at all: for instance, in the case of the Debye potential, r = e-r/r. Here are two categories of artificial collision kernels: - when one tames the singularity for grazing collisions and replaces the collision kernel by a locally integrable one, one speaks of cut-off collision kernel; - collision kernels of the form Iv- v,I • (Y > 0) are called variable hard spheres collision kernels. It is a common belief among physicists that the properties of the Boltzmann equation are quite a bit sensitive to the dependence of B upon the relative velocity, but very little to its dependence upon the deviation angle. True as it may be for the behavior of macroscopic quantities, this creed is definitely wrong at the microscopic level, as we shall see. In all the sequel, we shall consider general collision kernels B(Iv - v,I, cos0), in arbitrary dimension N, and make various assumptions on the form of B without always caring if it corresponds to a true interaction between particles (i.e., if there is a r whose associated collision kernel is B). Our goal, in a lot of situations, will be to understand how the collision kernel affects the properties of the Boltzmann equation. However, we shall always keep in mind the collision kernels given by physics, in dimension three, to judge how satisfactory a mathematical result is. 1.5. Boundary conditions Of course the Boltzmann equation has to be supplemented with boundary conditions which model the interaction between the particles and the frontiers of our domain X C R N (wall, etc.) 5See Boltzmann [93, Chapter 3]. 6Further recall that at the time, the "atomic hypothesis" was considered by many to be a superfluous complication.
- 88. A reviewof mathematical topics in collisional kinetic theory 85 The most natural boundary condition is the specular reflection: f (x, Rxv) -- f (x, v), Rxv = v - 2(v. n(x))n(x), x e OX, (16) where n(x) stands for the outward unit normal vector at x. In the context of optics, this condition would be called the Snell-Descartes law: particles bounce back on the wall with an postcollision angle equal to the precollision angle. However, as soon as one is interested in realistic modelling for practical problems, Equation (16) is too rough .... In fact, a good boundary condition would have to take into account the fine details of the gas-surface interaction, and this is in general a very delicate problem] There are a number of models, cooked up from modelling assumptions or phenomenological a priori constraints. As good source for these topics, the reader may consult the books by Cercignani [141,148] and the references therein. In particular, the author explains the relevant conditions that a scattering kernel K has to satisfy for the boundary condition f (x, rout) -- f K (Vin, rout) f (x, Vin)dvin to be physically plausible. Here we only list a few common examples. One is the bounce-back condition, f (x, - v) = f (x, v), x ~ OX. (17) This condition simply means that particles arriving with a certain velocity on the wall will bounce back with an opposite velocity. Of course it is not very realistic, however in some situations (see, for instance, [148, p. 41]) it leads to more relevant conclusions than the specular reflection, because it allows for some transfer of tangential momentum during collisions. Another common boundary condition is the Maxwellian diffusion, f (x, v) = p-(x)Mw(v), v . n(x) > O, (18) where p_(x) = fv.n<O f(x, v)dv and Mw is a particular Gaussian distribution depending only on the wall, Mw(v) = Ivl2 e 2Tw (2g)-~-2 T[ +l" 7Here we assumethat the finedetailsof the surfaceof the wall are invisible at the scaleof the spatial variable, so thatthe wallis modelled as a smooth surface,but we wishto takethese detailsintoaccountto predictvelocities aftercollisionwiththe wall. Anotherpossibilityis to assumethat the roughnessof the wallresultsin irregularities which can be seen at spatial resolution. Thenit is natural in many occasions to assume0X to be very irregular. For some mathematicalworksaboutthis alternativeapproach, see [33,272,273].
- 89. 86 C Villani In this model, particles are absorbed by the wall and then re-emitted according to the distribution Mw, corresponding to a thermodynamical equilibrium between particles and the wall. Finally, one can combine the above models. Already Maxwell had understood that a convex combination of (16) and (18) would certainly be more realistic than just one of these two equations. No need to say, since the work of Maxwell, much more complicated models have appeared, for instance the Cercignani-Lampis (CL) model, see [148]. In mathematical discussions, we shall not consider the problem of boundary conditions except for the most simple case, which is specular reflection. In fact, most of the time we shall simply avoid this problem by assuming the position space to be the whole of •N, or the torus qrN. Of course the torus is a mathematical simplification, but it is also used by physicists and by numerical analysts who want to avoid taking boundary conditions into account .... 1.6. Variants of the Boltzmann equation There are many variants. Let us only mention 9 relativistic models, see [50,158,199,200,233-235,14]; 9 quantum models, see [196,328,209] and the references therein. They will be discussed in Section 3 of Chapter 2E. Also we should mention that models of quantum Boltzmann equation have recently gained a lot of interest in the study of semi- conductors, see in particular [65,66] and the works by Poupaud and coworkers on related models [386-388,244,360], also [354,355] (in two dimensions) and [15] (in three dimensions); 9 linear models, in particular the linear Boltzmann equation, -- -+-v. Vxf = Ot N• dr, dcr B(v - v,, a)[F(v',) f (v') - F(v,) f (v)], where F is a given probability distribution. As a general rule, such linear equations model the influence of the environment, or background, on a test-particle (think of a particle in an environment of random scatterers, like in a random pinball game). The distribution F is the distribution of the background, and is usually assumed to be stationary, which means that the environment is in statistical equilibrium. Linear Boltzmann-like models are used in all areas of physics, most notably in quantum scattering [206] and in the study of transport phenomena associated with neutrons or photons [148, pp. 165-172]. A general mathematical introduction to linear transport equations can be found in [157, Chapter 21]; 9 diffusive models, like the Fokker-Planck equation, which is often used in its linear version, of Ot + v. Vxf = Vv. (Vvf + fv), (19)
- 90. A review of mathematical topics in collisional kinetic theory 87 or in its nonlinear form, 0f + v. Vxf-- paVv. [TVvf + f(v - u)], (20) Ot where o~E [0, 1], and p, u, T are the local density, velocity and temperature defined by (1). When o~= 1, this model has the same quadratic homogeneity as the Boltzmann equation. Of course it is also possible to couple the equation only via T and not u, etc. A classical discussion on the use of the Fokker-Planck equation in physics can be found in the important review paper by Chandrasekhar [153]. The Landau equation, which is described in detail in the next section, is another diffusive variant of the Boltzmann equation; 9 energy-dissipating models, describing inelastic collisions. These models are particu- larly important in the theory of granular materials: see Section 2 of Chapter 2E; 9 model equations, like the (simplistic) BGK model, see, for instance, [141,148]. In this model one replaces the complicated Boltzmann operator by M f - f, where M f is the Maxwellian distribution s with the same local density, velocity and temperature than f. Also variants are possible, for instance multiplying this operator by the local density p .... Another very popular model equation for mathematicians is the Kac model [283]. It is a one-dimensional caricature of the Boltzmann equation which retains some of its interesting features. The unknown is a time-dependent probability measure f on 11~,and the equation reads Of 1 - - dr. d0 f'f~, - f, (21) Ot 27c where (vt, vt.) is obtained from (v, v.) by a rotation of angle 0 in the plane •2. This model preserves mass and kinetic energy, but not momentum; 9 discrete-velocity models: these are approximations of the Boltzmann equation where particles are only allowed a finite number of velocities. They are used in numerical analysis, but their mathematical study is (or once was) a popular topic. About them we shall say nothing; references can be found, for instance, in [230,141,148] and also in the survey papers [384,62]. Among a large number of works, we only mention the original contributions by Bony [94,95], Tartar [416,417], and the consistency result by Bobylev et al. [367], for the interesting number-theoretical issues that this paper has to deal with. We note [148, p. 265] that discrete-velocity models were once believed by physicists to provide miraculously efficient numerical codes for simulation of hydrodynamics. But these hopes have not been materialized .... Also some completely unrealistic discrete-velocity equations have been studied as simplified mathematical models, without caring whether they would approximate or not the true Boltzmann equation. These models sometimes have no more than two or three velocities! Some well-known examples are the Carleman equation, with two 8SeeSection2.5 below.
- 91. 88 C. Villani velocities, the Broadwell model, with four velocities in the plane, or the Cabannes equation with fourteen velocities. For instance, the Carleman equation reads of 1 ~fl -S-+ = Of-1 Of-1 = f?__ f21. Ot Ox (22) For a study of these models see [230,384,141,148,60,61,108,25] and references included. Of course, these models are so oversimplified9 that they cannot be considered seriously from the physical point of view, even if one may expect that they keep some relevant features of the Boltzmann equation. By the way, as noticed by Uchiyama [432], the Broadwell model cannot be derived from a fictitious system of deterministic four-velocity particles ("diamonds") in the plane, see [149, Appendix 4C]. Only at the price of some extra stochasticity assumption can the derivation be fixed [117]. All the abovementioned models should be considered with appropriate boundary conditions. These conditions can also be replaced by the effect of a confinement potential V(x): this means that there is a macroscopic force of the form F(x) = -VV(x) acting on the system. Finally, it is important to note that Boltzmann's model is obtained as a result of the assumption of localized interaction; in particular, it does not take into account a possible interaction of long (macroscopic) range which would result in a macroscopic mean-field force, typically F(x) = -V~(x), qb(x) = p *x ~P, where 4~is the interaction potential and p the local density. The modelling of the interaction by such a coupled force is called a Vlasov description. When should one prefer a Vlasov, or a Boltzmann description? A dimensional analysis by Bobylev and Illner [88] shows that for inverse-power forces like 1/ rs, and under natural scaling assumptions, 9 for s > 3, the Boltzmann term should prevail on the mean-field term; 9 for s < 3, the Boltzmann term should be negligible in front of the mean-field term. The separating case, s = 3, is the so-called Manev interaction [88,279]. There are subtle questions here, which are not yet fully understood, even at a formal level. Also the uniqueness of the relevant scaling is not clear. From a physicist's point of view, however, it is generally accepted that a good description is obtained by adding up the effects of a mean-field term and those of a Boltzmann collision operator, with suitable dimensional coefficients. Another way to take into account interactions on a macroscopically significant scale is to use a description ?~la Povzner. In this model (see, for instance, [389]), particles interact 9As a word of caution, we should add that even if they are so simplified, their mathematical analysis is not trivial at all, and many problems in the field still remain open.
- 92. A review of mathematical topics in collisional kinetic theory 89 through delocalized collisions, so that the corresponding Boltzmann operator is integrated with respect to the position y of the test particle, and reads fR dy [ dr, B(v - - y)[f (y, v~,)f (x, v') - f (y, v,) f (x, v)]. V,,X JR (23) Note that the kernel B now depends on x - y. On the other hand, there is no collision parameter cr any more, the outgoing velocities being uniquely determined by the positions x, y and the ingoing velocities vt, v,.t It was shown by Cercignani [140] that this type of equations could be retrieved as the limit of a large stochastic system of "soft spheres". A related model is the Enskog equation for dense gases, which has never been clearly justified. It resembles Equation (23) but the multiplicity of the integral is 2N - 1 instead of 2N, because there is no integration over the distance [x - YI-Mathematical studies of the Enskog model have been performed by Arkeryd, Arkeryd and Cercignani [24,27,28] - in particular, reference [24] provides well-posedness and regularity under extremely general assumptions (large data, arbitrary dimension) by a contraction method. See Section 2.1 in Chapter 2E for an inelastic variant which is popular in the study of granular material. The study of these models is interesting not only in itself, but also because numerical schemes always have to perform some delocalization to simulate the effect of collisions. This explains why the results in [140] are very much related to some of the mathematical justifications for some numerical schemes, as performed in [453,396]. 1.7. Collisions in plasma physics The importance and complexity of interaction processes in plasma physics justifies that we devote a special section to this topic. A plasma, generally speaking, is a gas of (partially or totally) ionized particles. However, this term encompasses a huge variety of physical situations: the density of a plasma can be extremely low or extremely high, the pressures can vary considerably, and the proportion of ionized particles can also vary over several orders of magnitude. Nonelastic collisions, recombination processes may be very important. We do not at all try to make a precise description here, and refer to classical textbooks such as Balescu [46], Delcroix and Bers [164], the very nice survey by Decoster in [160], or the numerous references that can be found therein. All of these sources put a lot of emphasis on the kinetic point of view, but [160] and [164] are also very much concerned with fluid descriptions. A point which should be made now, is that the classical collisional kinetic theory of gases a priori applies when the density is low (we shall make this a little bit more precise later on) and when nonelastic processes can be neglected. Even taking into account only elastic interactions, there are a number of processes going on in a plasma: Maxwell-type interactions between ions and neutral particles, Van der Waals forces between neutral particles, etc. However, the most important feature, both from the mathematical and the physical point of view, is the presence of Coulomb
- 93. 90 C.Villani interactions between chargedparticles. The basic model for the evolution of the density of such particles is the Vlasov-Poisson equation I of 37 + v. Vxf + F(x). Vvf --O, e 2 F =-VV, V = 4rcsor*x P, p(t, x) = f f (t, x, v) dv. (24) For simplicity, here we have written this equation for only one species of particles, and also we have not included the effect of a magnetic field, which leads to the Vlasov-Maxwell system (see, for instance, [191]). We have kept the physical parameters e -- charge of the particle and e0 = permittivity of vacuum for the sake of a short discussion about scales. Even though this is not the topic of this review paper, let us say just a few words on the Vlasov-Poisson equation. Its importance in plasma physics (including astrophysics) cannot be overestimated, and thousands of papers have been devoted to its study. We only refer to the aforementioned textbooks, together with the famous treatise by Landau and Lipschitz [304]. From the mathematical point of view, the basic questions of existence, uniqueness and (partial) regularity of the solution to the Vlasov equation have been solved at the end of the eighties, see in particular Pfaffelmoser [382], DiPerna and Lions [191], Lions and Perthame [318], Schaeffer [402]. Reviews can be found in Glassey [233] and Bouchut [96]. Stability, and (what is more interesting!) instability of several classes of equilibrium distributions to the Vlasov-Poisson equation have recently been the object of a lot of studies by Guo and Strauss [264-268]. Several important questions, however, have not been settled, like the derivation of the Vlasov-Poisson equation from particle systems (see Spohn [410] and Neunzert [356] for related topics) and the explanation of the famous and rather mysterious Landau damping effect. There is no doubt that the Vlasov-Poisson equation is the correct equation to describe a classical plasma on a short time scale. However, when one wants to consider long periods of time, it is necessary to take into account collisions between particles. For this it is natural to introduce a Boltzmann collision operator in the fight-hand side of (24). However, the Boltzmann equation for Coulomb interactions does not make sense! Indeed, the collision integral would be infinite even for very smooth (or analytic) distribution functions, l~ This is due to the very slow decay of the Coulomb potential, and the resulting very strong angular singularity of the collision kernel given by Rutherford's formula (12). A standard remedy to this problem is to assume that there is a screening (due to the presence of two species of particles, for instance), so that the effective interaction potential between charged particles is not the Coulomb interaction, but the so-called Debye potential e-r/)~D ~b(r) -- ~ . (25) 4rceor 10More precisely, the natural definition of the collision operator would lead to the following nonsense: whateverf, Q(f, f) is an elementof {-oo, 0, +oo}, see [450,AnnexI, AppendixA].
- 94. A review of mathematical topics in collisional kinetic theory 91 Here Xo is the Debye length, i.e., a typical screening distance. In the classical theory of plasmas, ~eokT XD-- -~, where k is Boltzmann's constant, T is the temperature of the plasma (rigorously speaking, it should depend on x...) and p its mean density (same remark). The resulting collision kernel is no longer explicit, but at least makes sense, because the very strong angular singularity in Rutherford's formula (12) is tamed. The replacement of Coulomb by Debye potential can be justified by half-rigorous, half- heuristic arguments (see the references already mentioned). However, in most of the cases of interest, the Debye length is very largewith respect to the characteristic length r0 for collisions, called the Landaulength: r0 -- e2 4zceokT More precisely, in so-called classical plasmas (those for which the classical kinetic description applies), one has ro ~ p-l~3 ~ )~D. This means first that the Landau distance is very small with respect to other scales (so that collisions can be considered as localized), and secondly that the plasma is so dilute that the typical distance between particles is very small with respect to the screening distance, which is usually considered as the relevant space scale. By a formal procedure, Landau [291] showed that, as the ratio A -- 2XD/r0 --+ ec, the Boltzmann collision operator for Debye potential behaves as log A 27rA QL (f, f), where QL is the so-called Landaucollisionoperator: QL(f,f)--Vv.(fRsdv,a(v-v,)[f,(Vf)- f(Vf),]). (26) Here a(z) is a symmetric (degenerate) nonnegative matrix, proportional to the orthogonal projection onto z-L: L[ ziz,] aij(Z)---~l (~ij iz12 , (27) and L is a dimensional constant.
- 95. 92 C. Villani The resulting equation is the Landau equation. In adimensional units, it could be written as of Ot + v. Vxf + F(x). Vvf = QL(f, f). (28) Many mathematical and physical studies also consider the simplified case when only collisions are present, and the effect of the mean-field term is not present. However, the collision term, normally, should be considered only as a long-time correction to the mean- field term. We now present a variant of the Landau equation, which appears when one replaces the function L/Izl in (27) by Izl2, so that aij (Z) = Izl2~ij - zi zj. (29) This approximation is called "Maxwellian". It is not realistic from the physical point of view, but has become popular because it leads to simpler mathematical properties and useful tests for numerical simulations. Under assumption (29), a number of algebraic simplifications arise in the Landau operator (26); since they are independent on the dimension, we present them in arbitrary dimension N ~>2. Without loss of generality, choose an orthonormal basis of R N in such a way that f• fdv-1, N fR fvdv=O, N flvl 2dr = N N (unit mass, zero mean velocity, unit temperature), and assume moreover that fR f ViVj dv = Ti6ij N (the 7~'s are the directional temperatures; of course Ei T/ -- N). Then the Landau operator with matrix (29) can be rewritten as Z(N - Ti)Oiif + (N - 1)V. (f v) + As fi i (30) Here As stands for the Laplace-Beltrami operator, Asf - ~([v126ij -- vivj)Oijf - (N- 1)v. Vvf, ij i.e., a diffusion on centered spheres in velocity space. Thus the Landau operator looks like a nonlinear Fokker-Planck-type operator, with some additional isotropisation effect due to the presence of the Laplace-Beltrami operator. The diffusion is enhanced in directions
- 96. A reviewof mathematical topics in collisional kinetic theory 93 where the temperature is low, and slowed down in directions where the temperature is high: this is normal, because in the end the temperature along all directions should be the same. Formulas like (30) show that in the isotropic case, and under assumption (29), the nonlinear Landau equation reduces (in a well-chosen orthonormal basis) to the linear Fokker-Planck equation! By this remark [447] one can construct many explicit solutions (which generalize the ones in [296]). These considerations explain why the Maxwellian variant of the Landau equation has become a popular test case in numerical analysis [106]. Let us now review other variants of the Landau equation. First of all, there are relativistic and quantum versions of it [304]. For a mathematically-oriented presentation, see, for instance, [298]. There are also other, more sophisticated models for collisions in plasmas: see, for instance, [164, Section 13.6] for a synthetic presentation. The most famous of these models is the so-called Balescu-Lenard collision operator, whose complexity is just frightening for a mathematician. Its expression was established by Bogoljubov [90] via a so-called BBGKY-type hierarchy, and later put by Lenard under the form that we give below. On the other hand, Balescu [46,47] derived it as part of his general perturbative theory of approximation of the Liouville equation for many particles. Just as the Landau operator, the Balescu-Lenard operator is in the form Z.. ~ 3 dv, aij(v, v,) f*~vj -- f ~v,j ' tj (31) but now the matrix aij depends on f in a strongly nonlinear way: aij(v, v,) -- ~~R3,1kl~<Km.~ S[k. (V -- V,)] kikj 1 Ikl 4 lel 2 ~ dk, (32) where e is the "longitudinal permittivity" of the plasma, e=l f 1 k. (V f), dr,. Ikl~ k. (v - v.) - i0 Here 6 is the Dirac measure at the origin, and 1 = lim 1 _79(1~+i:r8 x - iO s~o+ x - it x] is a complex-valued distribution on the real line (79 stands for the Cauchy principal part). Moreover, Kmax is a troncature parameter (whose value is not very clearly determined) which corresponds to values of the deviation angles beyond which collisions cannot be considered grazing. This is not a Debye cut! Contrary to Landau, Balescu and Lenard derived the operator QL (26) as an approxima- tion of (31). To see the link between these two operators, let us set k = #w,/z ~>0, co E $2; then one can rewrite (32) as fS (f0KmaxdjJ~) dcoS[co. (v - v.)] COicoj 12 9 2 /ZI8 (33)
- 97. 94 C. Villani But 2 dco6(co, z)coicoj - -~ 6ij zizj ) izl2 9 So one can replace the operator (31) by the Landau collision operator if one admits that the integral in d/z in (33) depends very little on v, v., co. Under this assumption, it is natural to replace f by a Maxwellian, ll and one finds for this integral an expression of the form 1 + (/z2)//z 2, where #D has the same homogeneity as the inverse of a Debye length. Then one can perform the integration; see Decoster [160] for much more details. In spite of its supposed accuracy, the interest of the Balescu-Lenard model is not so clear. Due to its high complexity, its numerical simulation is quite tricky. And except in very particular situations, apparently one gains almost nothing, in terms of physical accuracy of the results, by using it as a replacement for the Landau operator. In fact, it seems that the most important feature of the Balescu-Lenard operator, to this date, is to give a theoretical basis to the use of the Landau operator! There exist in plasma physics some even more complicated models, such as those which take into account magnetic fields (Rostoker operator, see, for instance, [465] and the references therein). We refrain from writing up the equations here, since this would require several pages, and they seem definitely out of reach of a mathematical treatment for the moment .... We also mention the simpler linear Fokker-Planck operator for Coulomb interaction, derived by Chandrasekhar. This is a linear operator of Landau-type, which describes the evolution of a test-particle interacting with a "bath" of Coulomb particles in thermal equilibrium. A formula for it is given in Balescu [46, w in terms of special functions. 1.8. Physical validity of the Boltzmann equation Experience has shown that the Boltzmann equation and its variants realistically describe phenomena which occur in dilute atmosphere, in particular aeronautics at high altitude, or interactions in dilute plasmas. In many situations, predictions based on the Navier-Stokes equation are not accurate for low densities; a famous historical example is provided by the so-called Knudsen minimum effect in a Poiseuille flow [148, p. 99]: if the difference of pressure between the entrance and the exit of a long, narrow channel is kept fixed, then the flow rate through a cross-section of this channel is not a monotonic function of the average pressure, but exhibits a minimum for a certain value of this parameter. This phenomenon, established experimentally by Knudsen, remained controversial till the 1960's. Also the Boltzmann equation cannot be replaced by fluid equations when it comes to the study of boundary layers (Knudsen layer, Sone sublayer due to curvature...) and the gas-surface interaction. Nowadays, with the impressive development of computer power, it is possible to perform very precise numerical simulations which seem to fully corroborate the predictions based on the Boltzmann equation - within the right range of physical parameters, of course. 11Thisis the naturalstatisticalequilibrium,see Section2.5.
- 98. A reviewof mathematical topics in collisional kinetictheory 95 All these questions are discussed, together with many numerical simulations and experiments, in the recent broad-audience survey book by Cercignani [148] (see also the review paper [136] by the same author in the present volume). 2. Mathematical problems in collisional kinetic theory In this section, we try to define the most interesting mathematical problems which arise in the study of Boltzmann-like equations. At this point we should make it clear that the Boltzmann equation can be studied for the sake of its applications to dilute gases, but also as one of the most basic and famous models for nonequilibrium statistical mechanics. 2.1. Mathematical validity of the Boltzmann equation In the last section, we have mentioned that, in the right range of physical parameters, the physical validity of the Boltzmann equation now seems to be beyond any doubt. On the other hand, the mathematical validity of the Boltzmann equation poses a more challenging problem. For the time being, it has been investigated only for the hard-sphere model. Let us give a short description of the problem in the case of hard spheres. But before that, we add a word of caution about the meaning of "mathematical validity": it is not a proof that the model is the right one in a certain range of physical parameters (whatever this may mean). It is only a rigorous derivation of the model, in a suitable asymptotic procedure, from another model, which is conceptually simpler but contains more information (typically: the positions of all the particles, as opposed to the density of particles). Of course, the Boltzmann equation, just like many models, can be derived either by mathematical validation, or by direct modelling assumptions, and the second approach is more arbitrary, less interesting from the mathematical point of view, but not less "respectable" if properly implemented! This is why, for instance, Truesdell [430] refuses to consider the problem of mathematical validity of the Boltzmann equation. The validation approach to be discussed now is due to Grad [249], and it is particularly striking because the starting point is nothing but the model given by Newton's laws of classical mechanics. It was not before 1972 that Cercignani [139] showed Grad's approach to be mathematically consistent, in the sense that it can be rigorously implemented if one is able to prove some "reasonable" estimates on the solutions. 12 Grad's approach. The starting point is the equation of motion, according to Newton's laws, for a system of n spherical particles of radius r in Ii~3, bouncing elastically on each other with billiard reflection laws. The state of the system is described by the positions and velocities of the centers, Xl, U1, ..., Xn, Vn, and the phase space is the subset of (Rx 3 x R3) n (or (X x R3) n) such that Ixi - xjl >~r (i :/: j). On this phase space there is a flow (St)t>~o, well-defined up to a zero-probability set of initial configurations, which is neglected. We now consider symmetric probability densities fn(xl, Vl ..... Xn, Vn) on the phase space 12This remark is importantbecause many people doubted the possibility of a rigorous derivation, see the discussionin Section2.4.
- 99. 96 C Villani (symmetry reflects the physical assumption of undiscernability of particles). Of course, the flow (St) on the phase space induces a flow on such probability densities, the solution of which is denoted by (ff)t >1o.Moreover, by integrating ftn over all variables but the k first position variables and the k first velocity variables, one defines the k-particle distribution function Pkfn (X l, Vl ..... Xk, Vk) (think of Pk as a projection operator). In probabilistic terminology, P1fn is the first marginal of fn. Assume now that (i) n --+ o0, r --+ 0 (continuum limit) in such a way that nr 2 --+ 1 [the gas is sufficiently dilute, but not too much, so that only binary interactions play a significant role, and a typical particle collides about once in a unit of time. This limit is called the Boltzmann-Grad limit]; (ii) P1 f~ --+ fo, where fo is a given distribution function [this assumption means that the one-particle function at time 0 can be treated as "continuous" as the number of particles becomes large]; (iii) P2f~ --+ fo | fo, and more generally, for fixed k, P, f~ ~ fo | . . . | fo (34) [this is the chaos assumption at time 0]; the problem is then to prove that P1fP --+ ft, where ft = ft (x, v) is the solution of the Boltzmann equation with hard-sphere kernel, and with initial datum f0. This problem seems exceptionally difficult. The main result in the field is the 1973 Lan- ford's theorem [292]. He proved the result for small time, and under some strong assump- tions on the initial probability distributions Pkf~: they should be continuous, satisfy ap- propriate Gaussian-type bounds, and converge uniformly 13 towards their respective limits. Later his proof was rewritten by Illner and Pulvirenti, and extended to arbitrarily large time intervals, under a smallness assumption on the initial datum, which enabled to treat the Boltzmann equation as a kind of perturbation of the free transport equation: see [275, 276] and the nice reviews in [149,394,410]. For sure, one of the outstanding problems in the theory of the Boltzmann equation is to extend Lanford's result to a more general frame- work, without smallness assumption. Another considerable progress would be its extension to long-range interactions, which is not clear even from the formal point of view (see, for instance, Cercignani [137]). The Boltzmann-Grad limit is also often called the low-density limit, and presented in the following manner [149, p. 60]: starting from the equations of Newtonian dynamics, blow-up the scales of space and time by a factor e-1 (thus e is the ratio of the microscopic scale by the macroscopic scale), and require the number of particles to be of the order of e-z, then let e go to 0. In particular, the density will scale as e2/3 since the volume will scale as e-3, and this explains the terminology of "low density limit". Remarks about the chaos assumption. 1. Heuristically, the relevance of the chaos assumption in Boltzmann's derivation can be justified as follows: among all probability distributions fn which have a given marginal 13Oncompactsubsetsof (•3 • i~3)k whichis the setobtainedfrom(It~ 3 • R3)k by deletingall configurations r withxi = xj for somedistinctindicesi, j.
- 100. A reviewofmathematicaltopicsincollisionalkinetictheory 97 f = Plf n, the most likely, in some sense which we do not make precise here, 14 is the tensor product f | @ f. And as n --+ ec, this distribution becomes by far the most likely. Thus, Boltzmann's chaos assumption may be justified by the fact that we choose the most likely microscopic probability distribution fn which is compatible to our macroscopic knowledge (the one-particle distribution, or first marginal, f). 2. In fact, the chaos property is automatically satisfied, in weak sense, by all sequences of probability measures (fn) on (R 3 x R3) n which are compatible with the density f. More precisely: let us say that a microscopic configuration z -- (Xl, Vl ..... Xn, Vn) E (R3 X R3) n is admissible if its empirical density, 09 z --- __ 6(Xi,Vi), n i=1 is a good approximation to the density f(x, v)dx dr, and let (fn)neN be a sequence of symmetric probability densities on (Rx 3 x R3) n respectively, such that the associated measures/z n give very high probability to configurations which are admissible. In a more precise writing, we require that for every bounded continuous function q)(x, v) on R 3 x R 3, and for all s > 0, fR go(x, v)[o)z(dx dv) - f (t, x, v) dx dr] >s] )0. n ----~oo Then, (fn) satisfies the chaos property, in weak sense [149, p. 91]: for any k ~> 1, pkfn ~ f| n-----~oo in weak-, measure sense. This statement expresses the fact that fn is automatically close to the tensor product f| in the sense of weak convergence of the marginals. However, weak convergence is not sufficient to derive the Boltzmann equation, because of the problem of localization of collisions. Therefore, in Lanford's theorem one imposes a stronger (uniform) convergence of the marginals, or strong chaos property. 3. This is the place where probability enters the Boltzmann equation: via the initial datum, i.e., the probability density (fn)! According to [149, p. 93], the conclusion of Lanford's theorem can be reformulated as follows: for all time t > 0, if ftn is obtained 14This is related to the fact that the tensor product f| has minimal entropy among all n-particles probability distribution functions fn with given first marginal f, and to the fact that the negative of the entropy yields a measure of "likelihood", see Section2.4 below. [In thisreview,the entropy of a distribution function f is defined by the formula H (f) = f f log f; note the sign convention.] For hard spheres, a subtlety arises from the fact that configurations in which two spheres interpenetrate are forbidden, so fn cannot be a tensor product. Since however the total volume of the spheres, nr3, goes to 0 in the limit, it is natural to assume that this does not matter. On the other hand, contact points in the n-particle phase space play a crucial role in the way the chaos property is propagated in time.
- 101. 98 C.Villani from fff by transportation along the characteristics of the microscopic dynamics, and if/z n t is the probability measure on (Nx 3 x N3)n whose density is given by ftn, then for all e > 0, and for all bounded, continuous function ~o(x,v) on Nx 3 x N3, 3. , z (R3 xR ) ; fR ~o(x, dr)- ft(x, v)dv] V) [(.Oz (dx dx x~XR3 n-----)(x) where ft is the solution to the Boltzmann equation with initial datum f0. In terms of/z n" q)(x, v)[WS,z(dx dr) - ft(x, v)do] dx 3xR3 n-----)(x) In words: for most initial configurations, the evolution of the density under the microscopic dynamics is well approximated by the solution to the Boltzmann equation. Of course, this does not rule out the existence of "unlikely" initial configurations for which the solution of the Boltzmann equation is a very bad approximation of the empirical measure. 4. If the chaos property is the crucial point behind the Boltzmann derivation, then one should expect that it propagates with time, and that Yt>0, P~fF ~ ft|174 ft. (35) However, this propagation property only holds in a weak sense. Even if the convergence is strong (say, uniform convergence of all marginals) in (34), it has to be weaker in (35), say almost everywhere, see the discussion in Cercignani et al. [149]. The reason for this weakening is the appearance of microscopic correlations (under evolution by the microscopic, reversible dynamics). In particular, if the initial microscopic datum is "very likely", this does not imply at all that the microscopic datum at later times should be very likely! On the contrary, it should present a lot of correlations .... 5. In fact, one has to be extremely cautious when handling (35). To illustrate this, let us formally show that for t > 0 the approximation P2ft (x, v; y, w)"~ ft (x, v)ft (Y, w) (36) cannot be true in strong sense, uniformly in all variables, 15 as n ~ cx~ (the symbol here means "approaches, in L ~ norm, uniformly in all variables x, y, v, w, as n ~ ~"). Indeed, assume that (36) holds true uniformly in x, y, v, w, and choose y - x + rtr, 15Constrainedby Ix- Yl~>r.
- 102. A reviewof mathematical topics in collisionalkinetic theory 99 (v - w, o-) > 0, i.e., an ingoing collisional configuration in the two-particle phase space. Then, presumably P2f~(x, v; x + rcr, w) ~_ ft(x, v)ft(x + rcr, w) ~_ ft(x, v)ft(x, w) (37) as n --+ cx~.But from the specular reflection condition, for any t > 0, P2f~ (x, v; x + rcr, w) = P2f~ (x, v'; x + rcr, w'), where v' and w' are post-collisional velocities, v' = v - (v - w, cr) ~r, w' = w + ( v - w, cr ) cr. Applying (36) again, this would result in P2f~(x, v; x -q-rcr, w) = P2f~(x, v'; x nt- rcr, w') ~--ft(x, v')ft(x, w'), which is not compatible with (37) (unless ft solves Equation (53) below). This contra- diction illustrates the fact that (36) cannot be propagated by the dynamics of hard spheres. It is actually property (37), sometimes called one-sided chaos, which is used in the derivation of the Boltzmann equation, and which should be propagated for positive times: it means that the velocities of particles which are just about to collide are not correlated. But it is a very difficult problem to handle Equation (37) properly, because it involves the restriction of fn to a manifold of codimension 1, and may be violated even for initial data which satisfy the conditions of Lanford's theorem! So an appropriate generalized sense should be given to (37). Lanford's argument cleverly avoids any discussion of (37), and only assumes (36) at time 0, the approximation being uniform outside collisional configurations. So he plainly avoids discussing one-sided chaos, and does not care what is propagated for positive times, apart from weak chaos. 16 To sum up: the physical derivation of the Boltzmann equation is based on the propagation of one-sided chaos, but no one knows how this property should be expressed mathematically- if meaningful at all. An easier variant of the validation problem is the derivation of linear transport equations describing the behavior of a Lorentz gas: a test-particle in a random pinball game, with scatterers randomly distributed according to (say) a Poisson law. Under a suitable scaling, the law of this test-particle converges towards the solution of a linear Boltzmann equation, as was first formalized by Gallavotti [226], before several improvements appeared [409, 91]. See Pulvirenti [394] for a review and introduction of the subject. The convergence actually holds true for almost all (in the sense of Poisson measure) fixed configuration of scatterers, but fails for certain specific configurations, for instance a periodic array, as shown in Bourgain, Golse and Wennberg [102]. We also note that Desvillettes and Pulvirenti [179] are able to rigorously justify the linear Boltzmann equation for some interactions with infinite range. 16This is possible because he uses a perturbative proof, based on an iterative Duhamelformula, in which everythingis expressedin termsofthe initialdatum....
- 103. 100 C.Villani Kac's approach. To conclude this section, we mention another line of approach towards the mathematical justification of the Boltzmann equation. It goes via the construction of some many-particle stochastic system, such that the first marginal of its law at a given time t should be an approximation of the solution to the Boltzmann equation if the initial datum is chaotic. This subject was initiated by Kac 17 [283], and developed by Sznitman [412] in connection with the problem of propagation of chaos. Recent progress on this have been achieved by Graham and M616ard [256,344]. The main conceptual difference between both approaches lies in the moment where probability is introduced, and irreversibility 18 as well. In Lanford's approach, the starting point is a deterministic particle system; it is only the particular "chaotic" choice of the initial datum which leads to the macroscopic, irreversible Boltzmann equation in the limit. On the other hand, for Kac the microscopic particle system is already stochastic and irreversible from the beginning. Then the main effect of the limit is to turn a linear equation on a large n-particle phase space, into a nonlinear equation on a reduced, one-particle phase space. Of course Kac's approach is less striking than Grad's, because the starting point contains more elaborate modelling assumptions, since stochasticity is already built in. Kac formulated his approach in a spatially homogeneous 19 setting, while this would be meaningless for Grad's approach. In fact, it is as if Kac wanted to treat the positions of the particles (which, together with ingoing velocities, determine the outgoing velocities) as hidden probabilistic variables. Then, all the subtleties linked to one-sided chaos can be forgotten, and it is sufficient to study just propagation of (weak) chaos. Moreover, Kac's approach becomes important when it comes to make an interpretation of the Monte Carlo numerical schemes which are often used to compute approximate solutions of the Boltzmann equation. These schemes are indeed based on large stochastic particle systems. See Pulvirenti [394,453,396] for references about the study of these systems, in connection with the validation problem. We do not develop here on the problem of the rigorous justification of numerical schemes, but this topic is addressed in the companion review [136] by Cercignani. 2.2. The Cauchyproblem From the mathematical point of view, the very first problem arising in the study of the Boltzmann equation is the Cauchy problem: given a distribution function fo(x, v) on RN x R N (or X x RN), satisfying appropriate and physically realistic assumptions, show that there exists a (unique) solution of Of -+- v . Vx f = Q(f, f), Ot (38) f(O,., .)= fo. 17 SeeSection1.5inChapter2E. 18SeeSection2.4. 19SeeSection5.2.
- 104. A reviewof mathematicaltopicsin collisionalkinetictheory 101 Needless to say, the Boltzmann equation seems impossible to solve explicitly,2~except in some very particular situations: semi-explicit solutions by Bobylev [79], Bobylev and Cercignani [81]; self-similar solutions of infinite mass by Nikolskii, see [289, p. 286]; particular solutions in a problem of shear flow by Truesdell, see [430, Chapters 14-15], some simple problems of modelling with a lot of symmetries [148] .... Explicit solutions are discussed in the review paper [207]. These exact solutions are important in certain modelling problems, but they are exceptional. This justifies the study of a general Cauchy problem. Of course, the question of the Cauchy problem should be considered as a preliminary for a more detailed study of qualitative properties of solutions of the Boltzmann equation. The main qualitative properties in which one is interested are: smoothness and singularities, conservation laws, strict positivity, existence of Lyapunov functionals, long-time behavior, limit regimes. We shall come back on all of this in the next chapters. As recalled in Section 3, the properties of the solutions may depend heavily on the form of the collision kernel. As of this date, the Cauchy problem has still not received satisfactory answers. As we shall describe in Section 5, there are several "competing" theories which either concern (more or less) simplified cases, or are unable to answer the basic questions one may ask about the solutions. Yet this problem has spectacularly advanced since the end of the eighties. Another fundamental problem in many areas of modelling by Boltzmann equation, as explained, for instance, in Cercignani [148], is the existence of stationary solutions: given a box X, prove that there exists a (unique?) stationary solution of the Boltzmann equation in the box: v. Vxf=Q(f,f), x6X, VE]I~N, together with well-chosen boundary conditions (ideally, dictated by physical assumptions). The stationary problem has been the object of a lot of mathematical studies in the past few years; see, for instance, [31,25,34,36,26,37,38]. We shall not consider it here, except for a few remarks. This is first because the theory is less developed than the theory of the Cauchy problem, secondly because we wish to avoid the subtle discussion of boundary conditions for weak solutions. 2.3. Maxwell's weak formulation, and conservation laws , ' k) withk--(v-v,)/Iv-v,l, hasunit The change of variables (v, v,, cr) --+ (vt v,, , Jacobian and is involutive. Since cr = (vI f v~ - v,)/I - v,[, one can abuse terminology by referring to this change of variables as (v, v,) --+ (v~ v,). It will be called the pre- postcollisional change of variables. As a consequence of microreversibility, it leaves the collision kernel B invariant. 20Althoughno theoremofnon-solvabilityhas beenproven!
- 105. 102 C. Villani The fact that this change of variable has unit Jacobian is not a general feature of Boltzmann-like equations, actually it is false for energy-dissipating models21 .... Also the change of variables (v, v,) --+ (v,, v) is clearly involutive and has unit Jacobian. As a consequence, if q9is an arbitrary continuous function of the velocity v, fR Q(f' f) qgdv u f /. / dvdv, ] da B(v- v,, a)(f'ff, - ff,)9 (39) JRN• N JsN-1 ---f~,NxRN dvdv*fsN-1drrB(v-v*'rr)ff*(~~ (40) 1 dvdv, dcr B(v - v,, cr)ff,(qg' + qg, - q9- qg,). (41) N • N-1 This gives a weak formulation for Boltzmann's collision operator. From the mathematical point of view, it is interesting because expressions like (40) or (41) may be well-defined in situations where Q(f, f) is not. From the physical point of view, it expresses the change in the integral f f(t, x, v)qg(v)dv which is due to the action of collisions. Actually, this formulation is so natural for a physicist, that Equation (40) was written by Maxwell22 [335, Equation (3)] before Boltzmann gave the explicit expression of Q(f, f)! Let f be a solution of the Boltzmann equation (8), set in the whole space ~ff to simplify. By the conservative properties of the transport operator, v 9Vx, df f -~ f (t, x, v)qg(v) dx dv -- Q(f, f)q9 dx dr, (42) and the right-hand side is just the x-integral of any one of the expressions in formulas (39)- (41). As an immediate consequence, whenever q9satisfies the functional equation V(U, V,, O') E ][~N X ]1~N X S N-1 , qg(v') + q3(l/,) -- qg(v) -+-qg(v,) (43) then, at least formally, df dt f (t, x, v)99(v) dx dv - 0 along solutions of the Boltzmann equation. The words "at least formally" of course mean that the preceding equations must be rigorously justified with the help of some integrability estimates on the solutions to the Boltzmann equation. 21 See Section2 in Chapter2E. 22Actuallyit is not so easyto recognizethe Boltzmannequationin Maxwell'snotations!
- 106. A reviewof mathematical topics in collisionalkinetic theory 103 It can be shown under very weak conditions 23 [142,29], [149, pp. 36-42] that solutions to (43), as expected, are only linear combinations of the collision invariants: Ivl 2 qg(v)- 1, Vi, 2 ' 1 <.i <~N. This leads to the (formal) conservation laws of the Boltzmann equation, (1) -~ f (t, x, y) l)i dxdv = O, 1 <.i<.N, meaning that the total mass, the total momentum and the total energy of the gas are preserved. These conservation laws should hold true when there are no boundaries. In presence of boundaries, conservation laws may be violated: momentum is not preserved by specular reflection, neither is energy if the gas is in interaction with a wall kept at a fixed temperature. See Cercignani [141,148] for a discussion of general axioms of the classical modelling of gas-surface interaction, and resulting laws. If one disregards this possible influence of boundaries, then the preservation of mass, momentum and energy under the action of the Boltzmann collision operator is clearly the least that one can expect from a model which takes into account only elastic collisions. Yet, to this date, no mathematical theory has been able to justify these simple rules at a sufficient level of generality. The problem is of course that too little is known about how well behaved are the solutions to the Boltzmann equation. Conservation of mass and momentum are no problem, but no one knows how to obtain an a priori estimate which would imply a little bit more integrability than just finite energy. Another crucial topic for a fluid description is the validity of local conservation laws, i.e., continuity equations obtained by integrating the Boltzmann equation with respect to v only. With notations (1), these equations are Op + Vx. (pu)- o, o __~(tOU) .71_Vx " N f v | v d v =0, o ~--~(plul 2 + NpT) + Vx. g f[v[ 2 ray =0. (44) At this moment, only the first of these equations has been proven in full generality [308]. 23This problemwas first treated by Gronwall [259,260] and Carleman [119] under stronger conditions. Then people started to study it under weaker and weaker assumptions. Its interest lies not only in checking that there are no hiddenconservationlawsin the Boltzmannequation,but alsoin solvingthe importantEquation(53)below, for which simplermethodsare howeveravailable.
- 107. 104 C. Villani 2.4. Boltzmann's H theorem and irreversibility In this section, we discuss some of the most famous aspects of the Boltzmann equation. This will justify a few digressions to make the topic as clear as possible. Let us symmetrize the integral (39) once more, fully using all the symmetries of the collision operator. We obtain f Q(f, f)go dv 1 f dvdv, do- B(v - v,, a)(f'f" ff,)(go' -+-go', go go,) (45) 4 We shall refer to this formula as Boltzmann's weakformulation. Without caring about integrability issues, we plug go-- log f into this equation, and use the properties of the logarithm, to find f• f) log f - -D(f), (46) Q(f, dv N where D is the entropy dissipation functional, fR f'f*' 1 dvdv, da B(v- v,,a)(f'f',- ff,)log--f~-, ~>0. (47) D(f) = -~ 2N • That D(f) >,0just comes from the fact that the function (X, Y) ~ (X-Y)(log X- log Y) is nonnegative. Next, we introduce Boltzmann's H functional, /- H(f) / f log f. (48) JRN• Of course, the transport operator -v. Vx does not contribute in any change of the H functional in time. 24 As a consequence, if f = f (t, x, v) is a solution of the Boltzmann equation, then H (f) will evolve in time because of the effects of the collision operator: d ,(f(,, .))_ O(f(,,x .))dx<.O. dt ~ (49) This is the famous Boltzmann's H theorem: the H-functional, or entropy, is nonincreasing with time. This theorem is "proven" and discussed at length in Boltzmann's treatise [93]. Before commenting on its physical implications, let us give a few analytical remarks: 1. For certain simplified models of the Boltzmann equation, McKean [342] has proven that the H-functional is, up to multiplicative and additive constants, the only "local" (i.e., of the form f A(f)) Lyapunov functional. 24More generally, the transport operator does not contribute to any change of a functional of the form f a(f)dxdv.
- 108. A reviewofmathematicaltopicsincollisionalkinetictheory 105 2. There are some versions with boundary conditions; actually, it was emphasized by Cercignani that the H theorem still holds true for a modified H-functional (including the temperature of the wall, for instance, if the wall is kept at fixed temperature) as soon as a certain number of general axioms are satisfied. See [141] for precise statements. 3. The argument above, leading to formula (47), does not work for certain variants of the Boltzmann equation, like mixtures. Actually Boltzmann had also given a (false) proof in this case, and once the error was discovered, produced a totally obscure argument to fix it (see the historical references in [148]). As pointed out by Cercignani and Lampis [150], the most robust way to prove the H theorem is to use again Maxwell's weak formulation, and to note that if f,f~dvdv,da l f ( f, ff, -~ Bf f, log ff, -~ Bf f, log ff, f'f~ I-1/~<0, (50) ff, J because f Q(f, f) dv = 0 and log X - X + 1 <~0. This line of proof can be generalized to mixtures [148, w and other models. 4. The Landau equation also satisfies a H theorem. The corresponding entropy dissipation is (formally) 1 ~ ff, lv/a(v-v,)(V(log f)- [V(logf)],)12dvdv,. (51) DL (f) = ~ 3xR3 What is the physical sense of Boltzmann's H theorem? First of all, we note that the H functional should coincide with the usual entropy of physicists up to a change of sign. Also, it is a dynamical entropy, in the sense that is defined for nonequilibrium systems. Thus Boltzmann's H theorem is a manifestation of the second law of thermodynamics (Clausius' law) which states that the physical entropy of an isolated system should not decrease in time. In particular, it demonstrates that the Boltzmann model has some irreversibility built in. This achievement (produce an analytical proof of the second law for some specific model of statistical mechanics) was one of the early goals of Boltzmann, and was later considered as one of his most important contributions to statistical physics. But the H theorem immediately raised a number of objections, linked to the fact that the starting point of the derivation of the Boltzmann equation is just classical, reversible mechanics- so where does the irreversibility come from? Of course, since the existence of atoms was controversial at the time, Boltzmann's construction seemed very suspect .... About the controversy between Boltzmann and opponents, and the way to resolve apparent contradictions in Boltzmann's approach, one may consult [202,284,149,293]. Here we shall explain in an informal way the main arguments of the discussion, warning the reader that the following considerations have not been put on a satisfactory mathematical basis. Zermelo pointed out that Boltzmann's theorem seemed to contradict the famous Poincar6 recurrence theorem. 25 Boltzmann replied that the scales of time on which Poincar6's theorem applied in the present setting were much larger than the age of the 25For almostall choiceofthe initialdatum,a conservativesystemwitha compactphasespacewillalwayscome arbitrarily closeto its initialconfigurationforlargeenoughtimes.Thistheoremappliesto a systemof n particles obeyingthe lawsof classicalmechanics,interactingvia elasticcollisions,enclosedin a box.
- 109. 106 C. Villani universe, and therefore irrelevant. This answer is justified by the fact that the Boltzmann equation should be a good approximation of the microscopic dynamics, only on a time scale which depends on the actual number of particles - see the discussion in the end of Section 2.5. Then Loschmidt came out with the following paradox. Let be given a gas of particles, evolving from time 0 to time to > 0. At time to, reverse all velocities and let the gas evolve freely otherwise. If Boltzmann were fight, the same Boltzmann equation should describe the behavior of the gas on both time intervals [0, to] and [to, 2t0]. Since the reversal of velocities does not change the entropy of a distribution function, the entropy at time 2t0 should be strictly less than the entropy at time 0. But, by time-reversibility of classical mechanics, at time 2t0 the system should be back to its initial configuration, which would be a contradiction .... To this argument Boltzmann is reported to have replied "Go on, reverse the velocities !" The answer to Loschmidt's paradox is subtle and has to do with the probabilistic content of the Boltzmann equation. Starting with the classical monograph by P. and T. Ehrenfest [202], it was understood that reversible microdynamics and irreversible macrodynamics are not contradictory, provided that the right amount of probability is used in the interpretation of the macroscopic model. This view is very well explained in the excellent book by Kac [284]. In the case of the Boltzmann derivation, everything seems deterministic: neither the microscopic model, nor the macroscopic equation are stochastic. But the probabilistic content is hidden in the choice of the initial datum. As we mentioned earlier,26 with very high probability the Boltzmann equation gives a good approximation to the evolution of the density of the gas. And here randomness is in the choice of the microscopic initial configuration, among all configurations which are compatible with the density f (see Section 2.1 for more precise formulations). But for exceptional configurations which are not chaotic, the derivation of the Boltzmann equation fails. Now comes the tricky point in Loschmidt's argument. As we discussed in Section 2.1, the chaos property, in strong sense, is not preserved by the microscopic, reversible dynam- ics. What should be preserved is the one-sided chaos: ingoing collisional configurations are uncorrelated. More precisely, if the strong chaos assumption holds true at initial time, then chaos should be true for ingoing configurations at positive times, and for outgoing configurations at negative times. So, reversing all velocities as suggested by Loschmidt is an innocent operation at the level of the limit one-particle distribution (which has forgotten about correlations), but by no means at the level of the microscopic dynamics (which keeps all correlations in mind). It will transform a configuration which is chaotic as far as ingoing velocities are concerned, into a configuration which is chaotic as far as outgoing velocities are concerned27.... Then, the microscopic dynamics will preserve this property that outgoing velocities are not correlated, and, by repeating the steps of the derivation of the Boltzmann equation, we find that the correct equation, to describe the gas from time to on, should be the negative Boltzmann equation (with a minus sign in front of the collision operator). Therefore in 26Recall the remarks about propagation of chaos in Section 2.1. 27In particular, this configuration is very unlikely as an initial distribution. Thus Loschmidt's paradox illustrates very well the fact that the Boltzmann derivation works for most initial data, but not for all!
- 110. A review of mathematical topics in collisional kinetic theory 107 Loschmidt's argument, of course the entropy is unchanged at time to, but then it should start to increase, and be back at its original value at time 2t0. At the moment, a fully satisfactorily mathematical discussion of Loschmidt's paradox is not possible, since we do not know what one-sided chaos should really mean, mathematically speaking. But one can check, as is done in [149, Section 4.7], that strong chaos 28 is not propagated in time- so that it will be technically impossible to repeat Lanford's argument when taking as initial datum the microscopic configuration at time to, be it before or after reversal of velocities. With this in mind, one also easily answers to the objection, why would the Boltzmann equation select a direction of time? Actually, it does not, 29 and this can be seen by the fact that if strong 3~ chaos is assumed at initial time, then the correct equation should be the Boltzmann equation for positive times, and the negative Boltzmann equation for negative times. We have selected a direction of time by assuming the distribution function to be "very likely" at time 0 and studying the model for positive times. In fact, in Boltzmann's description, the entropy is maximum at time 0, and decreases for positive times, increases for negative times. As an amusing probabilistic reformulation: knowing the one-particle distribution function at some time to, with very high probability the entropy is a maximum at this time t0! Related considerations can be found in Kac [284, p. 79] for simpler models, and may explain the cryptic statement by Boltzmann that "the H-functional is always, almost surely, a local maximum". Most of the explanations above are already included in Boltzmann's treatise [93], in physicist's language; in particular Boltzmann was very well aware of the probabilistic content of his approach. But, since so many objections had been raised against Boltzmann's theory, many physicists doubted for a long time that a rigorous derivation of the Boltzmann equation, starting from the laws of classical mechanics, could be possible. This is one of the reasons why Lanford's theorem was so spectacular. After this digression about irreversibility, let us now briefly comment on Boltzmann's H-functional itself. Up to the sign, it coincides with Shannon's entropy (or information) quantity, which was introduced in communication theory at the end of the forties. 31 In the theory of Shannon, the entropy measures the redundancy of a language, and the maximal compression rate which is applicable to a message without (almost) any loss of information: see [156] and the many references therein. In this survey, we shall make precise some links between information theory and the kinetic theory of gases, in particular via some variants of famous information-theoretical inequalities first proven by Stam [165]. 28Roughly speaking, in the sense of uniform convergence of the marginals towards the tensor product distributions,recall Section2.1. 29Bythe way,Boltzmannhimselfbelievedthatthe directionof positivetimes shouldbe defined as the direction in whichthe H-functionalhas a decreasingbehavior.... 30"Double-sided" shouldbe the fightconditionhere! 31Hereis a quotationby Shannon,extractedfrom[331],whichwe learntin [16]. "My greatestconcernwas how to callit. I thoughtof callingit 'information'. But the wordwas overlyused, so I decidedto callit 'uncertainty'. When I discussedit with Johnvon Neumann,he had a better idea. He told me: "Youshouldcall it entropy,for two reasons.In firstplace youruncertaintyhas beenusedin statisticalmechanicsunderthat name, so it already has a name. In secondplace, and moreimportant,no one knowswhat entropyreally is, so in a debate you will always havethe advantage.""
- 111. 108 C. Villani From the physical point of view, the entropy measures the volume of microstates associated to a given macroscopic configuration. 32 This is suggested by the following computation, due to Boltzmann (see [93]). Let us consider n particles taking p possible different states; think of a state as a small "box" in the phase space •N • ii~N.Assume that the only information to which we have access is the number ni of particles in each state i. In other words, we are unable to distinguish particles with different states; or, in a probabilistic description, we only have access to the one-particle marginal. This macroscopic description is in contrast with the microscopic description, in which we can distinguish all the particles, and know the state of each of them. Given a macroscopic configuration (nl .... , n p), the number of compatible microscopic configurations is n! nl!" .np! Let us set j5 = ni In, and let all ni's go to infinity. By Stirling formula (or other methods, see [156, p. 282]), one shows that, up to an additive constant which is independent of the ft" 'S, 1 P - log ~2 --+ - E fi log j~. n i=1 This result explains the link between the H-functional and the original definition of the entropy by Boltzmann, as the logarithm of the volume of microstates. 33 So we see that it is the exponential of the negative of the entropy, which plays the role of a "volume" in infinite dimension. Up to a normalization, this quantity is known in information theory as the entropy power34: ./V'(f) = exp(-2H(f)/N). (52) More remarks about the physical content of entropy, or rather entropies, are formulated in Grad [251 ]. In the discussion of Boltzmann's derivation and irreversibility, we have seen two distinct entropies: the macroscopic entropy H(f), which is fixed by the experimenter at initial time, and then wants to decrease as time goes by; and the microscopic entropy, H(fn), which is more or less assumed to be minimal at initial time (among the class of microscopic distributions fn which are compatible with f), and is then kept constant in time by the microscopic dynamics. There is no contradiction between the fact that H (ft) is decreasing 32Here, the one-particle probability distribution f is the macroscopic description of the system, while the many- particle probability distribution fn is the microscopic state. 33This is the famous formula S = k In ~2, which was written on Boltzmann's grave. 34The analogy between power entropy and volume can be pushed so far that, for instance, the Shannon-Stam entropy power inequality, .A/'(f 9 g) ~>.A/'(f) + A/'(g), can be seen as a consequence of the Brunn-Minkowski inequality on the volume of Minkowski sums of compact sets IX + YI1/d >1IX[1/d + IF[1/d. This (very) nontrivial remark was brought to our attention by E Barthe.
- 112. A reviewof mathematical topicsin collisionalkinetictheory 109 and H(ff) is constant, because there is no link between both objects 35 if there are correlations at the level of ftn. As a last comment, the decrease of the entropy is a fundamental property of the Boltzmann equation, but the H-functional is far from containing all the information about the Boltzmann equation. This is in contrast with so-called gradient flows, which are partial differential equations of the form Of/Ot = -grad E(f), for some "entropy" functional E and some gradient structure. For such an equation, in some sense the entropy functional encodes all the properties of the flow .... The main, deep reason for the fact that the Boltzmann equation cannot be seen as a gradient flow, is the fact that the collision operator depends only on the velocity space; but even if we restrict ourselves to solutions which do not depend on space, then the Boltzmann equation is not (to the best of our knowledge) a gradient flow. Typical gradient flows, in a sense which will be made more precise later, are the linear Fokker-Planck equation (19), or the McNamara-Young model for granular media (see [70] and Section 2 in Chapter 2E). The lack of gradient flow structure contributes to the mathematical difficulty of the Boltzmann equation. 2.5. Long-time behavior Assume that B(v - v,, cr) > 0 for almost all (v, v,, cr), which is always the case in applications of interest. Then equality in Boltzmann's H theorem occurs if and only if for almost all x, v, v,, cr, f' f~ -- f f.. (53) Under extremely weak assumptions on f [149,307,377], this functional equality forces f to be a local Maxwellian, 36 i.e., a probability distribution function of the form f (x, V) = p(X) e-lV-U(X)12/2T(x) (2:rrT (x)) N/2 Thus it is natural to guess that the effect of collisions is to bring f (t, .) closer and closer to a local Maxwellian, as time goes by. This is compatible with Gibbs' lemma: among all distributions on R N with given mass, momentum and energy, the minimum of the entropy is achieved by the corresponding Maxwellian distribution. 35Except the inequality H(ft) <~liminfH(f~)/n .... By the way [149, pp. 99-100], the function ~b(X)= X logX is, up to multiplication by a constant or addition of an affine function, the only continuous function 4~which satisfies the inequality f 4~(P1fn) ~ f dp(fn)/n for all fn,s, with equality for the tensor product distribution. 36The result follows from the characterization of solutions to (43), but can also be shown directly by Fourier transform as in [377]. However,no proof is moreenlighteningthanthe one due to Boltzmann (see Section4.3 in Chapter 2C). His proof requires C1 smoothness,but Lions [307,p. 423] gavea beautiful proof that L1 solutions to (53) have to be smooth. And anyway, one can always easily reduce to the case of smooth densities by the remarks in Section4.7 of Chapter 2C.
- 113. 110 C. Villani Of course, Equation (53) implies Q(f, f) = 0. As a corollary, we see that solutions of the functional equation Q(f, f) = o, (54) where the unknown f(v) is a distribution function on ~N with finite mass and energy, are precisely Maxwellian distributions. As we have seen, local Maxwellian states are precisely those distribution functions for which the dissipation of entropy vanishes. But if the position space X is a bounded domain on R N (with suitable boundary conditions, like specular reflection, or Maxwellian re- emission), then one can show that there are very few time-dependent local Maxwellian distributions which satisfy the Boltzmann equation. Except in particular cases (domains with symmetries, see [254]), such a solution has to take the form e-lv-ulZ/2T f (t, x, v) = p (2JrT)N/2 , (55) for some parameters p, u, T which depend neither on t nor on x. A state like (55) is called a global Maxwellian, or global equilibrium state. It is uniquely determined by its total mass, momentum and energy. The problem of the trend to equilibrium consists in proving that the solution of the Cauchy problem (38) converges towards the corresponding global equilibrium as t --+ +c~, and to estimate, in terms of the initial datum, the speed of this convergence. On this subject, the (now outdated) paper by Desvillettes [168] accurately surveys existing methods up to the beginning of the nineties. Since that time, new trends have emerged, with the research for constructive estimates and the development of entropy dissipation techniques. To briefly summarize recent trends, we should say that the problem of trend to equilibrium has received rather satisfactory answers in situations where the Cauchy problem is known to have well-behaved solutions. We shall make a detailed review in Chapter 2C. On the other hand, if more complicated boundary conditions are considered, it may happen that the global equilibrium be no longer Maxwellian; and that the mere existence of a global equilibrium already be a very difficult problem. 37 Also, when the gas gets dispersed in the whole space, then things become complicated: in certain situations, and contrary to what was previously believed by many authors, solutions to the Boltzmann equation never get close to a local Maxwellian state [383,419, 327]. The main physical idea behind this phenomenon is that the dispersive effects of the transport operator may prevent particles to undergo a sufficient number of collisions. In the whole space setting, the relevant problem is therefore not trend to equilibrium, but rather dispersion: find estimates on the speed at which the gas is dispersed at infinity. We shall not develop on this problem; to get information the reader may consult Perthame [376] (see also [277] for similar estimates in the context of the Vlasov-Poisson equation). Dispersion estimates play a fundamental role in the modern theory of the Schr6dinger equations, and there is a strong analogy with the estimates appearing in this field [135]. 37 Seethe referencesin Section2.2.
- 114. A review of mathematical topics in collisional kinetic theory 111 We conclude this section with an important remark about the meaning of +ec in the limit t --+ +ec. It is only in a suitable asymptotic regime that the Boltzmann equation is expected to give an accurate description of, say, a system of n particles. But for a given, large number n of particles, say 1023, the quality of this description cannot be uniform in time. To get convinced of this fact, just think that Poincar6's recurrence theorem will apply to the n-particle system after a very, very long time. In fact, since the Boltzmann equation is established on physical scales such that each particle typically encounters a finite number of collisions in a unit of time, we may expect the Boltzmann approximation to break down on a time at most O(n), i.e., a time on which a typical particle will have collided with a nonnegligible fraction of its fellow particles, so that finite-size effects should become important. This means that any theorem involving time scales larger than 1023 is very likely to be irrelevant38 .... Such a conclusion would only be internal to Boltzmann's equation and would not yield any information about physical "reality" as predicted by the model. So what is interesting is not really to prove that the Boltzmann equation converges to equilibrium as t --+ +cx~, but rather to show that it becomes very close to equilibrium when t is very large, yet not unrealistically large. Of course, from the mathematical point of view, this may be an extremely demanding goal, and the mere possibility of proving explicit rates should already be considered as a very important achievement, as well as identifying the physical factors (boundary conditions, interaction, etc.) which should slow down, or accelerate the convergence. 2.6. Hydrodynamic limits The H theorem was underlying the problem of the trend to global equilibrium in the limit t --+ +oc. It also underlies the assumption of local thermodynamical equilibrium in the hydrodynamical limits. Generally speaking, the problem of the hydrodynamical limit can be stated as follows: pass from a Boltzmann description of a dilute gas (on microscopic scales of space and time, i.e., of the order of the mean free path and of the mean time between collisions, respectively) to a hydrodynamic description, holding on macroscopic scales of space and time. And the scaling should be such that f "looks like" a local Maxwellian, even if local Maxwellians cannot be solutions of the Boltzmann equation .... To make this more concrete, assume that one contracts the measurements of lengths and time by a factor e, the velocity scale being preserved (e can be thought of as the Knudsen number, which, roughly speaking, would be proportional to the ratio between the mean free path and a typical macroscopic length). Then, the new distribution function39 will be fe(t, x v) -- f e 38Besides beingof no practicalvalue,sincethesetime scalesare muchmuchlargerthanreasonablephysical scales. 39Notethat fE is not a probabilitydensity.
- 115. 112 C. Villani If f solves the Boltzmann equation, then f~ solves the rescaled Boltzmann equation 0fe 1 ~ + V . Vx fe = -Q(fe, fE). Ot s Hence the role of the macroscopic parameter s is to considerably enhance the role of collisions. In view of the H theorem, one expects fs to resemble more and more a local Maxwellian when s --+ 0: this is the assumption of local thermodynamical equilibrium, whose mathematical justification is in general a delicate, still open problem. Here is an equivalent, nonrigorous way of seeing the limit: the time scale of trend to local equilibrium should be of the order of the mean time between collisions, which should be much smaller than the macroscopic time. For fixed s, the macroscopic quantities (density, momentum, temperature) associated to fe via (1) satisfy the equations Ops -ST + v~. (p~u~) = o, 3t +Vx. u f~v | vdv =0, -~(p~lu, + NpsTe) + Vx . felvl 2 ray =0. N The assumption of local thermodynamical equilibrium enables one to close this system in the limit s ~ 0, and to formally obtain Op -~ + v~ . (pu)- o, O(pu) + 7x 9(pu | u + pTIN) = 0 Ot 0 ~(plul 2 + Npr) + Vx . (plulZu + (N + 2)pru) = 0 (56) with IN standing for the identity N • N matrix. System (56) is nothing but the system of the compressible Euler equations, when the pressure is given by the law of perfect gases, p = pT. See [431]. Other scalings are possible, and starting from the Boltzmann equation one can get many other equations in fluid mechanics [56]. In particular, by looking at perturbations of a global equilibrium, it is possible to recover Navier-Stokes-type equations. This is one of several possible ways of interpreting the Navier-Stokes equation, see [467] for remarks about other interpretations. From a physicist's point of view, the interesting aspect of this limit is the appearance of the viscosity from molecular dynamics. From a mathematician's perspective, another interesting thing is that there are some well-developed mathematical theories for the Navier-Stokes equation, for instance the famous theory of weak solutions by Leray [299-301], see Lions [313,314] for the most recent developments- so one can hope to prove theorems!
- 116. A review of mathematical topics in collisional kinetic theory 113 An interesting remark, due to Sone and coworkers [408,407], shows that sometimes a hydrodynamic equation which looks natural is actually misleading because some kinetic effects should have an influence even at vanishing Knudsen number; this phenomenon was called "ghost effect". There are formal procedures for "solving" the Boltzmann equation in terms of a series expansion in a small parameter e (like our e above), which are known as Hilbert and Chapman-Enskog expansions. These procedures have never received a satisfactory mathematical justification in general, but have become very popular tools for deriving hydrodynamical equations. This approach is described in reference textbooks such as [154, 430,141,148,48], and particularly [250, Section 22 and following]. But it also underlies dozens of papers on formal hydrodynamical limits, which we do not try to review. By the way, it should be pointed out that equations obtained by keeping "too many" (meaning 3 or 4) terms of the Hilbert or Chapman-Enskog series, like the so-called Burnett or super- Burnett equations, seem to be irrelevant (a discussion of this matter, an ad hoc recipe to fix this problem, and further references, can be found in Jin and Slemrod [282]). Also, these expansions are not expected to be convergent, but only "asymptotic". In fact, a solution of the Boltzmann equation which could be represented as the sum of such a series would be a very particular one4~ it would be entirely determined by the fields of local density, mean velocity and pressure associated with it. All these problems illustrate the fact that the Hilbert and Chapman-Enskog methods rely on very sloppy grounds. Rather violent attacks on their principles are to be found in [430]. In spite of this, these methods are still widely used. Without any doubt, their popularity lies in their systematic character, which enables one to formally derive the correct equations in a number of situations, without having to make any guess. An alternative approach, which is conceptually simpler, and apparently more effective for theoretical purposes, is a moment-based procedure first proposed by Grad [249]. For Maxwell molecules, this procedure leads to hydrodynamic equations whose accuracy could be expected to be of high order; the method has been developed in particular by Truesdell and collaborators [274,430] under the name of "Maxwellian iteration". The problem of rigorous hydrodynamical limits has been studied at length in the literature, but most of these results have been obtained in a perturbative setting. Recently, some more satisfactory results (in the large) have been obtained after the development of a spectacular machinery (see [441] for a presentation). We shall discuss a few references on both lines of approach in Section 5, as we shall go along the presentation of the mathematical theories for the Cauchy problem. On the other hand, the Boltzmann equation, as a model, does not capture the full range of hydrodynamical equations that one would expect from dynamical systems of interacting particles. At the level of the Euler equation above, this can be seen by the fact that the pressure law is of the form p = p T. See [363,397,467] for more general equations and partial results on the problem of the direct derivation of hydrodynamical equations from particle systems. This however does not mean that it is not worth working at the level of the Boltzmann equation; first because the limit should be simpler to rigorously perform, than the limit for "raw" particle systems; secondly because the Boltzmann equation is one 40"Normal"in the terminologyofGrad[250].
- 117. 114 c. Villani of the very few models of statistical physics which have been derived from mechanical first principles. This kind of preoccupations meets those expressed by Hilbert in the formulation of his sixth problem41 about the axiomatization of physics" can one put the equations of fluid mechanics on a completely rigorous basis, starting from Newton's laws of microscopic motion? 2.7. The Landau approximation This problem occurs in the kinetic theory of dilute plasmas, as briefly described in Section 1.7. Starting from the Boltzmann equation for screened Coulomb potential (Debye potential), can one justify the replacement of this operator by the Landau operator (26) as the Debye length becomes large in comparison with the space length scale? This problem can easily be generalized to an arbitrary dimension of space. A few remarks are in order as regards the precise meaning of this problem. First of all, this is not a derivation of the Landau equation from particle systems (such a result would be an outstanding breakthrough in the field). Instead, one considers the Boltzmann equation for Debye potential as the starting point. Secondly, if one wants to stick to the classical theory of plasmas, then - either one neglects the effects of the mean-field interaction; in this case the problem has essentially been solved recently, as we shall discuss in Section 5 of Chapter 2B; - or one takes into account this effect, and then the Landau equation should only appear as a long-time correction to the Vlasov-Poisson equation. To the best of our knowledge, no such result has ever been obtained even in simplified regimes. 2.8. Numerical simulations The literature about numerical simulation for the Boltzmann equation is considerable. All methods used to this day consider separately the effects of transport and collision. This splitting has been theoretically justified by Desvillettes and Mischler [178], for instance, but a thorough discussion of the best way to implement it seems to be still lacking. Dominant methods are based on Monte Carlo simulation, and introduce "particles" interacting by collisions. Of course, transport is no problem for a particles-based method: just follow the characteristics, i.e., the trajectories of particles in phase space. Then one has to implement the effect of collisions, and many variants are possible. Sometimes the dynamics of particles obey the Newton laws of collision only on the average, in such a way that their probability density still comply with the Boltzmann dynamics. We do not try to review the literature on Monte Carlo simulation, and refer to the very neat survey in Cercignani [148, Chapter 7], or to the review paper [136] by the same author. Elements of the theoretical justification of Monte Carlo simulation, in connection with chaos issues, are reviewed in Cercignani et al. [149, Chapter 10], Pulvirenti [394] and Graham and Mrlrard [257]. 41Directlyinspiredby Boltzmann'streatise,amongotherthings.
- 118. A reviewof mathematicaltopicsin collisionalkinetictheory 115 Let us however say a few words about deterministic methods which have emerged recently, thanks to the increase of computational capacity.42 For a long time these methods were just unaffordable because of the high computational cost of the (2N- 1)-fold integral in the Boltzmann collision operator, but they are now becoming more and more competitive. Deterministic schemes based on conservation laws have been devised in the last years by Buet, Cordier, Degond, Lemou, Lucquin [106,297,105]. In these works, the simulated distribution function is constrained to satisfy conservation of mass, momentum and energy, as well as decreasing of entropy. This approach implies very clever procedures, in particular to handle the discretization of the spheres appearing in the Boltzmann representation (this problem is pretty much the same than the consistency problem for discrete velocity models). For variants such as the Landau equation, these difficulties are less important, but then one has to hunt for possible undesirable symmetries, which may introduce spurious conservation laws, etc. Another deterministic approach is based on Fourier transform, and has been developed by Bobylev and Rjasanow, Pareschi, Perthame, Russo, Toscani [82,89,370,371,373,372]. We shall say a little bit more on these schemes in Section 4.8, when introducing Fourier transform tools. At the moment, both methods have been competing, especially in the framework of the Landau equation [107,215]. It seems that spectral schemes are useful to give extremely accurate results, but cannot beat conservation-based schemes in terms of speed and efficiency. Certainly more is to be expected on the subject. 2.9. Miscellaneous We gather here a few other basic questions about the Boltzmann equation, which, even though less important than the ones we have already presented, do have mathematical interest. Phenomenological derivation of the Boltzmann equation. We have seen two ways to introduce the Boltzmann equation: either by direct modelling assumptions (dilute gas, chaos, etc.) or by rigorous theorems starting from particle systems. There is a third way towards it, which is by making some "natural" phenomenological assumptions on the form of the collision operator, and try to prove that these assumptions uniquely determine the form of the collision operator. A classical discussion by Bobylev [84] exemplifies this point of view. To roughly sum up the very recent work by Desvillettes and Salvarani [180], it is shown that (essentially) the only smooth quadratic forms Q acting on probability densities such that (1) evolution by Otf = Q(f, f) preserves nonnegativity, (2) Q has Galilean invariance, (3) Q(M, M) = 0 for any Maxwellian M, 42Thesameincreaseofcomputationalcapacityseemsto makediscrete-velocitymodelslessandlessattractive.
- 119. 116 C. Villani have to be linear combinations of a Boltzmann and a Landau operator. Note the inversion of the point of view: in Section 2.4 we have checked that the form of Boltzmann's operator imposed a Maxwellian form to an equilibrium distribution, while here we see that the assumption of Maxwellian equilibrium states43 contributes to determine the structure of the Boltzmann operator. Image of Q. This problem is very simply stated: for any function f(v) with enough integrability, we know that f Q(f, f)go(v)dv = 0 whenever qg(v) is a linear combination of 1, vi (1 ~< i ~< N), Ivl2. Conversely, let h be a function satisfying f hq9 = 0 for the same functions qg,is it sufficient to imply the existence of a probability distribution f such that Q (f, f) = h ? This problem apparently has never received even a partial answer for a restricted subclass of functions h. Divergence form of Boltzmann's collision operator. Unlike several other operators, particularly Landau or Fokker-Planck, the Boltzmann operator is not written in divergence form, even though it is conservative. Physicists consider this not surprising, in as much as Boltzmann's operator models sudden (as opposed to continuous) changes of velocities. However, generally speaking, any function Q whose integral vanishes can be written as the divergence of something. This writing is in general purely mathematical and improper to physical interpretation. But a nice feature of Boltzmann's operator, as we have shown in [448], is that an explicit and (relatively) simple expression exists. We obtained it by going back to the physical interpretation in terms of collisions, as in Landau [291 ]. Since there is no well-defined "flux", one is led to introduce fictitious trajectories, linked to the parametrization of the pre-collisional variables. Here is one possible expression for the flux, which is a priori not unique: Q(f, f) = -Vv 9J (f, f), where J(f' f) = - fv-v,,o~)>o dr, do9B(v - v,, o9) fo (V-V*,~ x dr f(v + ro9)f (v, + ro9)o9 (57) ,-v,vo-v)<O dvo dr, f(vo)f(v,) V -- Vo ) V -- Vo xB Vo-V,, IV-Vol Iv-vo[ N" (58) The Boltzmann operator can even be written as a double divergence, 02 Q(f' f) = Z.. OVi OVj tj Aij (f, f), Z Aii (f, f) = -Vv" JE (f, f), i for some explicit functionals Aij and jE, 1 <<.i, j ~ N. Each of the "fluxes" J, A j, jE is related to a conservation law. 43Which can be taken as physically reasonable from outer considerations, like central limit theorem, etc.
- 120. A reviewof mathematicaltopicsin collisionalkinetictheory 117 Besides casting some more light on the structure of Boltzmann's operator and its relations with divergence operators, these formulas may be used for devising conservative numerical schemes via the "velocity diffusion" method (implemented in works by Mas- Gallic and others). Eternal solutions. Let us introduce this problem with a quotation from Truesdell and Muncaster [430, p. 191] about irreversibility: "In a much more concrete way than Boltzmann's H theorem, [the time-behavior of the moments of order 2 and 3 for the Boltzmann equation with Maxwellian44 molecules] illustrate[s] the irreversibility of the behavior of the kinetic gas. This irreversibility is particularly striking if we attempt to trace the origin of a grossly homogeneous condition by considering past times instead of future ones. Indeed the magnitude of each component of [the pressure tensor] and [the tensor of the moments of order 3] that is not 0 at t = 0 tends to cxz as t --+ -cxz. Thus any present departure from kinetic equilibrium must be the outcome of still greater departure in the past." Appealing as this image may be, it is our conviction that it is actually impossible45 to let t --+ -o~. More precisely, heuristic arguments in [450, Annex II, Appendix] make the following conjecture plausible: let f = f (t, v) be a solution of the Boltzmann equation for v 6 R N, t e •, with finite kinetic energy. Then f is a Maxwellian, stationary distribution. In the case of the spatially inhomogeneous Boltzmann equation, the conjecture should be reformulated by saying that f is a local Maxwellian (which would allow "travelling Maxwellians", of the form M (x - vt, v), where M (x, v) is a global Maxwellian in the x and v variables). This problem, which is reminiscent of Liouville-type results for parabolic equa- tions [468], may seem academic. But in [450, Annex II, Appendix] we were able to con- nect it to the important problem of the uniformity of trend to equilibrium, and to prove this conjecture for some simplified models; Cabannes [109,110] also was able to prove it for some simplified discrete-velocity approximations of the Boltzmann equation. At this stage we mention that the nonnegativity of the solution is fundamental, since nontrivial, partially negative solutions may exist. By the way, the nonnegativity of the solutions to the Boltzmann equation is also very important from the mathematical point of view: if it is not imposed, then irrelevant blow-up of the distribution function may occur. Very recently, Bobylev and Cercignani [81] made a considerable progress in this problem, proving the conjecture for so-called Maxwellian collision kernels, under the additional requirement that the solution has all of its moments finite for all time. Simultaneously, they discovered some very interesting self-similar eternal solutions of the Boltzmann equation with infinite kinetic energy, which may describe the asymptotic behavior of the Boltzmann equation in some physical regimes where the energy is very large. 44SeeSection3. 45Whichmaybe anevenmorestrikingmanifestationofirreversibility?
- 121. 118 C. Villani 3. Taxonomy After having introduced the main mathematical problems in the area, we are prepared for a more precise discussion of Boltzmann-like models. This section may look daunting to the non-specialist, and we shall try to spare him. The point is that there are, truly speaking, as many different Boltzmann equations as there are collision kernels. Hence it is absolutely necessary, before discussing rigorous results, to sketch a "zoological" classification of these. For simplicity, and because this case is much better understood, we limit the presentation to the classical, elastic case. For background about relativistic, quantum, nonelastic collision kernels, see Section 1.6, Chapter 2E, or the survey papers [209] and [146]. We shall conclude this section with a brief description of the main mathematical and physical effects that the choice of the collision kernel should have on the properties of the Boltzmann equation. 3.1. Kinetic and angular collision kernel Recall that the collision kernel B = B(v - v,, ~r) = B(lv - v,I, cos0) [by abuse of notations] only depends on Iv - v,I and cos0, the cosine of the deviation angle. Also recall that for a gas of hard spheres, B(Iv - v,l, cosO) -- glv - v,I, g>0, while for inverse s-power forces, B factors up like 8(1 - o,I, coso) = - I)b(cosO), (59) where ~(Izl) = Izl y, y = (s - 5)/(s - 1) in dimension N = 3, and b(cos0)sinN-2 0 ~, KO -(l+v), v = 2/(s - 1) in dimension N = 3 also. Exactly what range of values of s should be considered is by no means clear in the existing literature. Many authors [111,18,170] have restricted their discussion to s > 3. Klaus [288, p. 895] even explains this restriction by the impossibility of defining the Boltzmann linearized collision operator for s ~<3. However, as we shall explain, at least a weak theory of the Boltzmann equation can be constructed for any exponent s 6 (2, +cx~). The limit value s = 2 corresponds to the Coulomb interaction, which strictly speaking does not fit into the framework of the Boltzmann equation, as we have discussed in Section 1.7. REMARK. What may possibly be true, and anyway requires clarification, is that the derivation of the Boltzmann equation from particle systems may fail for s ~< 3, because of the importance of the mean-field interaction. But even in this case, the Boltzmann description of collisions should be rehabilitated in the investigation of the long-time behavior. Even though one is naturally led to deal with much more general collision kernels, products like (59) are the basic examples that one should keep in mind when discussing
- 122. A review of mathematical topics in collisional kinetic theory 119 assumptions. By convention, we shall call q0 the kinetic collision kernel, and b the angular collision kernel. We shall discuss both quantities separately. From the mathematical point of view, the control of Boltzmann's collision operator is all the more delicate that the collision kernel is "large" (in terms of singularities, or behavior as Iv- v,I ~ c~). On the contrary, when one is interested in such topics as trend to equilibrium, it is good to have a strictly positive kernel because this means more collisions; then the difficulties often come from the vanishings of the collision kernel. In short, one should keep in mind the heuristic rule that the mathematical difficulties encountered in the study of the Cauchy problem often come from large values of the collision kernel, those encountered in the study of the trend to equilibrium often come from small values of the collision kernel. 3.2. The kinetic collision kernel It is a well-established custom to consider the cases q~(Iv - v,I) = Iv - v,I • and to distinguish them according to 9 9/> 0: hard potentials; 9 Y = 0: Maxwellian potentials; 9 Y < 0: soft potentials. For inverse-power forces in dimension 3, hard potentials correspond to s > 5, soft potentials to s < 5. We shall stick to this convention, but insist that it is quite misleading. First of all, "hard potentials" are not necessarily associated to an interaction potential! It would be better to speak of "hard kinetic collision kernel". But even this would not be a neat classification, because it involves at the same time the behavior of the collision kernel for large and for small values of the relative velocity, which makes it often difficult to appreciate the assumptions really needed in a theorem. Sometimes a theorem which is stated for hard potentials, would in fact hold true for all kinetic collision kernels which are bounded below at infinity, etc. As typical examples, trouble for the study of the Cauchy problem may arise due to large relative velocities for hard potentials, or due to small relative velocities for soft potentials .... How positive may 9/be? For hard spheres, Y - 1, hence a satisfactory theory should be able to encompass this case. In many cases one is able to treat Y < 2 or 9/~< 2, or even less stringent assumptions. Conversely, how negative may 9/be? Contrarily to what one could think, critical values of the exponent s do not, in general, correspond to critical values of Y- As a striking example, think of Coulomb potential (s = N - 1), which normally should correspond to a power law Y = N/(N - 2) in dimension N. Besides the fact that this is meaningless when N - 2, this exponent is less and less negative as the dimension increases; hence the associated Cauchy problem is more and more easy because of the weaker singularity. The following particular values appear to be most critical: 9/ = -2, Y = -N. The appearance of the limit exponent -2 in the study of several mathematical properties [437, 247,248,446] has led us in [446] to suggest the distinction between moderately soft potentials (-2 < 9/< 0) and very soft potentials (Y < -2). It is however not clear whether
- 123. 120 C. Villani the border corresponds to a change of mathematical properties, or just an increase in difficulty. Note that dimension 3 is the only one in which the Coulomb potential coincides with the limit exponent -N, which makes its study quite delicate46 ! 3.3. The angular collision kernel We now turn to the angular collision kernel b(cos0) = b(k. or), k = (v - v.)/Iv - v.I. First of all, without loss of generality one may restrict the deviation angle to the range [0, zr/2], replacing if necessary b by its "symmetrized" version, [b(cos0) + b(cos(zr - 0))]10~<0~<~r/2. From the mathematical point of view, this is because the product ftf~ appearing in the Boltzmann collision operator is invariant under the change of variables ~r ~ -or; from the physical point of view this reflects the undiscernability of particles. As mentioned above, for inverse-power law forces, the angular collision kernel presents a nonintegrable singularity as 0 --+ 0, and is smooth otherwise. The fact that the collision kernel presents a nonintegrable singularity with respect to the angular variable is not a consequence of the choice of inverse-power forces, recall (15). By analogy with the examples of inverse-power forces, one would like to treat the following situations: b(cos0) sinN-2 0 ~ KO -(l+v) 0~<v<2. as 0 ~ 0, (60) Grad's angular cut-off [250,141 ] simply consists in postulating that the collision kernel is integrable with respect to the angular variable. In our model case, this means fo N_lb(k.cr)dff "-[sN-21 b(cosO)sinN-2OdO < 00. (61) The utmost majority of mathematical works about the Boltzmann equation crucially rely on Grad's cut-off assumption, from the physical point of view this could be considered as a short-range assumption. 3.4. The cross-section for momentum transfer Let M(lv - v,I) be defined by fs 1 U-, dcr B(v - v,, o')[v- v'] = -~(v- v,)M([v- v,[). (62) 46All the more that y = -3 also seems to have some special, bad properties independently of the dimension, see [429].
- 124. A review of mathematical topics in collisional kinetic theory 121 Indeed, by symmetry, the left-hand side is parallel to v - v,. The quantity M is called the collision kernel for momentum transfer. In our model case, M(Iv - v,[) -- #q0(lv - v,]), where f0 f /Z = [sN-2 [ b(cos0)(1 - cos0) sinN-2 0 d0. (63) From the physical point of view, the cross-section for momentum transfer is one of the basic quantities in the theory of binary collisions (see, for instance, [405,164]) and its computation via experimental measurements is a well-developed topic. On the other hand, the mathematical importance of the cross-section for momentum transfer has not been explicitly pointed out until very recently. From the mathematical point of view, finiteness of M (for almost all v - v,) is a necessary condition for the Boltzmann equation to make sense [450, Annex I, Appendix A], in the sense that if it is not satisfied, then Q(f, f) should only take values in {-oc, 0, +oc}. On the other hand, together with Alexandre we have recently shown [12] that suitable assumptions on M are essentially what one needs to develop a coherent theory for the Boltzmann equation without Grad's angular cut-off assumption. Note that in our model case, finiteness of M requires that b(cos0)(1 - cos 0), or equivalently b(cosO)O 2, be integrable on S N-1 . This precisely corresponds to the range of admissible singularities v ~ [0, 2). When v - 2, the integral (63) diverges logarithmically for small values of 0: this is one reason for the failure of the Boltzmann model to describe Coulomb collisions. Very roughly, the Debye truncature yields a finite/z, which behaves like the logarithm of the Debye length; this is what physicists call the Coulomb logarithm. And due to the fact that the divergence is only logarithmic, they expect the cross-section for momentum transfer to depend very little on the precise value of the Debye length. 3.5. The asymptotics of grazing collisions The mathematical links between the Boltzmann and the Landau collision operator can be made precise in many ways. As indicated for instance by Degond and Lucquin [162], for a fixed, smooth f, one can consider Q L(f, f) as the limit when e ~ 0 of a Boltzmann collision operator for Coulomb potential, with a truncated angular collision kernel, b~(cos0) -- log S-1 b(cos0) 10>e. The factor log e-1 compensates for the logarithmic divergence of the Coulomb cross- section for momentum transfer. As for the parameter e, it is proportional to the "plasma parameter", which is very small for classical plasmas, and actually goes to 0 as the Debye length goes to infinity. Also in the case of non-Coulomb potentials can one define an asymptotic regime in which the Boltzmann equation turns into a Landau equation. Such a formal study was
- 125. 122 C.Villani performed by Desvillettes [169]: he considered the limit of the Boltzmann operator when the collision kernel is of the form , sinN-2 0 be (cos 0) -- ~ sinn-2 - b cos 6 These two asymptotic procedures seem very different both from the mathematical and the physical point of view. Also, the truncation in the Coulomb case does not correspond to the Debye cut: indeed, for the Debye interaction potential, the collision kernel does not factor up. However, all limits of the type Boltzmann --+ Landau can be put into a unified formalism, first sketched in our work [446], then extended and made more precise in Alexandre and Villani [12]. The idea is that all that matters is that (1) all collisions become grazing in the limit, (2) the cross-section for momentum transfer keep a finite value in the limit. For simplicity we state precise conditions only in our model case where the collision kernel factors up as q~(Iv - v,l)be(cosO): u > 0, be(cos0) > 0 uniformly in 0 ~>00, 6-+0 f0 2" lZe ~ [sN-21 be(cos0) (1- cos0) sinN-2 0d0 >/z. e---~0 (64) This limit will be referred to as the asymptotics of grazing collisions. It can be shown that in this limit, the Boltzmann operator turns into the general Landau operator QL(f,f)--Vv'(fRNdV, a(v--v,)[f,(Vf)-- f(Vf),]), (65) ZiZj] aij(z) -- tit(Izl) 6ij izl2 , (66) taking into account the identity Izl2~(Izl) (67) q/(Izl) = 4(N- 1 - - - - - - - - - - - ~ According to (67), we shall use the terminology of hard, Maxwellian, or soft potentials for the Landau operator depending on whether q'(Izl) in (66) is proportional to Izl• with y > 0, y = 0, y < 0 respectively. Let us insist that the most relevant situation is the particular case introduced in (26)-(27) to describe Coulomb collisions. The general Landau operator (65) can be considered in several ways, - either like an approximation of the Landau equation with Coulomb potential: from the (theoretical or numerical) study of the corresponding equations one establishes results which might be extrapolated to the Coulomb case; - either as an approximation of the effect of grazing collisions in the Boltzmann equation without cut-off: one could postulate that such an operator may be well approximated by Q1 + Q2, where Q1 is a Boltzmann operator satisfying Grad's cut- off assumption and Q2 is a Landau operator (see [141] for similar considerations);
- 126. A reviewof mathematical topicsin collisionalkinetictheory 123 - or like a mathematical auxiliary in the study of the Boltzmann equation. This point of view will prove useful in Chapter 2C. To conclude this section, we mention that limits quite similar to the asymptotics of grazing collisions appear in the study of kinetic equations modelling quantum effects such as the Bose-Einstein condensation (like the Kompaneets equation, see Chapter 2E), as can be seen in the recent work of Escobedo and Mischler [209]. 3.6. What do we care about collision kernels ? We shall now try to explain in a non-rigorous manner the influence of the collision kernel (or equivalently, of the cross-section) on solutions to the Boltzmann equation. By the way, it is a common belief among physicists that the precise structure of the collision kernel (especially the angular collision kernel) has hardly any influence on the behavior of solutions. Fortunately for us mathematicians, this belief has proven to be wrong in several respects. We make it clear that the effects to be discussed only reflect the influence of the collision kernel, but may possibly come into conflict with boundary condition effects, 47 for instance. We also point out that although some illustrations of these effects are known in many regimes, none of them has been shown to hold at a satisfactory degree of generality. Distribution tails. First of all, let us be interested in the behavior of the distribution function for large velocities: how fast does it decay? can large distribution tails occur? The important feature here is the behavior of the kinetic collision kernel as the relative velocity Iv - v, I goes to infinity. If the collision kernel becomes unbounded at infinity, then solutions should be well-localized, and automatically possess finite moments of very high order, even if the initial datum is badly localized. 48 On the other hand, if the collision kernel decreases at infinity, then a slow decay at infinity should be preserved as time goes by. This effect is best illustrated by the dichotomy between hard and soft potentials. In certain situations, it has been proven that for hard potentials, no matter how slowly the initial datum decays, then at later times the solution has finite moments of all orders. For soft potentials on the other hand, a badly localized initial datum leads to a badly localized solution at later times. Precise statements and references will be given in Chapter 2B. As a consequence, it is also expected that the trend to equilibrium be faster for hard potentials than for soft potentials. We shall come back to this in Chapter 2C. Regularization effects. We now turn to the smoothness issue. Two basic questions are in order: if the initial datum is smooth, does it imply that the solution remains smooth? If the initial datum is nonsmooth, can however the solution become smooth? This time the answer seems strongly dependent on the angular collision kernel. If the angular collision kernel is integrable (Grad's cut-off assumption), then one expects that smoothness and 47In particular, it is not clear for us whether specular reflection in a non-convex domain would not entail appearanceof singularities.Thetroublescausedby non-convexdomainshavebeenwell-studiedin lineartransport theory,but not, to our knowledge,in the contextof the Boltzmannequation. 48See the heuristic explanationsin Section2.2 of Chapter 2B.
- 127. 124 C. Villani lack of smoothness are propagated in time. In other words, the solution at positive times should have precisely the same smoothness as the initial datum. The understanding of this property has made very significant progress in the past years. On the other hand, when the collision kernel presents a nonintegrable singularity, then the solution should become infinitely smooth for positive times, as it does for solutions to the heat equation. This idea emerged only in the last few years, and its very first mathematical implementation was done by Desvillettes [171] for a one-dimensional caricature of the Boltzmann equation. Since then this area has been very active, and nowadays the regularization effect begins to be very well understood as well. Yet much more is to be expected in this direction. Effects of kinetic singularity. If we summarize our classification of collision kernels, there are only three situations in which they can become very large: small deviation angles (as illustrated by the cut-off vs. non-cutoff assumption), large relative velocities (as illustrated by hard vs. soft potentials), and small relative velocities (as illustrated by hard vs. soft potential, but in the reverse way). If the influence of the former two is now fairly well understood, it is not so at all for the latter. It is known that a singularity of the collision kernel at small relative velocities is compatible with propagation of some smoothness, but no one knows if it preserves all the smoothness or if it entails blow-up effects in certain norms, or conversely regularization phenomena. In Chapter 2E, we shall say a little bit more on this issue, so far inexistent, and which we believe may lead to very interesting developments in the future. 4. Basic surgery tools for the Boltzmann operator Here we describe some of the most basic, but most important tools which one often needs for a fine study of the Boltzmann operator, Q(f, f) = ~ dv. fs N N-1 da B(v - v,, a)(f' ff, - f f,). Later on, we shall describe more sophisticated ingredients which apply in specific situations. 4.1. Symmetr&ation of the collision kernel In view of formulas (5), the quantity f'f',-ff, is invariant under the change of variables cr --~ -cr. Thus one can replace (from the very beginning, if necessary) B by its "symmetrized" version -B(z, or)- [B(z, or) + B(z,-a)] lz.~>0.
- 128. A review of mathematical topics in collisional kinetic theory 125 In other words, one can always assume the deviation angle 0 to be at most zr/2 in absolute value. This is why all spherical integrals could be written with an angular variable going from 0 to zr/2, instead of zr. From the physical point of view, this constatation rests on the undiscernability of particles (and this principle does not hold for mixtures). From the mathematical point of view, this trick is very cheap, but quite convenient when one wants to get rid offrontal collisions (deviation angle close to zr, which almost amounts to an exchange between the velocities). 4.2. Symmetric and asymmetric point of view There are (at least) two entirely different ways to look at the Boltzmann operator Q(f, f). The first is the symmetric point of view: the important object is the "tensor product" f f, - f | f, and the Boltzmann operator is obtained by integrating (f @ f)(v', v~,)- (f | f)(v, v,) with respect to the variable v, and the parameter or. This point of view is often the most efficient in problems which have to do with the trend to equilibrium, because the H theorem rests on this symmetry. On the other hand, one can consider Q(f, f) as the action upon f of a linear operator which depends on f: Q(f, f)= l~f(f). This introduces an asymmetry between f, (defining the operator) and f (the object on which the operator acts). This point of view turns out to be almost always the most effective in a priori estimates on the Boltzmann equation. For many asymmetric estimates, it is important, be it for the clarity of proofs or for the methodology, to work with the bilinear (but not symmetric!) Boltzmann operator Q(g' f) -- f•u dv*fsu-, dcr B(v - v,, cr)(g',f' - g, f). (68) Note that we have reversed the natural order of the arguments to make it clear that Q(g, f) should be understood as s (f) .... 4.3. Differentiation of the collision operator The following simple identities were proven in Villani [445] (but certainly someone had noticed them before): V Q+ (g, f) = Q+ (Vg, f) + Q+ (g, v f). (69) These formulas enable one to differentiate the collision operator at arbitrary order via a Leibniz-type formula.
- 129. 126 C. Villani 4.4. Joint convexity of the entropy dissipation Remarkably, Boltzmann's entropy dissipation functional if D(f)- -4 dvdv, da B(v - v,,a)(f' f~ - ff,) log ~ f'f', ff, is a convex functional of the tensor product f f, - but not a convex functional of f! This property also holds for Landau's entropy dissipation, which can be rewritten as if DL(f)- -~ dvdv, q~(Iv- v,I)I/7(v- v,)(V- V,)(ff,) [2 ff, so that convexity of DL results from convexity of the function (x, y)~ [xl2/y in ~N X]~+. Such convexity properties may be very interesting in the study of some weak limit process, because weak convergence is preserved by tensor product. But beware! f f, is a tensor product only with respect to the velocity variable, not with respect to the x variable. 4.5. Pre-postcollisional change of variables A universal tool in the Boltzmann theory is the involutive change of variables with unit Jacobian49 ' k) (v, v,, a) ~ (v', v,, , (70) where k is the unit vector along v - v,, k l) m U, [v - - V,[ Since a (vt I vt = - v,)/[ - v,[, the change of variables (70) formally amounts to the exchange of (v, v,) and (vt, v~,).As a consequence, under suitable integrability conditions on the measurable function F, f F(v, v,, v', v',)B(lv- v,l,k.a)dvdv, da -f -f F(v, v,, v', vt,)B(lv - v,[, k. a)dr' dye,dk ! F(v', v,, v, v,)B(Iv- v,l,k.a)dvdv, da. 49A way to see that this change of variables has unit Jacobian is to use the w-representation of Section 4.6. In this representation, very clearly the pre-postcollisional change of variables has unit Jacobian; and the Jacobian from the a-representation to the co-representation is the same for pre-collisional and for post-collisional velocities.
- 130. A review of mathematical topics in collisional kinetic theory 127 Here we have used Iv' ' - v,I- Iv- v,I, a .k -- k.a to keep the arguments of B unchanged; also recall the abuse of notations B(v - v,, a) = B(Iv - v,I, k. a). ! Note that the change of variables (v, v,) --->(v~, v,) for given a is illegal V 4.6. Alternative representations There are other possible parametrizations of pre- and post-collisional velocities. A very popular one is the co-representation, v' = v - (v - v,, o9)o9, ! V, -- V, + (V -- V,, CO) CO, COE S N-1 . (71) In this representation, the bilinear collision operator5~ reads 1/ Q(g, f) - -~ dr, do) B(v - v,, co)(g~,f' - g, f), (72) where - B (z, co) -- 2 N-2 B(z,a). ! We have kept a factor 1/2 in front of B to recall that each pair (v~, v,) corresponds to two distinct values of co. One of the advantages of the co-representation is that it is possible to change variables l (v, v,) +, (v~, v,) for fixed co and this is again an involutive transformation with unit Jacobian. Another advantage is that it is a linear change of variables. Yet, as soon as one is interested in fine questions where the symmetries of the Boltzmann operator play an important role, the o--representation is usually more convenient. A third representation is the one introduced by Carleman [119], particularly useful for the study of the gain operator Q+ when the collision kernel satisfies Grad's angular cut- the off. The principle of Carleman's representation is to choose as new variables v~and v,, I pre-collisional velocities. Of course, not all values of v~and v, are admissible. If v and v~ is the hyperplane Evv,, orthogonal to are given, then the set of admissible velocities v, v - v~and going through v. Using the identity v - v, -- 2v - v~- v,,~one gets Q(g, f) 1 , , -v, - dv ~ dye, B 2v-v -v,, N IV l)tl N- , IVt t vv t -- __ V,I x [g(v~,)f(v ') - g(v' + v~, - v)f(v)]. (73) 50All the representations formulas below for Q also work just the same for its gain and loss terms (Q + and Q- ) separately, with obvious changes.
- 131. 128 C. Villani To conclude this section, we mention Tanaka's representation [415], which is equivalent to Maxwell's weak formulation: Q+ (g, f) = f dv dr, g, f (Fl~v,v, - 17v,v,), (74) where Fly,v, (resp./7v~,v,) is a measure on the sphere S N-l, l~v,v, = B(v - v,, o")do"6v (resp. Fl~,v, = B(v - v,, o") do" av,). 4.7. Monotonicity Each time one has to handle an expression involving a nonnegative integrand and the collision kernel, it may be useful to consider it as a monotone function of the collision kernel. This point of view is particularly interesting for the entropy dissipation (47), which obviously is an increasing function of the collision kernel. Therefore, to bound (47) from below for a given collision kernel B, it is sufficient to bound it below for an auxiliary, simplified collision kernel B0 such that B ~>B0. Most of the time, the "simplified" collision kernel will be a Maxwellian one. As we shall see in Chapter 2D, Maxwellian collision kernel have specific properties. 4.8. Bobylev's identities We now turn to more intricate tools introduced by Bobylev. Even though the Boltzmann operator has a nice weak formulation (Maxwell's formula in Section 2.3), it is a priori quite painful to find out a representation in Fourier space. It turns out that such a representation is not so intricate, at least when the collision kernel is Maxwellian! This fact was first brought to the attention of the mathematical community by Bobylev, who was to make Fourier transform an extremely powerful tool in the study of the Boltzmann operator with Maxwellian collision kernel (see the review in [79]). Here is Bobylev's identity: let b(cos 0) be a collision kernel depending only on the cosine of the deviation angle, and let Q(f' f)-- ~u • dr, do" b(cosO)[f' f~, - f f,] be the associated Boltzmann operator. Then its Fourier transform is :)d: (75) where f stands for the Fourier transform of f, ~ is the Fourier variable, and ~i_ ~ + I~1~ - --------~--- 9 (76)
- 132. A reviewof mathematicaltopicsin collisionalkinetictheory 129 Note that I~+ 124- 1~-12 -- 1. A remarkable feature about (75) is that the integral is now (N - 1)-fold, instead of (2N - 1)-fold. This formula is actually a particular case of a more general one which does not assume Maxwellian collision kernel [10, Appendix]" 1 , ) f'[Q(g, f)](,e) _ (2n') N/2 N• , ]-~ 9O" [~(,e- 4- ~,)f(~+ _ ~,) -- i(~,)f(~ -- ~,)] d~, dcr, (77) A where the Fourier transform B of B = B(Iz[, cos0) is with respect to the variable z only. Of course, in the particular case B(Izl, cos0) = b(cos0), we have B(I~.I, cos0) -- (27r)N/26[~. -- 0] b(cos0), and this entails formula (75). Thus we see that the reduction of the multiplicity in the integral is directly linked to the assumption of Maxwellian collision kernel. As a consequence of (75), results Bobylev's lemma 51" if Q is a Boltzmann operator with Maxwellian collision kernel, then, whatever the Maxwellian probability distribution M, Q(g 9 M, f 9 M) -- Q(g, f) 9 M. This is a very useful regularization lemma when dealing with Maxwellian collision kernels. 4.9. Application of Fourier transform to spectral schemes Here we digress a little bit to briefly discuss numerical schemes based on Fourier transform, which are related to Bobylev's ideas. Here are the main ideas of these "spectral schemes": (1) truncate the support of the distribution function f, then extend f into a periodic function on ]I~N; (2) expand f in Fourier series, and compute the expression of the collision operator Q(f, f) in terms of the Fourier coefficients of f. Special attention must be given to the way the support is truncated! As explained in [372], for instance, if the support of f is reduced to a compact set with diameter R, then it should be extended by periodicity with period T ~> (2 4- ~/2)R, in order to avoid overlap problems in the computation of the collision integral. Assume, for instance, T -- Jr, R = Mr, )~= 2/(3 + ~/2). After passing in Fourier representation, m (27r)N ~,Yr] N f (v)e -ik'v dr, 51Thislemmawas actuallyproven,for aconstantcollisionkernel,byMorgenstern[351,Section 10] in the fifties! However,Bobylevwas the author who devised a generalproof, madethis lemmawidely known and linked it to otherpropertiesof Maxwellian collisionkernels.
- 133. 130 C. Villani and truncating high Fourier modes, a very simple expression is obtained for the k-th mode Q(k) of Q(f, f)" Q'(k) = Z fg J~/~(g' m), Ikl ~<K, g§ Igl,lml <<. K where /~(g, m) -- fzl~2)vTr dzfSN-1do"B(z, o")[e i~'z++im'z- - eiz'm], zi z+lzl~ = ~ . (78) 2 Of course, this formula is very much reminiscent of (77). Spectral schemes have several advantages: once in Fourier space, the numerical simulation is immediate Oust a simple first-order system of coupled ODE's). Moreover, all coefficients/~(g, m) can be computed once for all with extreme precision; this may demand some memory space, but is quite a gain in the speed of the computation. They are able to give extremely precise results. However, these schemes are rather rigid and do not allow for all the numerical tricks which can be used by other methods to reduce computational time. Moreover, since conservation laws are not built in the schemes, numerical simulations have to be conducted with a certain minimal precision in order to get realistic results. From the mathematical point of view, this method presents very interesting features, think that it simulates the Boltzmann equation in weak formulation. In particular, the method works just the same with or without Grad's angular cut-off assumption. This can be used in theoretical works like [374] (analysis of the behavior of the method in the asymptotics of grazing collisions). 5. Mathematical theories for the Cauchy problem In this section, we try to survey existing mathematical frameworks dealing with the Cauchy problem (and also more quantitative issues) for Boltzmann-like equations. These theories are all connected, but more or less tightly. Before this, in the first section we describe the most apparent problems in trying to construct a general, good theory. 5.1. What minimalfunctional space ? In the full, general situation, known a priori estimates for the Boltzmann equation are only those which are associated to the basic physical laws: 9 formal conservation of mass and energy, 9 formal decrease of the entropy. Of course, the latter means two estimates! an estimate on the entropy, and another one on the dissipation of entropy.
- 134. A reviewof mathematicaltopicsin collisionalkinetictheory 131 When the position space is unbounded, say the whole of ~N, then the properties of the transport operator also make it rather easy to get local (in time) estimates on f f (t, x, v)Ix 12dx all3. For this it suffices to use the identity df dt f (t, x, v)lx - vtl 2 dx dv - O. On the whole, one gets for free the a priori estimates finN • RN f(t,x, v)[1 + Ivl2 -t-Ixl 2 -t-logf(t,x, v)]dxdv fots + D(f(r, x, .)) dr dx N fRN • f(O,x, v)[1 + 21xl2 + (2t 2 + 1)lvl 2 + log f(0, x, v)]dxdv. (79) Apart from the term in D(f), estimate (79) does not depend on the collision kernel B. Disregarding this entropy dissipation estimate which is difficult to translate in terms of size and smoothness of the distribution function, all that we know for the moment about f is f E L~ T]; L~(IRN x NN) n LlogL(R N x IKf)). Here IIf IIL~ - fRN ~RN f(x, v)(1 + Ixl2 -+-Ivl2) dx dr. It is easily seen that the estimate in L log L, combined with the moment estimate, entails an L 1 control of f (log f)+. REMARK. One may be interested in situations where the total mass is infinite (gas in the whole space). The estimates can then be adapted to this situation: see in particular Lions [310]. By the Dunford-Pettis compactness criterion, sequences of solutions to the Boltz- mann equation with a uniform entropy estimate will be weakly compact, say in N L p ([0, T]; L l(lt{x N x 11~ v )), and this ensures that their cluster points cannot be singular measures. This looks like a mathematical convenience: after all, why not try to handle the Boltzmann equation in a genuine measure setting, which seems to best reflect physical intuition? It turns out that the space of measures is not stable 52 - and thus irrelevant- for the study of the Boltzmann equation. This can be seen by the following remark. Consider 52On the other hand, the space of functions with bounded entropy is stable, as shown by the works of DiPerna and Lions discussed below.
- 135. 132 C. Villani as initial datum a linear combination of Dirac masses approximating a continuous profile (think of each of these Dirac masses as clusters of particles with a common velocity, each cluster being located at a different position). Even though the meaning of the Boltzmann equation for so singular data is not a priori clear, it is easy to figure out what weak solution we should expect: each cluster should keep moving with constant velocity, until two clusters happen to be in the same position and collide. Should this eventuality occur, particles in each cluster would be scattered in all directions, with respective probabilities given by the Boltzmann collision kernel. But the point is, by modifying very slightly the initial positions of the clusters, one can always make sure (in dimension N/> 2) that no collision ever occurs! Therefore, this example shows that one can construct a sequence of weak solutions of the Boltzmann equation, converging in weak measure sense towards a continuous solution of the free transport equation. Thus there is no stability in measure space .... It would be tedious, but quite interesting, to extend this counterexample to absolutely continuous initial data, say linear combinations of very sharply peaked Maxwellians. The aforegoing discussion is strongly linked to the fact that both x and v variables are present in the Boltzmann equation. If one is interested in solutions which do not depend on the space variable, then it is perfectly possible to construct a meaningful theory of measure solutions, as first noticed by Povzner [389]. To come back to our original discussion, we have just seen that it is in general impossible to deal with arbitrary measures, and therefore the entropy estimate is welcome to prevent concentration. But is it enough to handle the Boltzmann operator? Very very roughly, it seems that to control the Boltzmann operator, we would like to have a control of f dr, ff,, which is just IIf IILLf. The L lx,v norm of this quantity is just the norm of f in L2(L 1). This is the kind of spaces in which we would like to have estimates on f. But such estimates are a major open problem! It seems that the entropy dissipation estimate is not sufficient for that purpose, one of the troubles being caused by local Maxwellian states, which make the entropy dissipation vanish but can have arbitrarily large macroscopic densities. Thus the only a priori estimates which seem to hold in full generality do not even allow us to give a meaningful sense to the equation we wish to study.., this major obstruction is one of the reasons why the Cauchy problem for the Boltzmann equation is so tricky - another reason being the intricate nature of the Boltzmann operator. Due to this difficulty, most theories only deal with more or less simplified situations. On the other hand, for simpler models like the BGK equation, in which the collision operator is homogeneous of degree 1, the Cauchy problem is much easier [375,378,398]. By the way, the lack of a priori estimates for the Boltzmann equation is also very cumbersome when dealing with boundary conditions, especially in the case of Maxwellian diffusion. Indeed, the treatment of boundary conditions in a classical sense requires the trace of the solution to be well-defined on the boundary, which is not trivial [271,149]. Weak formulations are available (see in particular Mischler [347]), but rather delicate. We distinguish six main theories, inequally represented and inequally active - and by no means hermetically independent, especially in recent research. By order of historical appearance: spatially homogeneous theory, theory of Maxwellian molecules, perturbative theory, solutions in the small, renormalized solutions, one-dimensional problems.
- 136. A review of mathematical topics in collisional kinetic theory 133 5.2. The spatially homogeneous theory In the spatially homogeneous theory one is interested in solutions f(t, x, v) which do not depend on the x variable. This approach is rather common in physics, when it comes to problems centered on the collision operator: this looks reasonable, since the collision operator only acts on the velocity dependence. Moreover, the spatially homogeneous theory naturally arises in numerical analysis, since all numerical schemes achieve a splitting of the transport operator and the collision operator. Finally, it is expected that spatial homogeneity is a stable property, in the sense that a weakly inhomogeneous initial datum leads to a weakly inhomogeneous solution of the Boltzmann equation. Under some ad hoc smallness assumptions, this guess has been mathematically justified by Arkeryd et al. [32], who developed a theory for weakly inhomogeneous solutions of the full Boltzmann equation. Thus, the spatially homogeneous Boltzmann equation reads af = Q(f, f), (80) ot and the unknown f = f (t, v) is defined on IK+ • ]~N. Note that in this situation, the only stationary states are Maxwellian distributions, by the discussion about Equation (54) in Section 2.5. The spatially homogeneous theory was the very first to be developed, thanks to the pioneering works by Carleman [118] in the thirties. Carleman proved existence and uniqueness of a solution to the spatially homogeneous problem for a gas of hard spheres, and an initial datum f0 which was assumed to be radially symmetric, continuous and decaying in O(1/Ivl 6) as Ivl --+ +c~. He was also able to prove Boltzmann's H theorem, and convergence towards equilibrium in large time. Later he improved his results and introduced new techniques in his famous treatise [119]. Then in the sixties, Povzner [389] extended the mathematical framework of Carleman and relaxed the assumptions. In the past twenty years, the theory of the Cauchy problem for spatially homogeneous Boltzmann equation for hard potentials with angular cut-off was completely rewritten and extensively developed, first by Arkeryd, then by DiBlasio, Elmroth, Gustafsson, Desvillettes, Wennberg, A. Pulvirenti, Mischler [17,20,23,186,187,204,269,270,170,458, 456,457,392,393,349]. It is now in a very mature state, with statements of existence, uniqueness, propagation of smoothness, moments, positivity .... Optimal conditions for existence and uniqueness have been identified by Mischler and Wennberg [349,463]. Recent works by Toscani and the author [428] have led to almost satisfactory results about the H theorem and trend to equilibrium, even though some questions remain unsettled. Also, one can work in a genuine measure framework. The study of singular kernels (be it soft potentials or potentials with nonintegrable angular singularities) is much more recent [18,172,171,173,446,449,10]. This area is still under construction, but currently very active. In view of the last advances, it is quite likely that very soon, we shall have a fairly complete picture of the spatially homogeneous theory, with or without cut-off, at least when the kinetic collision kernel is not too singular for small relative velocities. A review on the state of the art for the spatially homogeneous theory is performed in Desvillettes [174].
- 137. 134 C. Villani A task which should be undertaken is to systematically extend all of these achievements to the framework of weakly inhomogeneous solutions [32]. 5.3. Maxwellian molecules After spatial homogeneity, a further simplification is the assumption that the collision kernel be Maxwellian, i.e., do not depend on the relative velocity but only on the (cosine of the) deviation angle. The corresponding theory is of course a particular case of the preceding one, but allows for a finer description and presents specific features. It was first developed by Wild and Morgenstern in the fifties [464,350,351]. Then Truesdell [274] showed that all spherical moments of the solutions could be "explicitly" computed. Simple explicit solutions, important for numerical simulations, were produced independently by Bobylev [78] and Krook and Wu. Later, Bobylev set up and completed an ambitious program based on the Fourier transform (see the survey in [79]). Included were the classification of several families of semi-explicit solutions, and a fine study of trend to equilibrium. Key tools in this program were the identities of Section 4.8. As of now, the theory can be considered as complete, with the exception of some non-standard problems like the identification of the image of Q or the classification of all eternal solutions53 .... 5.4. Perturbation theory Another regime which has been extensively studied is the case when the distribution function is assumed to be very close to a global Maxwellian equilibrium, say the centered, unit-temperature Maxwellian M. Under this assumption, it is natural to try to linearize the problem, in such a way that quadratic terms in Boltzmann's operator become negligible. In order to have a self-adjoint linearized Boltzmann operator, the relevant change of unknown is f = M(1 4- h). Then, one can expand the Boltzmann operator Q(f, f) by using its bilinearity, and the identity Q(M, M) = 0. This leads to the definition of the linearized Boltzmann operator, Lh -- M-I[Q(M, Mh) 4- Q(Mh, M)]. L is a symmetric, nonpositive operator in L2(M) (endowed with the scalar product (hi, h2)M -- f hlh2M), nonpositivity being nothing but the linearized version of the H theorem. Moreover, the remainder in the nonlinear Boltzmann operator, Q(Mh, Mh), can be considered small if h is very close to 0 in an appropriate sense. Note that smallness of h in LZ(M) really amounts to smallness of f - M in L2(M-1), which, by the standards of all other existing theories, is an extremely strong assumption! The linearized Boltzmann equations also have their own interest, of course, and the spectral properties of the linearized Boltzmann operator have been addressed carefully; 53Theseproblemshavebeenexplainedin Section2.9.
- 138. A review of mathematical topics in collisional kinetic theory 135 see [141] for discussion and references. The different analysis required by the linearized Landau equation is performed in [161,298]. A remarkable feature is that when the collision kernel is Maxwellian, then the spectrum can be computed explicitly (and eigenfunctions too: they are Hermite polynomials). This calculation was first performed in a classical paper by Wang Chang and Uhlenbeck [454], then simplified by Bobylev [79, p. 135] thanks to the use of Fourier transform. Grad has set up the foundations for a systematic study of the linearized Boltzmann equation, see [250,252,255]. At the same time he initiated in this perturbative framework a "rigorous" study of hydrodynamical equations based on Chapman-Enskog or moments expansion [249,253]. Later came the pioneering works of Ukai [434,435] on the nonlinear perturbation [434,435], followed by a huge literature, among which we quote [333,359, 111,437,403,436,286]. This theory is now in an advanced stage, with existence and uniqueness theorems, and results of trend to equilibrium. The proofs often rely on the theory of linear operators, abstract theory of semigroups, abstract Cauchy-Kowalewski theorems. As we just said, with the development of this branch of the Cauchy problem for the Boltzmann equation, came the first rigorous discussions on the transition to hydrodynamical equations, in a perturbative framework, after a precise spectral analysis of the linearized operator was performed. On this approach we quote, for instance, [112,113, 159,203,358,438,44,334,58,43,212,213 ]. Actually, in the above we have mixed references dealing with stationary and with evolutionary problems, for which the setting is rather similar .... An up-to-date account of the present theory can be found in [214]. However, all known results in this direction deal with smooth solutions of the hydrodynamic equations. Also related to this linearized setting is a large literature addressing more qualitative issues, like half-space problems, to be understood as a modelization for kinetic layers,54 or the description of simple shock waves. Among many works, here are a few such papers: [42,357,115,258,52,51,332,155,151,188,214] (the work [37] is an exception, in the sense that it deals with the Milne problem in the fully nonlinear case). It is certainly mathematically and physically justified to work in a perturbative setting, as long as one keeps in mind that this only covers situations where the distribution function is extremely close to equilibrium. Thus this theory can in no way be considered as a general answer to the Cauchy problem. Even taking this into account, some criticisms can be formulated, for instance, the use of abstract spectral theory which often leads to nonexplicit results. Also, we note that the great majority of these works is only concerned with hard potentials with cut-off, and most especially hard spheres. Early (confusing) remarks on the Boltzmann operator without cut- off can be found in Pao [369] and Klaus [288], but this is all. Soft potentials with cut-off have been studied by Caflisch [111], Ukai and Asano [437]. 54A kinetic layer describes the transitionbetween a domainof space where a hydrodynamicdescriptionis relevant, and anotherdomainin which a moreprecisekinetic descriptionis in order;for generalbackground see [148].
- 139. 136 C. Villani Theoretical research in the area of linearized or perturbative Boltzmann equation is not so intensive as it used to be.* We shall not come back to this theory and for more details we address the reader to the aforementioned references. 5.5. Theories in the small Another line of approach deals with short-time results. This may seem awkward for statistical equations which are mostly interesting in the long-time limit! but may be interesting when it comes to validation issues. Conversely- but this is more delicate-, it is possible to trade the assumption of small time for an assumption of small initial datum expanding in the vacuum. This case could be described as perturbation of the vacuum, and exploits the good "dispersive" properties of the transport operator. The modern approach starts with the classical papers of Kaniel and Shinbrot for the small-time result [285], Illner and Shinbrot for the small-datum result [278]. Also it should be clear that Lanford's theorem55 contains a proof of local in time existence under rather stringent assumptions on the initial datum. It was computed that Lanford's bounds allowed about 15% of the particles to collide at least once! The extensions of Lanford's results by Illner and Pulvirenti [275,276] were also adaptations of small-datum existence theorems. Theories in the small were further developed by Toscani and Bellomo [64,425,418- 420,368], some early mistakes being corrected by Polewczak [385] who also proved smoothness in the x variable on this occasion. See the book of Bellomo, Palczewski and Toscani [63] for a featured survey of known techniques at the end of the eighties. Then the results were improved by Goudon [246,247] (introducing some new monotonicity ideas), and also Mischler and Perthame [348] in the context of solutions with infinite total kinetic energy. One of the main ideas is that if the initial datum is bounded from above by a well-chosen Maxwellian (or squeezed between two Maxwellians), then this property remains true for all times, by some monotonicity argument. Therefore, solutions built by these methods usually satisfy Gaussian-type bounds. To get an idea of the method, a very pedagogical reference is the short proof in Lions [308] which covers smooth, fast-decaying collision kernels. Bellomo and Toscani have also studied cases where the decay of the solution is not Gaussian, but only polynomial. It is in this framework that Toscani [419] was able to construct solutions of the Boltzmann equation in the whole space, which do not approach local equilibrium as time becomes large.56 From the mathematical point of view, these theories cannot really be considered in a mature stage, due to a certain rigidity. For instance, it is apparently an open problem to treat boundary conditions: only the whole space seems to be allowed. Also, since the *Note addedinproof: Aftercompletionofthisreview,twoimportantworksbyGuo,abouttheLandauequation and the Boltzmannequation for soft potentials, in a close-to-equilibrium, periodic setting, popped up just to contradictthisstatement. 55See Section2.1. 56SeeSection2.5.
- 140. A review of mathematical topics in collisional kinetic theory 137 proofs strongly rely on Grad's splitting between the gain and loss part, the treatment of non-cutoff potentials is open. Finally, the limitations of small time, or small initial datum, are quite restrictive, even though not so much as for the linearized theory (the bounds here have the noticeable advantage to be explicit). However, as mentioned above, theories in the small have led to two quite interesting, and physically controversial, results: the validation of the Boltzmann equation for hard-spheres via the Boltzmann-Grad limit, at least in certain cases; and a simple construction of solutions of the Boltzmann equation which do not approach local equilibrium as time becomes large. Recent works by Boudin and Desvillettes [101] in this framework also resulted in interesting proofs of propagation of regularity and singularities, which were implicitly conjectured by physicists [148]. We shall not develop further on this line of approach, and address the reader to the above references for more. 5.6. The theory of renormalized solutions Introduced by DiPerna and Lions at the end of the eighties, this theory is at the moment the only framework where existence results for the full Boltzmann equation, without simplifying assumptions, can be proven [190,192,194,167,307-309,311,310,306,316,12, 13]. Apart from a high technical level, this theory mainly relies on two ingredients: 9 the velocity-averaging lemmas: under appropriate conditions, these lemmas, initiated by Golse, Perthame and Sentis [243] and further developed in [242,232,195,379, 99,312,315], yield partial regularity (or rather regularization) results for velocity- averages f g(t, x, v)qg(v) dv of solutions of transport equations Og/Ot + v. Vxg = S. The regularity is of course with respect to the time-space variables, and thus the physical meaning of these lemmas is that macroscopic observables enjoy some smoothness properties even when the distribution function itself does not. 9 the renormalization: this trick allows one to give a distributional sense to the Boltzmann equation even though there does not seem to be enough a priori estimates for that. It consists in formally multiplying (8) by the nonlinear function of the density, fll (f), where 13belongs to a well-chosen class of admissible nonlinearities. By chain- rule, the resulting equation reads Off(f) Ot + v. Vxfl (f) = fl' (f) Q (f, f). (81) Assume now that Ifi'(f)l ~ C/(1 -+-f), for some C > 0. Then, since the Boltzmann operator Q(f, f) is quadratic, one may expect fl'(f)Q(f, f) to be a sublinear operator of f... in which case the basic a priori estimates of mass, energy and entropy57 would be enough to make sense of (81). Distribution functions satisfying (81) in distributional sense are called renormalized solutions. Strictly speaking, these solutions are neither weaker, nor stronger than distributional solutions. Typical choices for/3 are fl (f) = 3-1 log(1 + 6f) (3 > 0) or/3 (f) = f~ (1 + 6f). 57See Section5.1.
- 141. 138 C. Villani Apart from the study of the Boltzmann equation, renormalization and velocity-averaging lemmas have become popular tools for the study of various kinetic equations [191, 375,318,176,229], ordinary differential equations with nonsmooth (Sobolev-regular) coefficients [193,97], or the reformulation of some hyperbolic systems of conservation laws as kinetic systems [319,320,280,381,380].* The idea of renormalizafion has even been exported to such areas of partial differential equations as nonlinear parabolic equations (see [76,77] and the references therein). In fact, renormalization is a general tool which can be applied outside the field of renormalized solutions; in this respect see the remark at the level of formula (130). As regards the Boltzmann equation, many fundamental questions are still unsolved: in particular uniqueness, propagation of smoothness, energy conservation, moment estimates, positivity, trend to equilibrium .... Therefore, as of this date, this theory cannot be considered as a satisfactory answer to the Cauchy problem. However, it provides a remarkable answer to the stability problem. The techniques are robust enough to adapt to boundary-value problems [271,144, 30,347,346] (be careful that some of the proofs in [271] are wrong and have been corrected in [30]; the best results are those of Mischler [347]). As an important application of the theory of renormalized solutions, Levermore [302] proved the validity of the linearizafion approximation if the initial datum is very close to a global Maxwellian. Also the hydrodynamical transition towards some models of fluid mechanics can be justified without assumption of smoothness of the limit hydrodynamic equations: see, in particular, Bardos, Golse and Levermore [57,55,54,53], Golse [239], Golse et al. [241], Golse and Levermore [240], Lions and Masmoudi [317], Golse and Saint-Raymond [245], Saint- Raymond [401]. The high point of this program is certainly the rigorous limit from the DiPerna-Lions renormalized solutions to Leray's weak solutions of the incompressible Navier-Stokes equation, which was performed very recently in [245]; see [441] for a review. The original theory of DiPerna and Lions heavily relied on Grad's cut-off assumption, but recent progress have extended it to cover the full range of physically realistic collision kernels [12]. This extended theory has set a framework for the study of very general effects of propagation of "regularity", in the form of propagation of strong compactness [308], or "smoothing", in the form of appearance of strong compactness [311,316,12,13]. Moreover, even if a uniqueness result is not available, it appears that renormalized solutions are strong enough to prove some results of weak-strong uniqueness [308,324]: under certain assumptions on the collision kernel, if we know that there exists a strong solution to the Boltzmann equation, then there exists a unique renormalized solution, and it coincides with the strong solution. On the occasion of this study, Lions [308] pointed out the possibility to construct very weak solutions, called "dissipative solutions", which are of very limited physical value, but have been used in various areas as a powerful tool for treating some limit regimes, be it in fluid mechanics for such degenerate equations as the three-dimensional Euler equation [313], in hydrodynamical limits [239,241,401 ] or stochastic fully nonlinear partial differential equations [321,322]. Thus ramifications of the DiPerna-Lions theory have been a source of inspiration for problems outside the field. *Note added in proof: Recently,Bouchuthas shownhow to use velocity-averaginglemmasto studyclassical hypoellipticityin certainkineticequations.
- 142. A review of mathematical topics in collisional kinetic theory 139 In view of these achievements and of the current vitality of the theory of renormalized solutions, we shall come back to it in more detail in the next chapters. 5.7. Monodimensional problems It is of course impossible to speak of a monodimensional Boltzmann equation, since elastic collisions are meaningless in dimension 1. But in many problems of modelling [148], symmetry assumptions enable one to consider solutions depending on the position in space, x, through only one variable. From the mathematical point of view, such problems seem to present specific features, one of the reasons being that the dispersive power of the transport operator is very strong in dimension 1, so that dispersion estimates can be used to (almost) control the collision operator. In the end of the eighties, Arkeryd [22] was able to apply a contraction method similar to the one in [24] in order to get existence results for the Boltzmann equation in one dimension of space, however he needed a physically unrealistic damping in the collision operator for small relative velocities in the space direction. Then, building on original works by Beale [61] and especially Bony [94,95] on discrete-velocity Boltzmann equations, Cercignani [145,147] was able to extract some new estimates in this one-dimensional situation, and prove existence of "strong" solutions to the Boltzmann equation, under rather stringent assumptions on the collision kernel. Here "strong" means that Q+(f, f) E L~oc(RX • RN). For some time this line of research was quite promising, but it now seems to be stalled ....
- 143. CHAPTER 2B Cauchy Problem Contents 1. Use of velocity-averaging lemmas ...................................... 143 1.1. Reminders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 1.2. How to use velocity-averaging lemmas in the Boltzmann context? .................. 145 1.3. Stability/propagation/regularization ................................... 146 2. Moment estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 2.1. Maxwellian collision kernels ...................................... 148 2.2. Hard potentials .............................................. 148 2.3. Soft potentials .............................................. 150 2.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 3. The Grad's cut-off toolbox .......................................... 153 3.1. Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.2. Control of Q + by Q- and entropy dissipation ............................ 154 3.3. Dual estimates .............................................. 155 3.4. Lions' theorem: the Q+ regularity ................................... 156 3.5. Duhamel formulas and propagation of smoothness .......................... 157 3.6. The DiPerna-Lions renormalization .................................. 159 3.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4. The singularity-hunter's toolbox ....................................... 165 4.1. Weak formulations ............................................ 166 4.2. Cancellation lemma ........................................... 168 4.3. Entropy dissipation estimates ...................................... 170 4.4. Boltzmann-Plancherel formula ..................................... 172 4.5. Regularization effects .......................................... 173 4.6. Renormalized formulation, or F formula ................................ 175 4.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5. The Landau approximation .......................................... 180 5.1. Structure of the Landau equation .................................... 180 5.2. Reformulation of the asymptotics of grazing collisions ........................ 181 5.3. Damping of oscillations in the Landau approximation ......................... 183 5.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6. Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.1. Mixing effects .............................................. 185 6.2. Maximum principle ........................................... 186 6.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 141
- 144. Preface The motion of fluids has intrigued scientists since antiquity and we may say that the field of mathematical fluid dynamics originated more than two centuries ago. In 1755 Euler [2] gave a mathematical formulation of the principle of conservation of mass in terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear and angular momentum by PDEs. Material symmetry and frame invariance were used by Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint- Venant [11] and Stokes [12]. In the 19th century no sharp distinction was drawn between mathematicians and physicists as we sometime see in more recent times. The formulation of the equations of fluid motion could be considered as either mathematics or physics. The first work in fluid dynamics that has a "modern" mathematical flavor may have been done by Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous problem. Riemann recognized the mathematical nature of the entropy. This notion led him to his duality method for solving the non-characteristic Cauchy problem for linear hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his contemporaries. What we now call the Cauchy problem for a PDE and the search for its solution did not have the significance that it is accorded nowadays. Even Poincar6 did not raise that kind of question in his Th~orie des tourbillons. For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28. Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much as Riemann, Leray developed new mathematical tools which proved to have independent interest: e.g., weak solutions (that we now call Leray's solutions in this context) and topological degree (a joint work with Schauder [4]). Since the 1930s, the interest that mathematicians devote to fluid dynamics has unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic displays such a variety of models and questions that thousands of scientists, among them many mathematicians, focus their research on fluid dynamics.
- 145. A review of mathematical topics in collisional kinetic theory 143 The meaning of"Cauchy problem" in this chapter is to be understood in an extended sense: we shall not only be concerned in existence and uniqueness of solutions, but also in a priori estimates. Three main issues will be addressed: decay of the solutions at large velocities (and also at large positions, but large velocities are the main concern), smoothness, and strict positivity. As we explained above, the decay of the solutions mainly depends on the behavior of the kinetic collision kernel, while their smoothness heavily relies on the angular collision kernel. As for the strict positivity, the matter is not very clear yet. We have adopted the following presentation: first, we recall a bit about velocity- averaging lemmas, which have become a universal tool in the study of transport equations, and we shall comment on their use in the particular context of the Boltzmann equation. In Section 2, we address moment estimates, and discuss the influence of the kinetic collision kernel. Then in Section 3, we first enter the core of the study of Boltzmann's operator, and we discuss issues of propagation of smoothness and propagation of singularities when the angular collision kernel is integrable (Grad's angular cut-off). Conversely, in Section 4, we explain the structure of Boltzmann's operator when the angular collision kernel presents a nonintegrable singularity for grazing collisions, and associated theorems of regularization. Since the Landau equation is linked to the Boltzmann equation via the emphasis on grazing collisions, this will lead us to discuss the Landau approximation in Section 5. We conclude in Section 6 with lower bound estimates. In many places the picture is incomplete, especially in the full, spatially inhomogeneous situation. Our discussion is mainly based on a priori estimates. We have chosen not to discuss existence proofs, strictly speaking. Sometimes these proofs follow from the a priori estimates by rather standard PDE arguments (fixed point, monotonicity, compactness), sometimes they are very, very complicated. In any case they are unlikely to be of much interest to the non-specialist reader, and we shall skip them all. Complete proofs of the most famous results can be found in [149]. Also, in this review we insist that a priori estimates should be explicit, but we do not care whether solutions are built by a constructive or non-constructive method. This is because we are mainly concerned with qualitative statements to be made about the solutions, and their physical relevance. If we were more concerned about practical aspects like numerical simulation, then it would be important that existence results be obtained by constructive methods. As a last remark, we note that we have excluded from the discussion all references which include nonstandard analysis [19,21,25] -just because we are not familiar with these techniques. 1. Use of velocity-averaging lemmas 1.1. Reminders Velocity-averaging lemmas express the local smoothness in macroscopic variables (t, x) of averages of the distribution function with respect to the microscopic variable (the velocity).
- 146. 144 C. Villani Here is a basic, important example: assume that f satisfies af at --+v.Vxf --S, f EL 2 SEL 2 (Hv s) t,x,v, t,x 9 (82) Then, for any q99 C~ (RN), 1 /4 2(l+s) f (t, x, v)qg(v) dv 9 "'t,x 9 N Here H c~is the Sobolev space of order c~, and when we write "e", this really means "lies in a bounded subset of". From the physical point of view, averaging lemmas express the fact that observables (typically, the local density) are smoother than the distribution function f itself. From the mathematical point of view, they are consequences of a "geometric" fact which we shall describe briefly. Consider the Fourier transform of f with respect to the variables t and x, write (r, ~) for the conjugate variables, then (82) becomes so that ill 2 - i 12 Ir+v.~l 2" Since the numerator vanishes for well-chosen values of v, this does not tell us much about the decay of f as r and ~ go to infinity. But when v varies in a compact set of NN, the set of values of v such that r + v 9~ is small will itself be very small; this is why on the average Ifl will decay at infinity faster than IS'l. Many variants are possible, see in particular [242,195]. A pedagogical introduction about velocity-averaging lemmas is provided by Bouchut [96]. Let us make a few comments: 1. The L 2 a priori bound for f may be replaced by a L p bound, p > 1 (then the regularization holds in some W ~'q Sobolev space), but not by an L 1 bound. Some replacements with L 1 estimates can be found, e.g., in Saint-Raymond [400].* 2. It is possible to cover cases in which the fight-hand side also lies in a negative Sobolev space with respect to the x variable, provided that the exponent of differentiation be less than 1. Obviously, if the exponent is greater than 1, then the transport operator, which is first-order differential in x, cannot regularize .... The case where the exponent is exactly 1 is critical, see Perthame and Souganidis [379]. 3. The above theorem considers time and space variables (t, x) E R • R N, but there are local variants, see in particular [99]. 4. The transport operator v 9Vx may be replaced by a(v) 9Vx under various conditions on a. 9Note added in proof: See also a recent note by Golse and Saint-Raymond.
- 147. A review of mathematical topics in collisional kinetic theory 145 5. Some vector-valued variants show that convolution products of the form f *v q9 are smooth in all t, x, v variables. A remarkable aspect of averaging lemmas is that they do not rely on the explicit solution to the linear transport equation (82) (at least nobody knows how to use the explicit solution for that purpose !). Instead, they are usually based on Fourier transform, or more generally harmonic analysis. There are variants of averaging lemmas which do not lead to smoothness but to a gain of integrability, with estimates in L p (L q (Lr)), and sometimes apply in a larger range of exponents. Developed by Castella and Perthame [135] with a view of applications to the Vlasov-Poisson equation, these estimates are analogous to a famous family of inequalities due to Strichartz for the Schrrdinger equation. Even though these estimates also give more information about the transport operator (which appears to be much more complex than it would seem!), it is still not very clear what to do with them. A discussion of the links between these Strichartz-like estimates and velocity-averaging lemmas can be found in Bouchut [96]. In the next two sections, we briefly describe the interest of averaging lemmas in the context of the theory of renormalized solutions for the Boltzmann equation. 1.2. How to use velocity-averaging lemmas in the Boltzmann context? No need to say, it would be very useful to get regularity results on averages of solutions of the Boltzmann equation. Since the Boltzmann collision operator looks a little bit like a convolution operator with respect to the v variable, we could hope to recover partial smoothness for it, etc. However, if we try to rewrite the Boltzmann equation like (82), with S = Q(f, f), we run into unsurmountable difficulties. First of all, we do not have the slightest a priori estimate on S! Something like integrability would be sufficient, since measures can be looked as elements of negative Sobolev spaces, but even this is not known in general. 1 Next, we only have f E L log L, and this seems to be a limit case where averaging lemmas do not apply 2 .... As pointed out to us by E Bouchut, L logl+e L would be feasible, although extremely technical, but for L log L this seems to be linked with deep unsolved questions of harmonic analysis in Hardy spaces. This is the place where the clever DiPerna-Lions renormalization trick will save the game. After rewriting the Boltzmann equation in renormalized formulation, Off(f) Ot + v. Vxfl(f)= fl'(f)Q(f, f), we see an opportunity to apply averaging lemmas to the function fl(f), which lies, for instance, in L 1 N L ~ as soon as fl(f) <~Cf/(1 + f). If we take it for granted that we shall find a meaningful definition of fl'(f)Q(f, f), i.e., a renormalizedformulation of the 1Except when the x-variable is one-dimensional, see Section 5.7 in Chapter 2A. 2Note however the following result by Golse and Saint-Raymond [245]. If Otfn + v. Vx fn = Sn with (fn) weakly compact in L 1 and (Sn) bounded in L 1, then f ~o(v)fn dv is strongly compact in (t,x).
- 148. 146 C. Villani Boltzmann operator, withbounds like fl'(f)Q(f, f) ~ L1 ([0, T] x IRN; H-c~(IRN)), then we shall get a smoothness bound for f # (f)~ do, or t~(f) *~ ~ .... Of course, smoothness for averages of fl(f) does not mean smoothness for averages of f. But since fl may be chosen to vary over a large range of admissible nonlinearities, by using approximation arguments combined with the a priori bounds on mass, energy, entropy, it is not difficult to show that averages like f f (v) q)(v) dr, f 9 q9 (g) ~ L ~ will lie in a uniformly strongly compact set of L 1. The interest of such approximation arguments is that they are robust and easy, and retain the softest information, which is gain of compactness. However, their nonexplicit nature is one of the drawbacks of the theory, and one can expect that important efforts will be devoted in the future to turn them into quantitative statements (as in [101,176], for instance). 1.3. Stability/propagation/regularization Some of the main principles in the theory of renormalized solutions are (1) as a starting point, try to work with just the basic known a priori estimates of mass, energy, entropy, entropy dissipation, (2) treat the Cauchy problem as a stability problem, and (3) replace smoothness by strong compactness. Point (2) means the following: consider a sequence fn(t, x, V)n~r~ of solutions, or approximate solutions, satisfying uniform a priori estimates of mass, energy, entropy, entropy dissipation. Without loss of generality, fn ~ f weakly in L p ([0, T]; L I(]~N • N R v )), 1 ~< p < (x~. Then one would like to prove that f also solves the Boltzmann equation. As a corollary, this will yield a result of existence and stability of solutions. Since a priori estimates are so poor, this is a very bad, unsatisfactory existence result. But for the same reason, this is an extremely good stability result. Now for point (3), it consists in replacing the statement "f is a smooth function", which is meaningless in a framework where so little information is available, by the statement ,,fn lies in a strongly compact set of L l''. For instance, 9 "smoothness propagates in time" is replaced by "if fn(o,., .) lies in a strongly N compact set of L I(~ N • R v ), then for all t > 0, so does fn (t,., .)"; 9 "singularities propagate in time" is replaced by "if fn (0,., .) does not lie in a strongly N ,, compact set of L l(R u x I~v ), then for all t > 0, neither does fn (t,., .) ; 9 "there is an immediate regularization of the solution" is replaced by "for (almost all) N ,, t > O, fn (t,., .) lies in a strongly compact set of L 1(RN x IRv ) . Note that the second item in the list can be rephrased as "smoothness propagates backwards in time". One of the nice features of the theory of renormalized solutions is that, with the help of averaging lemmas, these goals can be achieved by a good understanding of the structure of the Boltzmann operator alone. This approach has been developed by Lions, especially in [307] and in [311]. As a typical example, if we suspect some regularization effect due to collisions and wish to prove appearance of strong compactness, then, it essentially suffices to derive some smoothness estimate in the velocity variable, coming from an a priori estimate where the effect of collisions would be properly used,3 together with 3Most typically, the entropy dissipation estimate.
- 149. A review of mathematical topics in collisional kinetic theory 147 a meaningful renormalized formulation. Indeed, the velocity-smoothness estimate would imply that whenever q9is an approximation of a Dirac mass, then fn *v q9should be close to fn, in strong sense. On the other hand, from the use of averaging lemmas one would expect something like: fn , 99is "smooth" in t, x, v. Then the strong compactness would follow. This strategy was introduced by Lions [311 ]. Of course, the technical implementation of these fuzzy considerations turns out to be very intricate. All of the statements of the previous lines are only approximately satisfied: for instance we will not know that fn *v ~Pis close to fn, but rather that y(f n) *v q9is close to y(fn), and we will not know whether this holds for almost all t, x, but only for those t, x at which the local mass, energy, entropy are not too high, etc. In all the sequel we shall conscientiously wipe out all of these difficulties and address the reader to the references above for details. On the other hand, we shall carefully describe the structure of the Boltzmann operator, its renormalized formulation, and how these properties relate to statements of propagation of smoothness or regularization. 2. Moment estimates Moment estimates are the first and most basic estimate for the Boltzmann equation. Since one wants to control the energy (= second moment), it is natural to ask for bounds on moments higher than 2. In fact, if one wants to rigorously justify the identity f• Q(f, f)lv[ 2 dv - 0, N and if the kinetic collision kernel in the Boltzmann operator behaves like Iv - v,I • , then it is natural to ask for bounds on the moments of order 2 + y. Of course, once the question of local (in time) estimates is settled, one would like to have information on the long-time behavior of moments. In the spatially homogeneous situation, moment estimates are very well understood, and constitute the first step in the theory. In the case of the full, spatially-inhomogeneous Boltzmann equation there is absolutely no clue of how to get such estimates. This would be a major breakthrough in the theory. As for perturbative theories, they are not really concerned with moment estimates: by construction, solutions have a very strong (typically, Gaussian) decay at infinity. As for the long-time behavior of moments, it is also well controlled in the spatially homogeneous case. In the full setting, even for much simpler, linear variants of the Boltzmann equation, the problem becomes much trickier, and satisfactory answers are only beginning to pop out now. In all the sequel, we shall only discuss the spatially homogeneous situation. The starting point of most estimates [389,204,170,460,349] is the weak formulation J~ Q(f , f)qg(v) dv u _1 ::, 2N N-1
- 150. 148 C. Villani applied to r = Ivls, s >2, or more generally to ~r(lvl2), where ~ is an increasing convex function. 2.1. Maxwelliancollisionkernels The most simple situation is when the collision kernel is Maxwellian. As noticed by Trues- dell [274], integral expressions like fsN_~b(cosO)~o(v')dcrcan be explicitly computed when ~0is a homogeneous polynomial of the velocity variable. As a consequence, the in- tegral in (83) can be expressed in terms of moments of f and angular integrals depending on b. This makes it possible to establish a closed system of differential equations for all "homogeneous" moments. So in principle, the exact values of all moments can be explic- itly computed for any time. Then, Truesdell showed that all moments which are bounded at initial time converge exponentially fast to their equilibrium values. Moreover, if some moment is infinite at initial time, then it can never become finite. 2.2. Hardpotentials In the case of hard potentials, or more generally when the kinetic collision kernel grows unbounded at infinity, then the solution to the Boltzmann equation is expected to be well- localized at infinity, even if the decay at initial time is relatively slow. Heuristically, this can be understood as follows: if the collision kernel diverges for very large relative velocity, then very fast particles have a very high probability to collide with rather slow particles, which always constitute the majority of the gas. Thus, these fast particles will certainly be slowed down very quickly. At the level of weak formulations like (83), this means that the "dominant" part will be negative (as soon as q9 is a convex function of Ivl2). More precisely, if, say, B(lv- v.I, cos0)= Iv- v.lZb(cosO),y > 0, then, for some constants K > 0, C < +cx~ depending only on s, N, y and b, f•u Q(f' f)lvlSdv ~ --K(fRN f dv) (fRNflvlS+• dv) +c(fRu flvl• dv) (fRu flvlSdv)" (84) This inequality is just one example among several possible ones. It easily follows from the Povznerinequalitiesor their variants, introduced in [389] and made more precise by Elmroth [204], Wennberg [460], Bobylev [85], Lu [328]. Here is a typical Povzner inequality from [328]: for any s > 2, and y ~<min(s/2, 2), [01 ~<zr/2, ,i s s s is Iv'l s + Iv, -Ivl -Iv, I ~<-x~(0)lv + Cs(IvlS-ylv, Iy + Ivl• lS-y), where xs(O)is an explicit function of 0, strictly positive for 0 < 0 ~<rr/2.
- 151. A review of mathematical topics in collisional kinetic theory 149 Let us now look at the applications to the solutions of the spatially homogeneous Boltzmann equation. As a consequence of (84) and the conservation of mass (f f = 1), Elmroth [204] proved uniform boundedness of all moments which are finite at initial time. This kind of estimates has been simplified and lies at the basis of spatially homogeneous theory. Let us explain the argument without entering into details, or looking for best possible constants. Multiplying the inequality above by Iv - v, I• (0 < y < 2, say), and integrating against the angular collision kernel, one easily gets 1 is b(cosO)[v v,[ • s -q-[v~,[s Iv]s Iv, Is) do- 4 N-1 <<-girl s+• + C(IvlSlv, Iy + Iv, lSlol • (85) (additional terms, like Ivls-y Iv, I2y, are easily absorbed into the last term in the right-hand side by Young's inequality). Then, let us integrate (85) against ff,: after application of Fubini's identity, we find (84), which can be rewritten as d f• f(t, Olvl s dv (86) dt N The last integral is bounded because of the energy bound and y < 2. Since f f = 1, one finds that the s-order moments, Ms(t) -- f f (t, Olvl s dr, satisfy some system of differential inequalities d dtMs <~-KsMs+• + CsMs. (87) Now, by HOlder's inequality and f f = 1 again, Ms+• <<.M~+• (88) Since solutions of dX/dt <<,--CS l+~ .qt_CX, ol > 0, are uniformly bounded, Elmroth's theorem follows. Desvillettes [170] pointed out that the conclusion is much stronger: if at least one moment of order s > 2 is finite at initial time, then all moments immediately become finite for positive times - and then of course, remain uniformly bounded as time goes to infinity. The result was further extended by Wennberg [460], Wennberg and Mischler [349], in particular the assumption of finite moment of order s > 2 at initial time can be dispended with. Moreover, these results hold for cut-off or non-cutoff angular collision kernels.
- 152. 150 C. Villani Bobylev [85] has given a particularly clear discussion of such moment estimates, with various explicit bounds of Ms(t) in terms of Ms(O) and s. A very interesting by- product of this study was the proof of Gaussian tail estimates. By precise estimates of the growth of the bounds on Ms (t), he was able to prove that if the initial datum satisfies f exp(c~olvl2) fo(v)dv < +c~ for some oto > O, then, at least when g = 1 (hard spheres), there exists some ot > 0 such that sup L exp(~ f (t, v) dv < +cx~. t/>0 N Anticipating a little bit on precise results for the Cauchy problem, we can say that moment estimates have been a key tool in the race for optimal uniqueness results in the context of hard potentials with cut-off In fact, progress in this uniqueness problem can be measured by the number of finite moments required for the initial datum: Carleman needed 6, Arkeryd [17] only 4, Sznitman [412] was content with 3, Gustafsson [270] with 2 + V, Wennberg [458] needed only 2 + e (e > 0). Finally, Mischler and Wennberg [349] proved uniqueness under the sole assumption of finite energy. On this occasion they introduced "reversed" forms of Povzner inequalities, which show that the kinetic energy of weak solutions to the Boltzmann equation can only increase or stay constant; hence the uniqueness result holds in the class of weak solutions whose kinetic energy is nonincreasing. More surprisingly, these moment estimates can also be used for proving nonuniqueness results! in the class of weak solutions whose kinetic energy is not necessarily constant, of course. The idea, due to Wennberg [463], is quite simple: consider a sequence (f~)neN of initial data, made up of a Maxwellian (equilibrium) distribution, plus a small bump centered near larger and larger velocities as n ~ cx~.The bump is chosen in such a way that its contribution to the total mass is negligible as n --+ cx~,but not its contribution to the kinetic energy; so that the total kinetic energy of fn is, say, twice the energy E of the Maxwellian. For each n, one can solve the corresponding Boltzmann equation with hard potentials, and it has energy 2E. One can check that, as n ~ cx~,this sequence of solutions converges, up to extraction of a subsequence, to a weak solution of the Boltzmann equation, with Maxwellian initial datum. But, by means of some precise uniform moment bounds, one can prove that for positive times, the kinetic energy passes to the limit: u lim fR fn(t, v)lvl2 dv = fR f (t, v)lvl2 dv. n---+~ N N Hence this weak solution f of the Boltzmann equation has energy E at time 0, and energy 2E for any time t > 0, in particular it is not the stationary solution .... 2.3. Soft potentials When the kinetic collision kernel decays as Iv - v,I--+ c~, or more generally when it is uniformly bounded, then local in time moment estimates are much easier to get. On
- 153. A review of mathematical topics in collisional kinetic theory 151 the other hand, the result is much weaker, since only those moments which are initially bounded, can be bounded at later times. When the collision kernel presents a singularity for zero relative velocity, say Iv - v,] • with 7 < 0, then additional technical difficulties may arise. When ~, <~ -1, and even more when ~, < -2, it is not a priori clear that power laws ]vis (or their mollified versions, (1 4- IVI2)s/2) are admissible test-functions for the spatially homogeneous Boltzmann equation. This difficulty is overcome for instance by the method in Villani [446], or the remarks in [10]. In all cases, anyway, one proves local in time propagation of moments. But now, due to the decay of the collision kernel at infinity, it becomes considerably more difficult to find good long time estimates. Obtaining polynomial bounds is quite easy, but this is not always a satisfactory answer. The main result on this problem is due to Desvillettes [170]" he showed that when the kinetic collision kernel behaves at infinity like Iv - v,] • -1 < 7 < 0, then all those moments which are initially bounded, can be bounded by C (14- t), as the time t goes to infinity. By interpolation, if the initial datum has a very good decay at infinity (many finite moments), then the growth of "low"-order moments will be very slow. Thus, even if uniform boundedness is not proven, the "escape of moments at infinity" has to be slow. The results of Desvillettes can be extended to the case where 7 > -2, though there is no precise reference for that (at the time where Desvillettes proved his result, weak solutions were not known to exist for 7 <~ -1). On the other hand, when 7 ~< -2, then it is still possible to prove local in time propagation of moments of arbitrary order, but the bounds are in general polynomial and quite bad .... In the case of the Landau equation where things are less intricate, one can derive the following estimate [429] when the decay at infinity is like Iv - v,[ • -3 ~< y ~< -2" then the moment of order s grows no faster than O((1 + t);~), with X -- (s - 2)/3. 2.4. Summary We now sum up all the preceding discussion in a single theorem. As we said above, for the time being it is only in the spatially homogeneous setting that relevant moment estimates have been obtained. The conditions of the following theorem are enough to guarantee existence of weak solutions, but not necessarily uniqueness. THEOREM 1. Let B(iv-v,], cos0)= ]v-v,[• be acollisionkernel, -3 <. }, <. 1, fo b(cos0)(1 - cos0) sinN-20 dO < +cx~. Let fo E L~(R N) be an initial datum with finite mass and energy, and let f (t, v) be a weak solution of the Boltzmann equation, f (O, .)- fo, whose kinetic energy f f (t, v)iv] 2 dv is nonincreasing. Then this kinetic energy is automatically constant in time. Moreover, if Ms(t) =--f f (t, v)lvl s dr, then,
- 154. 152 C. Villani (i) If y = O, then for any s > 2, Vt >0, [Ms(t) < +o0 ~ Ms(O) < +~], Ms(O) < +~------> sup Ms(t) < +cx~. t />o Moreover, under the sole assumption M2 < +cx~, there exists a convex increasing function 4~, 4~(Ivl) ~ c~ as Ivl ~ ~, such that supf f (t, v)qb(lvl)lv[ 2dr < +c~. t />o (ii) If F > O, then for any s > 2, Vto > O, sup Ms (t) < +cx~. t/>to Moreover, if y = 1, and the initial datum fo satisfies f exp(c~olvl 2) fo(v) dv < +c~ for some oto > O, then there exists some ot > 0 such that sup f ] e~ (t, v) dv < +c~. t/>o J (iii) If Y < O, then for any s > 2, vt > o, [Ms(t) < +~ ~ M~(O) < +~]. Moreover, (a) if F > -2, then Ms(O) < +cx~ ~ 3C > O, Ms(t) <. C(1 + t); in particular, for any e > O, Ms/s < +cxz----> 3C > O, Ms(t) <~C(1 + t)s; (b) if y < - 2, then Ms(O) < +cx~ ~ 3C > O, 3X > O, Ms(t) <. C(1 + t) x (X = 1for s <~4).
- 155. A review of mathematical topics in collisional kinetic theory 153 REMARKS. (1) All the constants in this theorem are explicit. (2) The statement about finiteness of f f(t, p)r 2 dv in point (i) is interesting because it implies lim sup fly e--+ cx)t>/0 I>~R f(t, v)lvl 2 dv = 0; in other words, no energy leaks at infinity. Such an estimate is obvious in situation (ii); it is a seemingly difficult open problem4 in situation (iii). (3) The range y 6 [-3, 1] has been chosen for convenience; it would be possible to adapt most of the proofs to larger values of y, maybe at the expense of slight changes in the assumptions. Values of y which would be less than -3 pose a more challenging problem, but do not correspond to any physical example of interest. The first part of point (i) is due to Truesdell [274], while the statement about point (ii) is mainly due to Desvillettes [170] and improved by Wennberg [458], Wennberg and Mischler [349]; the estimate for exponential moments is due to Bobylev [85]. As for point (iii), it is proven in Desvillettes [170] for y > - 1, and elements of the proof of the rest can be found in [446,444]. For the Landau equation (with q/(Iv- v,I) = KIv- v,I • the very same theorem holds, with the following modification: point (ii) is known to hold only if there exists so > 2 such that Ms0(0) < +oo (see [182]). As for point (iii)(c), the more precise estimate )~= (s - 2)/3 holds [429], at least if the collision kernel is replaced by a mollification which decreases at infinity like Iv - v,I • but does not present a singularity for Iv - v,I --~0. 3. The Grad's cut-off toolbox We now present several tools which are useful to the study of Boltzmann's equation when the collision kernel satisfies Grad's angular cut-off assumption. This means at least that whenever Iv - v, I # 0, A(lv - v,]) = f B(v - v,, a)do- < +cx~. (89) Typical examples are Iv - v, I• b(cos0), where f0 rr b(cos0) sinN-2 0 dO < +~. We shall mainly insist on two ingredients: the important Q+ regularity theorem, and the DiPerna-Lions renormalization. 4See Section 5.3 in Chapter 2C for more, and some results.
- 156. 154 C. Villani 3.1. Splitting When Grad's assumption holds true, then one can split the Boltzmann collision operator into the so-called "gain" and "loss" terms, and the loss term is then particularly simple. We give this splitting in asymmetric form: Q(g, f) = Q+ (g, f) - Q-(g, f) -- Q+ (g, f) - f (A 9 g). (90) Clearly, the delicate part in the study is to understand well enough the structure of the complicated integral operator L /, Q+ (g, f)- dr, da B(v - v,, a)g(v,) f (v'). N N-1 As early as in the thirties, this problem led Carleman to the altemative representation u1) ! [1) -- Vii N-1 Vt t ) -- V, t B 2v- v'- v~,, [v' , g(v,)f(v'), (91) with Evv, standing for the hyperplane going through v, orthogonal to vI - v. In Sections 3.3 and 3.4, we shall expand a little bit on the structure of the Q+ operator. Before that, we give an easy lemma about the control of Q+ by means of the entropy dissipation. 3.2. Control of Q+ by Q- and entropy dissipation Using the elementary identity 1 X (X-Y) log K>I X ~ K Y + log K -Y' ' with X = flf~, and Y = ff,, we find, after integration against B dr, da, Q+(f, f) ~<K Q-(f, f)+ 4 d(f), log K (92) where 1s d(f) = -~ N• (f,ft, _ ff,) log f'ff* B dr, da ff, is a nonnegative operator satisfying f d(f) do = D(f), the entropy dissipation functional. Inequality (92) was first used by Arkeryd [21], and has proven very useful in the DiPema-Lions theory [192].
- 157. A review of mathematical topics in collisional kinetic theory 155 3.3. Dual estimates Many estimates for Q+ are best performed in dual formulation, with the help of the pre-postcollisional change of variables. For instance, to bound IIQ+ (g, f)IIL,(RN), it is sufficient to bound LN Q+(g' f)q)dv- fiR2,,,,dvdv*g*f(LN-1B(Iv-v,l,cosO)q)(v')do') uniformly for I1~oIILp' ~< 1. So the meaningful object is the linear operator ~o~ LN-' B(Iv -- v,I, cos0)q)(v')do'. (93) Pushing the method a little bit, one easily arrives at the following abstract result: let X, Y be two Banach spaces of distributions, equipped with a translation-invariant norm. Assume that the linear operator ~T""(D1""~LN_I B(Ivl, cos 0)~o( v+ 21 vI~r) dCr is bounded (as a linear map) from Y to X. Then, the following estimate on Q+ holds, IIQ+':.f, g)llw CIIgIIL,Ilfllx'. Actually, Y' (resp. X') does not really need to be the dual of Y (resp. X), it suffices that IIQ+llr, = sup{f Q+g; Ilgllr = 1} (resp. f fO <~Ilfllxllgtllx,). As an example of application of this result, consider the simple situation B(v - v., o.) = 4"(Iv - v.l)b(cosO), where 4, is bounded and b(cos0) sinN-2 0 is integrable with support in [0, re/2]. Obviously, 7" is bounded L ~ ~ L ~ and by the change of variables v ~ v' (which is valid for fixed o. because we have restricted ourselves to 0 6 [0, 7r/2]), one can prove that 7" is bounded L 1 --+ L 1. By interpolation, 7" is bounded L p ~ L p for 1 ~<p ~<oc, and therefore we obtain the estimate ][Q+(g,f)ncp ~CIIgllLlllfll zp, 1 ~p~. (94) A variant of the argument when B(v - v,, o.) = Iv - v,l• y > 0, leads to [1Q+(g' f)[[Lp <-CIIgllL~IlfllL~, 1 ~ p ~ ~, (95) where we use the notation (L IIflIL,P -- NfP(1 4-IriS) p dv (96)
- 158. 156 C. Villani More sophisticated variants of estimates like (95) have been studied by more compli- cated means in Gustafsson [269,270]. They constitute a first step in the L p theory for the spatially homogeneous Boltzmann equation with hard potentials and cut-off. These esti- mates show that, in first approximation, the Q+ operator resembles a convolution operator. But we shall see in the next paragraph that a stronger property holds. To conclude this paragraph, we mention that the case where there is a singularity in the kinetic collision kernel for Iv - v, I -~ 0 (soft potentials ...) has never been studied very precisely from the point of view of LP integrability. 3.4. Lions' theorem: the Q+ regularity One of the main ideas of Grad [252], when developing his linear theory and making clear his assumptions, was that the Q+ term should be considered as a "perturbation". This may sound strange, but think that the linear counterpart of Q+ is likely to be an integral operator with some nice kernel, while the linear counterpart of Q- will contain a multiplicative, noncompact operator. Generally speaking, since Q+ is more "mixing" than Q-, we could expect it to have a smoothing effect. In a nonlinear context, the idea that Q+ should be smoother than its arguments was made precise by Lions [307]. In this paper he proved the following estimate. PROPOSITION 2. Let B be a C ~ collision kernel, compactly supported as a function of Iv - v, l, vanishing for Iv - v, I small enough, compactly supported as a function of 0 9 (0, Jr). Then, there is afinite constant C depending only on B and N such that for any g 9 L1 (RN), f 9 L2(RN), IIa+<g II.-- , N~I ~<Cllgllc, Ilfllc2, (97) and similarly [IQ+(g,f ll,, CllgllLzllfllL '' (98) The proof was based on a duality method quite similar to the one above, and very sophisticated tools about Fourier integral operators; this estimate is actually linked to the regularity theory of the Radon transform. In fact, as a general rule [406], operators of the form 7"qg(x) = fsx b(x, y)qg(y) dcrx(y), where b is a smooth kernel and Sx is a hypersurface varying smoothly with x 9 ]I~N , satisfy an estimate like IIT~ollH~N-,)/2 <<.CIl~ollL2, under some nondegeneracy condition 5 about the way Sx varies with x. 5In the case of operator (93), Sx is the sphere with diameter [0, x] and crx is just the uniform measure on Sx. The nondegeneracy condition is not satisfied at y = 0 (which is fixed), and this is why Lions' theorem does not
- 159. A review of mathematical topics in collisional kinetic theory 157 Lions' theorem has become one of the most powerful tools in the study of fine properties of the Boltzmann equation with cut-off (see [101,307,308,349,429,393]). Wennberg [459] gave a simplified proof of this result, with explicit(able) constants, by using Carleman's representation (91). Then, Bouchut and Desvillettes [98], and, independently, Lu [325] devised an elementary proof, with simple constants, of a slightly weaker result: IIa+<g f)I1 , N21 <~Cllgllc2 II/11c2. (99) However, the qualitative difference between (97) and (99) is significant in some applica- tions: see, for instance, our a priori estimates in LP norms for collision kernels which decay at infinity [429]. Also a relativistic variant of this result has been established by Andrrasson [14]. In a more recent paper, Wennberg [462] has put both the classical and the relativistic estimates in a unified context of known theorems for the regularity of the Radon transform. He noticed that these two cases are the only ones, for a whole range of parameters, where these theorems apply. Of course, by Sobolev embedding (and interpolation), Lions' theorem yields refinements of (94) when the collision kernel is smooth. A task which should be undertaken is to use a precise version of Lions' theorem, like the one by Wennberg, to derive nice, improved weighted Lp bounds for realistic collision kernels. This, combined with the methods in [429], should enable one to recover the main results of Gustafsson in a much more elegant and explicit way.6 As a last remark, the fact that the collision kernel vanish at v - v. = 0 is essential in Lions' theorem (not in the weaker version (99)). If this is not the case, then smoothness results similar to the ones obtained by (97) require additional integrability conditions on g. In the case of hard potentials however, it is still possible to prove a weaker gain of smoothness or integrability with respect to f, without further assumptions on g. 3.5. Duhamelformulas andpropagation of smoothness The idea to consider Q+ as a perturbation can be made precise by the use of Duhamel-type formulas. For instance, in the spatially homogeneous case, the Boltzmann equation can be rewritten as af Ot + (Lf) f = Q+ (f, f), f(O, .) = fo, where we use the notation A 9 f = Lf. Then the solution can be represented as fot f(t, v)-- fo(v)e -foLf(r'v)dr + e-ftLf(r'v)drQ+(f, f)(s, v)ds. (100) apply in presenceof frontalcollisions,and needs the collisionkernel to be vanishingcloseto 0 = Jr. Another degeneracyariseswhen v - v. goesto 0, so the Q+ smoothnessalsoneedsvanishingof the collisionkernelat zerorelativevelocity. 6At the timeof writing,thistaskhasjust beenperformedby C. Mouhotwiththe helpof the author'sadvice.
- 160. 158 C. Villani In the spatially inhomogeneous case, one can write similarly ( 0 ) tLf(r,x-(t-r)v,v)dr _ Q+(f, f)ef oLf(r,x-(t-r)v v)dr -~ + v" Vx fefo -- ' or f (t,x, v) = fo(x -tv, v)e- fd Lf(r,x-(t-r)v,v)dr fo + Q+(Z, f)(s,x - (t - s)v, v)e-f's LI(~,x-(,-~)~,~)d~. (101) By these formulas, one can understand, at least heuristically, the phenomenon of propagation of regularity and singularities. Let us illustrate this in the case of the spatially homogeneous Boltzmann equation, starting with formula (100). If f0 E L2, then, at least for collision kernels which become very large at infinity, f(t, .) is uniformly bounded in L 2. Now, assume for simplicity that the collision kernel B is sufficiently smooth that estimate (97) applies, then it becomes clear that the second term in the right-hand side of (100) is H 1-smooth (when N = 3). Indeed, thanks to the convolution structure, Lf is also smooth, say C ~176 if B is C ~176 On the other hand, the first term on the right-hand side has exactly the same smoothness as f0. Thus both regularity and singularities are propagated in time. More precise theorems of this kind, for some realistic collision kernels, e.g., hard spheres, are to be found in Wennberg [459]. In principle one could actually prove that f (t, v) = G(t, v) fo(v) + H (t, v), (102) where, at least if B is very smooth, G is a positive C ~176 function of t, v and H is smoother than f0. This would follow by an iteration of Duhamel's formula, in the spirit of [349]. Alternatively, one expects that f (t, .) can be decomposed into the sum of a part which is smooth (at arbitrarily high order) and a part which is just as singular as f0, but decays exponentially fast. In the spatially inhomogeneous situation, the same kind of results is expected. In particular, in view of (101) it is believed that singularities of the initial datum are propagated by the characteristics of free transport, (x, v) ~ (x + tv, v). Such a result was recently proven by Boudin and Desvillettes for small initial data: they showed the following generalization of (102), f (t, x, v) = G(t, x, v) fo(x - tv, v) + H (t, x, v), where G and H are not necessarily smooth, but at least possess a fractional Sobolev regularity. Their proof is based on a combination of the Q+ smoothness and averaging lemmas. We should make an important point here about our statement above about propagation of singularities. If we consider an initial datum which is very smooth apart from some singular set S, then it is not expected that the solution stay very smooth apart from the image St of this singular set by the characteristic trajectories. It is possible that the smoothness of
- 161. A review of mathematical topics in collisional kinetic theory 159 the solution deteriorate, even very far from St. But one expects that the worst singularities always lie on St. And in any case, one always expects that the singular part of the solution be decaying very fast as time goes by. 3.6. The DiPerna-Lions renormalization The Q+ smoothing effect cannot be applied directly to the spatially inhomogeneous equation, ~f Ot + v. Vxf = Q(f, f), (103) because of the lack of nice a priori estimates. As explained above, the idea of renormalization consists in giving a meaningful definition of fl'(f)Q(f, f) under very weak a priori estimates. Solutions are then defined as follows: N DEFINITION 1. Let f e C([0, T]; Ll(Ii~x N x ~U))A L~(R+, L~_(Rx N x II~v )). It is said to be a renormalized solution of the Boltzmann equation if for any nonlinearity fl CI(~+, R+), such that fl(0) =0, Ifl'(f)l ~<C/(1 + f), one has 3fl(f) Ot -b v. Vxfl (f) = fl' (f) Q (f, f), (104) in distributional sense. The DiPerna-Lions renormalization [190,192] achieves this goal by using the split- ting (90). First of all, fl' (f) Q- (f, f) = ffl' (f)(A 9 f). (105) Following Section 5.1, let us assume that sup f•zu f(t,x, V)(1 + IV[2 + IX[2)dxdv < -q-cx~, O~t~r and that A(z) = o(Izl2). Then A 9 I L (t0, L' (Rf; L?oc(Rf))). Further assume that the nonlinearity fl satisfies 0 <~fl'(f) ~< C l+f
- 162. 160 C.Villani Then, ffl'(f) 9 L~, and as a consequence (105) is well-defined in L~oc. As for the renormalized gain term, it can be handled easily because it is nonnegative. As a general "rule", when one manages to give sense to all terms but one in some equation, and when this last term has a sign, then the equation automatically yields an a priori estimate. In our situation this is accomplished by integrating Equation (104) in all variables on [0, T] x IR~ x IR~; using the bounds on mass and energy, one gets foTf• fl'(f) Q+(f, f) dt dx dv 2N <~ fl(fo) dx do + (f) f Lf dt dx dv < +cx~, 2N 2N whence fl'(f) Q+ (f, f) e L1 ([0, T] x JRN x NN). By the way, there is another, more widely known, version of this estimate, based on the entropy dissipation [192]; but this variant is more complicated and has the disadvantage to use a symmetric estimate, while the renormalization procedure can also be done from an asymmetric point of view: fl'(f)Q(g, f) = fl'(f)Q+(g, f) - fl'(f)fLg. On the other hand, the entropy dissipation has been very useful, both in the proof of stability of renormalized solutions, and in certain refinements of the theory by Lions [308]. Also, by using the entropy dissipation one can define a renormalized formulation with a stronger nonlinearity, namely fl(f) = ~/1 + f - 1. This is implicit in [308], and has been shown by an elementary method in [444, Part IV, Chapter 3], as an outgrow of the idea of H-solutions explained in Section 4.1. Renormalization and averaging lemmas were the two basic tools in the DiPema-Lions theorem, which was the very first existence/stability result in the large for the spatially inhomogeneous Boltzmann equation. More precisely, these authors have proven [192] that the renormalized Boltzmann equation (104) is stable under weak convergence (a priori estimates of mass, energy and entropy being used). This theorem has remained a singular point in the field, due to the complexity of the proof and the use of technicalities which have no real counterpart for the rest of the theory .... Since then, Lions [307] found a simpler proof of existence, using the Q+ regularity. In fact, he proved that strong compactness propagates in time for the Boltzmann equation with cut-off: this is the analogue of the spatially homogeneous results discussed in the section above. Once strong compactness is established, passing to the limit is almost straightforward. By the way, it is rather easy to prove the converse property, namely that the sequence of initial data has to be strongly compact if the sequence of solutions is. This theorem is a (very weak) illustration of the principle of propagation of singularities.
- 163. A review of mathematical topics in collisional kinetic theory 161 3.7. Summary To conclude this section, we give explicit theorems illustrating the discussion above. As of this date, all of them are best in their category, but not optimal. We begin with the spatially homogeneous situation. THEOREM 3. Let B(v- v,,rr)= Iv- v,l• be a collision kernel for hard potentials, satisfying Grad's cut-off assumption: fo rr b(cos0) sinN-2 0 dO < +cx~, y>0, and let fo be a probability distribution function with finite second moment, fR f~ < +cx~. N Then, (i) there exists a unique energy-preserving solution to the Cauchy problem associated to B and fo. This solution is unique in the class of weak solutions whose energy does not increase; (ii) if fo ~ L~1 (q LPs for some p E [1,+cxz], Sl > 2, 1 < s <. Sl - V/P, then supt>~0 Ilf(t, ")ILL; < +~; (iii) ifmoreover y > 1/2, s >~2 and fo ~ HI(RN), N -- 3, then supt,>0 Ilf(t, ")lln~ < +cx~; (iv) if on the other hand, F > 1/2, s ~ 2and fo q~HI(~N), N -- 3, then, for all t ~ 0, f (t, .) ~ HI(RN). But f (t, .) = g(t, .) + h(t, .) where Ilg(t, ")llts~ = O(e-~t) for some lZ > O, and supt,>0 IIh(t, ")lln~ < +~. Point (i) is due to Mischler and Wennberg [349]; point (ii) to Gustafsson [270] for p < cx~ and to Arkeryd [20] for p = cx~; as for point (iii), is is due to Wennberg [459], point (iv) being an immediate consequence of the proof. A recent work by Mouhot and the author recovers the conclusion of (ii) under slightly different assumptions on sl, with the advantage of getting explicit constants; we are working on extending the allowed range of exponents sl. Further work is in progress to extend also the range of validity of the conclusion of (iii) (arbitrary dimension, more general collision kernels) as well as to treat propagation of H ~ smoothness for arbitrary k. We now turn to the inhomogeneous theory in the small. THEOREM 4. Let B = B(z, rr) be a collision kernel, B E L ~ (S N-1 , W l'cx~(~N)), N = 3. Let fo(x, v) be a nonnegative initial datum satisfying the Maxwellian bound fo(x v) <. Coexp(-Ixl2-+- Ivl2) ' 2 ' where Co = 1/(81[IBIIL~(~N;LI(SN-1))). Then,
- 164. 162 C. Villani (i) there exists a global solution to the spatially inhomogeneous Boltzmann equation with collision kernel B and initial datum fo, and for all t ~ [0, T] (T > 0), it satisfies a Maxwellian bound of the form f (t x v) <. C;r exp(- lx - vtlZ + lvl2 ) ' ' 2 ' (106) with C7~depending only on T and Co; (ii) there exist "smooth"functions R and S in//lo c~(I~+ x IRxNx I~vN)for any ot < 1/25, such that f (t,x, v) = fo(x - vt)R(t,x, v) + S(t,x, v); (iii) if moreover fo ~ W k'c~for some k ~ N (or k = oo), then I W, o'c (R+ • Rx • This theorem is extracted from Boudin and Desvillettes [101]. Part (i), inspired from Mischler and Perthame [348], is actually an easy variation of more general theorems by Illner and Shinbrot [278]. One may of course expect the smoothness of R and S to be better than what this theorem shows! But this result already displays the phenomenon of propagation of singularities along characteristic trajectories. Moreover, it conveys the feeling that it will be possible to treat propagation of smoothness and singularity for very general situations, as soon as we have solved the open problem of finding nice integrability a priori estimates for large data. The main idea behind the bound (106) is the following: the left-hand side satisfies the differential inequality ~f Ot + v. Vxf = Q(f, f) ~< Q+ (f, f), while one can devise a fight-hand side of the form g(t,x, v)-C(t)exp(-Ix- vtl2 + which satisfies the differential inequality Og Ot + v. Vxg ~ Q+(g, g), so that it is natural to expect f ~< g if f0 ~< go. Note that since g is a Maxwellian, Q+ (g, g) = Q- (g, g). Finally, we consider the DiPerna-Lions theory of renormalized solutions.
- 165. A reviewofmathematicaltopicsin collisionalkinetictheory 163 THEOREM 5. Let B(lv - v,I, cos0) = r - v,l)b(cosO) be a collision kernel satisfying Grad's angular cut-off together with a growth condition at infinity: f0 zrb(cos0) sinN-2 0 dO < +c~, qS(lz[)=o(lzl 2) aslzl--+cx~. Let (f~)nel~ be a sequence of initial data with uniformly bounded mass, energy, entropy, supfR nr N• f~(x, v)[1 +Ix] 2 -+-Iv[2 + log f~(x, v)]dxdv < +oo. Let fn (t, x, v) be a sequence of solutions7 of the Boltzmann equation of'+ Ot v. Vxfn = Q(fn fn), fn (0,., .) = f~) . t ~0, X e]I~N, VE]t~N, (107) Assume that the fn's satisfy uniform bounds of mass, energy, entropy and entropy dissipation: sup sup f~ n~Nt~[O,T] Nx•N fn(t,x,v)[1 + [x[2 + Iv[2 +logfn(t,x,v)]dxdv < +oo, (lo8) fo T sup D(fn(t,x, .)) dx dt < +c~. hEN (109) Without loss of generality, assume that fn __+f in L p ([0, T]; L ~(~N • ~N)), 1 ~<p < ~, T < oo. Then, (i) f is a renormalized solution of the Boltzmann equation. It satisfies global conservation of mass and momentum, and the continuity estimate f ~ C ([0, T]; L 1(i~u x N R~)); (ii) moreover, for all t > O, fn __+f strongly in L 1 if and only if f~ --+ fo strongly; in this case the convergence actually holds in C ([0, T], L I(I~u x I~u)); (iii) if moreover there exists a strong, classical solution of the Boltzmann equation with initial datum fo, then f coincides with fo. COROLLARY 5.1. Let fo be an initial datum with finite mass, energy and entropy: f~N• fo(x, v)[1 + Iv[2 + Ix[2 + log fo(x, v)] dx dv < +c~. 7Renormalizedsolutions,or strongsolutions,or approximatestrongsolutions.
- 166. 164 C. Villani Then there exists a renormalized solution f (t,x, v) of the Boltzmann equation, with f (O, ., .)-- fo. REMARKS. (1) A typical way of constructing approximate solutions is to solve the equation ofn Q(fn, fn) + V.Vxf n-- 1 fn Ot 1 + n f dv which is much easier than the "true" Boltzmann equation because the collision operator is sublinear. (2) If the fn's are strong, approximate solutions, then the bounds (108)-(109) automatically hold, provided that the initial data have sufficient regularity. This remark, combined with the preceding, explains why the corollary follows from the theorem. (3) Point (iii) as stated above is slightly incorrect: for this point it is actually necessary to assume that the fn's are strong, approximate solutions, or are constructed as limits of strong, approximate solutions. Point (i) is due to DiPerna and Lions [192], points (ii) to Lions [307,308]. For the sake of simplicity, we have stated unnecessarily restrictive assumptions on the collision kernel in points (i) and (ii). Point (iii) was first proven by Lions under an assumption which essentially implies q~ ~ L~, then extended by Lu [324]. We have not made precise what "classical" means in point (iii): in Lions' version, g should satisfy the Boltzmann equation almost everywhere on [0, T] • RN • RN, and also satisfy the dissipative inequalities introduced in Lions [308]. The discussion of dissipative inequalities is subtle and we preferred to skip it; let us only mention that this concept is based on the entropy dissipation inequality, and that it led Lions to a clean proof of local conservation of mass, as well as to the concept of dissipative solutions. In Lu's theorem, much more general collision kernels are included, at the price of slightly more restrictive (but quite realistic) assumptions imposed on the strong solution g. Lu also uses results from the theory of solutions in the small [63] to show existence of such strong solutions when q'(Izl) = O(Izl• Z > -1, and the initial datum is bounded by a well-chosen, small function. So these results bridge together the theory of renormalized solutions and the theory of solutions in the small. At the moment, point (ii) is the most direct way towards the corollary. The scheme of the proof is as follows. In a first step, one uses the uniform bounds and the Dunford-Pettis criterion to get weak compactness of the sequence of solutions in L 1. This, combined with the renormalized formulation and averaging lemmas, implies the strong compactness of velocity-averages of fn. Since the operators L = A. and Q+ are velocity averaging operators in some sense (remember the Q+ regularity theorem), one can then prove the strong compactness of Lf n and Q+ (fn, fn). This is combined with a very clever use of Duhamel formulas to prove the strong compactness of the sequence fn itself, provided that the sequence of initial data f~ is strongly compact. Finally, one easily passes to the limit in the renormalized Q- operator, and then in the renormalized Q+ operator by a variant of the dominated convergence theorem which involves the domination of Q+ by Q- and (a little bit of) the entropy dissipation, as in Section 3.2.
- 167. A review of mathematical topics in collisional kinetic theory 165 4. The singularity-hunter's toolbox In this section, we now examine the situation when the collision kernel presents a nonintegrable angularsingularity. This branch of kinetic theory, very obscure for quite a time, has undergone spectacular progress in the past few years, which is why we shall make a slightly more detailed exposition than in the case of Grad's angular cut-off. The starting point for recent progress was the work by Desvillettes [171] on a variant of the Kac model, which was devised to keep some of the structure of the Boltzmann equation without cut-off. The study of these regularizing effects was first developed in the spatially homogeneous theory, and later in the theory of renormalized solutions, in the form of strong compactification. As explained previously, the main qualitative difference with respect to the situation where Grad's assumption holds, is that one expects immediate regularization of the solution. From the mathematical point of view, the first clear difference is that the splitting Q(f, f)= Q+ (f, f)- Q-(f, f) is impossible: both terms should be infinite. From the physical point of view, one can argue that when particles collide, there is an overwhelming probability for the change in velocity to be extremely small, hence the density in probability space should spread out, like it does in a diffusion process. The main analytical idea behind the regularization effect is that Q(f, f) should look like a singular integral operator. As we shall see, it resembles a fractional diffusion operator; this illustrates the physically nontrivial fact that collision processes for long- range interactions are neither purely collisional in the usual sense, nor purely diffusive, but somewhat in between. There is an important body of work due to Alexandre about the study of the non-cutoff Boltzmann operator, with the help of pseudo-differential formalism [2-9], and on which we shall say almost nothing, the main reason being that most of the results there (some of which have been very important advances at the time of their appearance) can be recovered and considerably generalized by means of the simpler techniques described below. Generally speaking, there are two faces to singular operators in partial differential equations. On one hand one would like to control them, which means (i) find weak formulations, or (ii) find if, in some situations, they induce compensations due to symmetries. On the other hand, we would like to have (iii) simple estimates expressing the fact that they really are unbounded operators, and that the associated evolution equation does have a regularizing effect. To illustrate these fuzzy considerations, think of the Laplace operator, and the formulas (i) f Aft0 = f fA~0, (ii) (Af) 9 ~0 = f 9 (A~0), (iii) f f(Af) -- -IIf[121 . Keeping this discussion in mind may help understanding the interest of the weak formulations in Section 4.1, the cancellation lemma in Section 4.2, and the entropy dissipation estimate in Section 4.3, respectively. Finally, as we already mentioned several times, another singularity problem will come into play: when one is interested in soft potentials, then the kineticcollision kernel presents a singularity for Iv - v, I -~ 0. When the strength of this singularity is high, this will entail additional technical difficulties, but it is not clear at the moment that this feature is related to physically relevant considerations.
- 168. 166 C.Villani 4.1. Weakformulations In presence of a nonintegrable singularity, Boltzmann's collision operator is not a bounded operator between weigthed L1 spaces; it is not even clear that it makes sense almost everywhere. Thus one should look for a distributional definition. The most natural way towards such a definition (both from the mathematical and the physical points of view) is via Maxwell's weak formulations" f Q(f, f) do = f dvdv*ff*[f su-, B(v-v,,a)(qg'-qg)da]. (110) As pointed out by Arkeryd [18], if ~0 is a smooth test-function, then ~ot - q9 will vanish when 0 ~ 0 (because then v~_~ v), and this may compensate for a singularity in B. This circumstance actually explains why one is able to compute relevant physical quantities, such as the cross-section for momentum transfer, even for non-cutoff potentials .... For the sake of discussion, we still consider the model case B(v - v., a) = Iv - v,l• Using moment estimates, and the formula ~0' - ~o= O([v - v,10) (which holds true when q9is smooth), Arkeryd [18] was able to prove existence of weak solutions for the spatially homogeneous Boltzmann equation as soon as y~-l, f b(cosO)O sinN-2 0 dO < 00. The use of the more symmetric form obtained by the exchange of variables v +-~ v, does not a priori seem to help a lot, since one has only Iqg'+ qgt,- q9- go,[ ~ C(go)lv - 0,120, so there is no gain on the angular singularity. But, as noted independently by several authors (see, for instance, [248,446]), an extra order of 0 can be gained by integrating in spherical coordinates. More precisely, use the standard parametrization of cr in terms of 0, r (r 6 SN-2), then fs (~o'+ ~o~,- ~p- ~p,)de N-2 C(99)1v- 0,120 2. (111) This simple remark enables one to extend Arkeryd's results to y ~>-2, f b(cosO)(1 - cos0) sinN-2 0 dO < ~. In dimension 3, these assumptions are fulfilled by inverse-power forces 1/r s when s ~>7/3 (to compare with s > 3 for Arkeryd's original result.., the reader may feel that the gain is infinitesimal, but remember that s = 2 should be the truly interesting limit exponent!). However, this point of view, which relies on the v +-~ v. symmetry, is in part misleading. The same control on the angular singularity (but worse in the kinetic singularity) can be
- 169. A reviewof mathematical topics in collisional kinetic theory 167 obtained without using the symmetry v ~ v,, as shown in Alexandre and Villani [12] by the use of more precise computations. When one is only interested in weak solutions in a spatially homogeneous problem, this remark is of no interest, but it becomes a crucial point in spatially inhomogeneous situations, or in the study of fine regularization properties. Here is a precise bound from [12]. Introduce the cross-section for momentum transfer, formulas (62) or (63). Then Q(g, f)q9 dv u ~<~ll~0llw2,~ 2N dvdv, g,f[v - v,[(1 + Iv - v,I)M(Iv - v,I). (112) To treat values of y below -2 with the help of formula (111) and others, it seems that one should require nontrivial a priori estimates like f~, f(v)f(v,) 2N ]U- U,[ -(y+2) dv dr, < +oc (the exponent of Iv - v,I in the denominator is positive!). As we shall see in Section 4.3, such estimates are indeed available in most cases of interest. But they are by no means easy! At the time when these extra estimates were not yet available, the search for a treatment of values of y below -2 led the author [446] to introduce a new weak formulation (H- solutions), based on the a priori bound f Q(f, f)q9 ~< ~D(f) 1 Bff,(go' nt- g)', - q) - g0,) 2 Here D is Boltzmann's entropy dissipation functional (47). This new bound was based on Boltzmann's weak formulation (45), and the elementary estimate D(f) >~s • xsN-1 B(v - v,,o)(v/f'ff, - fx~,)edvdv, da (113) (which follows from (X - Y)(log X - log Y) >~4(~/-X - ~/-~)2). It enabled the author to prove existence of weak solutions under the assumptions y > -4, f b(cosO)(1 - cos0) sinN-2 0 dO < e~, which allow the three-dimensional Coulomb potential as a limit (excluded) case. A main application was the first proof of the Landau approximation 8 for realistic potentials in a spatially homogeneous setting. 8See Section 5.
- 170. 168 C. Villani This use of the entropy dissipation for the study of grazing collisions had the merit to display some interesting feature: a partial regularity estimate associated to the entropy dissipation. More precisely, finiteness of the entropy dissipation, when the collision kernel is singular, implies a partial smoothness estimate for f f. in the tensor velocity space NN x R N. This effect is best seen at the level of the Landau equation: Landau's entropy dissipation can be rewritten as DL(f) = 2 f~,NxRN IFI(v -- v*)(V - V*)v/ ff*O(Iv - v*l)12dvdv*" (114) Recall that II(v- v,) is the orthogonal projection on (v- v,) • Equation (114) is a regularity estimate on the function ff,, but only in the variable v- v,, and only in directions which are orthogonal to v- v,. On the whole, this means N- 1 directions out of 2N. At the level of Boltzmann's entropy dissipation, for each point (v, v,) E ]1~2N, these N - 1 directions are precisely the tangent plane to the (N - 1)-dimensional manifold Svv, = {(v', v',)satisfying (5)}. One may conclude to the simple heuristic rule: entropy dissipation yields a smoothness estimate along collisions. These entropy dissipation bounds have a lot of robustness in a spatially homogeneous context, due to the tensorial structure of the entropy dissipation functional. For instance, one can prove that if De is the entropy dissipation functional associated to a Boltzmann operator QE "converging" in a suitable sense to Landau's operator, and DL is Landau's entropy disipation, and fe ~ f in weak L l, then D L(f) ~<lim inf DE(f~). e--+O On the other hand, precisely because they rely so much on the symmetry v +-~ v, and the tensor product structure, these methods turn out to be inadapted to more general problems. More efficient approaches will be presented in the sequel. 4.2. Cancellation lemma In this and the next two sections, we shall introduce more sophisticated tools for fine surgery on Boltzmann's operator. As discussed above, integrals such as f• dv.da B(v - v.,a)(g' - g) (115) NxSN-1 are well-defined for a smooth function g, at least if the collision kernel is not too much singular. When g is not smooth (say L1), it is not clear at all that such an integral should converge. This is however true with great generality, due to symmetry effects. A precise
- 171. A reviewofmathematicaltopicsin collisionalkinetictheory 169 quantitative version was introduced by the author in [449] (related estimates are to be found in Desvillettes [173] and Alexandre [6]). The estimate in [449] shows that when the collision kernel B in (115) depends smoothly on v - v, and presents an nonintegrable angular singularity of order v < 2, then the integral (115) converges. The need to cover more singular situations motivated further refinement of this estimate; here we present the sharp version which is proven in [12]. It only requires finiteness of the cross-section for momentum transfer, M, and a very weak regularity assumption with respect to the relative velocity variable. PROPOSITION 6. Let B(Izl, cos0) be a collision kernel with support in 0 E [0, re~2], and let S be defined by Sg =--fi[~N• N-I dr, do" B(v - v,, a)(g~, - g,). Then, for any g E L 1 (]t~vN), Sg=g*vS, where the convolution kernel S is given by f zr/2 dO sinu-2 0 S(Izl)-- IsN- l o E ' ( 'z' ) ,] x B ~ , c o s 0 - B(lzl,cos0 . (116) COSN (0/2) cos(0/2) Recall from Section 4.1 that the assumption about the deviation angle is no loss of generality. The proof of this lemma is rather elementary and relies on the change of ' which for fixed a E S N-1 is allowed if the integration domain avoids variables v, ~ v,, frontal collisions (0 ~_ -+-jr). Here is an easy corollary: COROLLARY 6.1. With the notations z = v - v,, k = (v - v,)/lv - v,I, let B'(z, or) - IB()~z, a) - n(z, a)l sup l<X~x/2 (Z- 1)[Z[ fs B'(z,a)(1 -k.a)da. M'(Iz[) = n-, (117) Then Is(Izl)L CN[M(Izt)+ tzIM'(Izl)],
- 172. 170 C. Villani where CN is a constant depending only on N. In particular, if IzIM'(Izl) L' (118) then S E t~oc(]l~N). What typical collision kernels are allowed by this lemma? It turns out that the quantity M I measures the regularity of B with respect to the relative velocity variable in a very weak sense. For instance, assumption (118) is satisfied for all potentials of the form B(v - v,, a) = Iv - v,l• y > -N. Of course, this excludes the borderline case where y = -N, which may be the most interesting, because in dimension 3 it corresponds to Coulomb interactions .... However, now it is homogeneity which will save the game. A quick glance at formula (116) may give the impression that if B is homogeneous of degree -N in the relative velocity variable, then S = 0!! Of course, this is a trap: S should be defined as a principal value operator, and after a little bit of algebra, one finds the following corollary to Proposition 6: COROLLARY 6.2. If B(v - v,, or)- Iv- v,l-N flo(cosO), then S=~60, where 3o is the Dirac measure at the origin, and f ro~2 fl0(cos0) logcos(0/2) sinN-2 0 dO. Z=--IsN-2I[sN-1IjO Note that ~ is finite as soon as the cross-section for momentum transfer (63) with b = rio, is. The compensation lemma of Proposition 6 has been a crucial tool, (1) to obtain sharp entropy dissipation estimates, see next section; (2) to derive a sharp renormalized formulation for the Boltzmann operator without cut-off, see Section 4.6. 4.3. Entropy dissipation estimates As we have explained in Section 4.1, under certain assumptions the entropy dissipation estimate yields a partial regularity bound on ff, when the collision kernel is singular. But does it imply a true regularity estimate on f itself? The first result in this direction is due to Lions [316]. He proved that if B(v - v,, or) >~ q~(lv - v,l)b(cosO), where q~ is smooth and positive, and sinN-2 0 b(cos0) ~> KO -l-v, then for all R > 0 there is a constant CR such that
- 173. A review of mathematical topics in collisional kinetic theory 171 [Iv/f(, .)112 , ns(Ivl<e) <~CR[D(f) 1/2 + IlfllL~], v( 1 ) s<s0=~ l+NV___~_ 1 9 (119) The exponent so here is not optimal. The idea of the proof was clever, and a very unexpected application of the Q+ regularity in L2 form (formula (99), with explicit bounds needed). Starting from the entropy dissipation and formula (113), one finds some smoothness-type estimate on v/f'f~ - ~ in ~;~U • ~U )< sN-1. Then one integrates this estimate with respect to the variables v, and a after multiplication by an artificial, well-chosen collision kernel B~(v - v,, a). An estimate on Qe (Vc-f, v/f) follows, where A A Qe is the Boltzmann operator associated with Be. Then one writes ,/7(L 9 ,/7) = 0 + (,/7, ,/7) - (,/7, ,/7), where AAe-- fsU_~ Be do. The regularity of 0 + and the estimates on Qe (~/-f, ~/c-f) yield the conclusion in the limit e ~ 0, after quite a bit of intricate computations. A completely different method [449], based on the compensation lemma of last paragraph and on Carleman's representation (91), led the author to a better estimate under more stringent assumption on f: assume that 8(v - v,, ~)/> ~0(Iv - v,l)bo(cosO), (120) where ~o(Izl) is continuous, ~0(Izl) > 0 if Izl > 0, (121) and (as usual) b0(cos0) sinN-2 0 >/KO -(l+v), K > O, v >/O. (122) Further assume that f is positive, then II f(t, )112 [D ], nv/Z(lvl<R ) ~ Cg (f) + Ilfll2{ (123) where CR depends on f only via inf{f(v); Ivl ~ R}. Exponent v/2 is optimal, as shown by a variant of the proof. But the assumption of local bound below for f is too strong. In a joint work with Alexandre, Desvillettes and Wennberg [10], we were able to extend the scope of (123), and eventually obtain sharp entropy dissipation estimates. The main difference with the previous argument was the replacement of Carleman's representation for the use of Fourier transform. In the next paragraph we shall make this a little bit more precise, for the moment we precisely state our main result in [10]. To this date this may be considered as the most general manifestation of the regularizing effects of grazing collisions.
- 174. 172 C. Villani PROPOSITION 7. Let f ~ L~ A L log L(~ N) be a nonnegative distribution function, and let B be a collision kernel satisfying assumptions (120)-(122). Then, for all R > 0 there is a constant C -- CR such that 11,/711H./2(IvI<R) <~CR[D(Z) + 11/112~], (124) where IlfllL~ = f f(v)(1 + Ivl2) dv and CR (which is explicit) depends only on N, ~o, K, v, R, a lower boundfor f f dr, and an upper boundfor f f(1 + Ivl2 + [log f[). REMARKS. (1) In the limit case v = 0, one recovers local estimates in log H. As a very general fact, as soon as the function b0(cos0) sinN-2 0 is not integrable, i.e., when s Z'Oo w-~ b0(cos0) sinN-2 0 dO diverges as O0 -+ O, then one can estimate x/'-f in Xloc, where X is the functional space defined in Fourier representation by X={F6L2(~N); fill>>.1 (1)} (2) This entropy dissipation estimate is asymmetric, and this is quite in contrast with the estimates which we shall discuss in the study of trend to equilibrium. In fact, the estimate holds just the same for f Q(g, f) log f (which is not always a nonnegative expression!), if one imposes B = ~0b0 instead of B ~>~0b0, and replaces [IfllL~ in (124) by IIfIIL~ + IlgllL~; then the constant CR would not depend on f but only on g. By (124), one can guess a precise heuristic point of view for the regularity properties associated with the non-cutoff Boltzmann operator: if the angular collision kernel is singular of order v (assumption (122)), and g is a fixed distribution function with finite mass, energy and entropy, then the linear operator f w-~ Q(g, f) "behaves" in the same way as thefractional diffusion operator --(--A) v/2. For Maxwellian molecules, the intuition of this result goes back to Cercignani [138], who had noticed, thirty years ago, that the eigenvalues of the linearized Boltzmann operator behaved like those of the power 1/4 of the Fokker-Planck operator. 4.4. Boltzmann-Plancherel formula A key step in the proof of Proposition 7 is the use of Fourier transform. As we said earlier, in the context of Boltzmann equation, it is only for Maxwellian collision kernels that the
- 175. A review of mathematical topics in collisional kinetic theory 173 Fourier transform leads to simple expressions. So one step of the proof (based on a lot of fine surgery) is the reduction to the purely Maxwellian case: 45o = 1 in the previous notations. Then, it all reduces to a sharp estimate from below of expressions of the form dvdv, fs do" g(v,)[F(v') - F(v)] 2 2N N-1 where g is an approximation of f (say, f multiplied by a smooth cut-off function), and F is an approximation of ~/-f. The following Plancherel-type formula, established in [10] after the ideas of Bobylev, is the appropriate ingredient. PROPOSITION 8. With the notations ~+ = (~ 4-I~lrr)/2, f• fs b(k.cr)g,(F'- F)2dvdv, dcr (125) 2N N-1 (2rr) N N N-1 ) 2 2 m - 29i (~(~-)F'(~+)F'(~)))d~ &r, with 92standingfor realpart. This general formula can be used in many other regularity problems, in particular to establish Sobolev-regularity estimates for the spatially homogeneous Boltzmann equation without cut-off [185]. 4.5. Regularization effects As a consequence of Proposition 7, one can derive some (rather weak) regularization theorems. This is immediately seen in the spatially homogeneous situation. Combining the entropy dissipation estimate fO T D(f (t, .))dt + H(f (T, .)) <<.H(fo) with Proposition 7, the a priori bound v/2 N ,/7 L2([o,r]; Hloc(< 11 (126) follows at once. Weak as it is, this regularization estimate is already useful to the existence theory for singular collision kernels. Indeed, assume that B(v - v,, or) = Iv - v,l• where
- 176. 174 C. Villani b satisfies the same assumptions as b0 in (122). As we saw in Section 4.1, when y is very negative one runs into trouble to define relevant weak solutions of the spatially homogeneous Boltzmann equation. But now, with this new entropy dissipation bound, one can get sufficient a priori estimates if the angular singularity is strong enough compared to -y. More precisely, if y + v + 2 ~>0, (127) then, by the Hardy-Littlewood-Sobolev and Sobolev inequalities, f ff, lv - v,I • dvdv, dt <~C IIf IIL~ (L1)IIf IIL](Lq) <. cllfoll , 11r L2(Hv/2) for some well-chosen q > 1 (everything being understood in local sense). It is worth pointing out that inequality (127) always holds for collision kernels coming out from inverse-power forces in dimension 3. We also note that the case which appears the most delicate to treat now, is the one of a collision kernel which is singular in the relative velocity variable but not in the angular variable; soon we shall encounter a similar problem in the spatially inhomogeneous setting. Other variants of these entropy dissipation estimates lead to (strong) compactness results. For instance, let (fn (t, l)))n6N be a sequence of probability distributions, satisfying sup{f0 T n Dn(fn)dt + sup (llfnllL~ogL + IlfnllL~)} < -+-~, t6[0,T] where Dn is the entropy dissipation associated to a collision kernel Bn, approximating (in almost everywhere sense for instance) a singular collision kernel B. Then, (fn) is strongly compact in L1. This holds true even if there is not necessarily a uniform smoothness estimate. The smoothing effects which we just discussed are rather weak, but using the same kind of techniques one can bootstrap on the regularity again and again, at least if the collision kernel is smooth with respect to the relative velocity variable. The key inequality can be formally written as d dt Ilfll2H~~ -KIIfll2 2 - - HOt+v~ 2 + C IIf IIH~. This easily leads, after integration on [0, T], to the immediate appearance of the H a+v/2 norm of f if the H a norm of f is initially finite- and, by induction, to immediate C ~ regularization. 9 9In fact this methodis but an adaptationof the "energymethod"in the studyof parabolic regularity,where similarestimateswouldholdwiththe constantv replacedby 2.
- 177. A reviewof mathematical topics in collisionalkinetic theory 175 Such a study is currently worked out by Desvillettes and Wennberg, who have announced a proof of C ~ instantaneous regularization for solutions to the spatially homogeneous Boltzmann equation, starting from an initial datum which has finite entropy. This result had already been proven in certain particular cases by Desvillettes [172,171, 173], and his student Prouti~re [390], with the use of Fourier-transform techniques. 4.6. Renormalized formulation, or F formula The treatment of regularizing effects for the full, spatially inhomogeneous Boltzmann equation without cut-off requires an additional tool because of the difficulty of defining the collision operator. As we explained in Section 1.3, a renormalized formulation of the collision operator, together with entropy dissipation estimates (in the sharp form of Proposition 7), is enough to prove appearance of strong compactness. For a long time this problem stood open, until Alexandre came up with a very clever idea [6]. The implementation of Alexandre's ideas, based on pseudo-differential theory, suffered from intricate computations and the impossibility to cover physically realistic collision kernels. In a joint work [12] with Alexandre, we have given a very general definition, based on the use of the cancellation lemma of Section 4.2, and the idea of using the asymmetric Boltzmann operator. Here is the renormalized formulation of [12], given in asymmetric formulation: fl'(f) Q(g, f) -- [ffl'(f)- fl(f)] ~NxSN_ , + Q(g' fl(f)) -- s215 1 dr, do- B(v - v,, o-) (g', - g,) dr, do- Bg~,F(f, f'), (128) where r (f, f') = fl (f') -- fl (f) -- fl' (f) (f' -- f). (129) If/3 is a strictly concave function or strictly convex function, then F has a fixed sign. In the context of the study of renormalized solutions, it will be convenient to choose/3 to be concave (sublinear), satisfying fl'(f) ~<C/(1 + f). Let us explain why each of the three terms in (128) is then well-defined. For the first one, we may assume f~'(f) - fl(f) 6 L ~, and then this term will satisfy an L1oc bound as a result of the cancellation lemma of Section 4.2. As for the second term in (128), it can be given a distributional sense, by means of formula (112). The estimate works in a spatially inhomogeneous context because the arguments of the collision operator are g (6 L 1, say) and fl(f) (6 L ~, say). This is the point where it is very important to have an asymmetric weak formulation! In the end, the third term is nonnegative as soon as/3 is concave, and since all other terms are well-defined, it satisfies an a priori estimate in L~oc for free -just as in the argument for the gain term in the DiPerna-Lions renormalization. 1~ l~ Section3.6.
- 178. 176 C. Villani Notice that this renormalization procedure can be understood as a "commutator" problem: find a nice expression for fl'(f)Q(g, f)- Q(g, fl(f)). (130) Such commutators are widely used in the study of linear diffusion operators. When L is a diffusion operator, then fl'(f) Lf - Lfl (f) = -fl" (f) I"(f), where F is the "Dirichlet form" associated with L. This justifies our terminology of"F formula". As a matter of fact, the renormalization procedure above presents some similarities with the renormalization of parabolic equations by Blanchard and Murat [76,77], and is also very close to the renormalization of the Landau operator given in Lions [311 ]. As an illustration of the drawbacks of "soft" theories, we note that the construction of renormalized solutions with the preceding definition is still an open problem. Instead, one is led to introduce the following, slightly weaker, definition: DEFINITION 2. Let f e C(~ +, 79t(]1~ xNX ]1~ vN))fq L~ (II~+ LI(~ u x N , R v )). It is said to be a renormalized solution of the Boltzmann equation with a defect measure, if for any nonlinearity fl e C2(~ +, R +) such that fl(0) = 0, 0 ~<fl'(f) ~<C/(1 + f), fl"(f) < O, one has 0 --fl(f) + v . Vxfl(f) ) fl'(f)Q(f, f), (131) Ot in distributional sense, and moreover f satisfies the law of mass-conservation: Yt~>0, fR f(t'x'v)dxdv= f~ 2N (132) We insist that this is really a notion of weak solution, not just sub-solution. Indeed, if f were smooth, then the combination of (131) and (132) shows that there is equality in (131). See [12], and also DiPerna and Lions [190] for similar situations. Finally, we note that formula (128) is a general tool which finds applications outside the theory of renormalized solutions, for instance in the study of regularity for the spatially homogeneous equation (or even for the spatially inhomogeneous one, if suitable integrability bounds are assumed). In this context, it is convenient to choose fl to be convex when studying regularization for initial data which belong to L log L or LP spaces, and concave when studying regularization for initial data which are only assumed to be probability measures. In fact, in some sense the F formula plays for the Boltzmann equation the same role as integration by parts plays in the energy method for diffusion operators; therefore one should not be surprised of its great utility.
- 179. A review of mathematical topics in collisional kinetic theory 177 4.7. Summary The following two theorems summarize our current knowledge of regularizing effects, respectively in the spatially homogeneous setting and in the framework of renormalized solutions. We restrict to the model cases B(v - v,, or) -- r - v,l)b(cosO), (133) where q0(Izl) > 0 for Izl ~ 0, and b satisfies the usual singularity condition, sinN-2 0 b(cos0) ~ KO -(l+v) as 0 --+ 0. (134) THEOREM 9. Let B satisfy Equations (133)-(134), and let fo be a probability density on R N, with bounded energy; fo may have a singular part, but should be distinctfrom a Dirac mass. 11 Then, (i) if cp is smooth and bounded from above and below, then there exists a solution f (t, v) to the Boltzmann equation with initial datum fo, which lies in C~((0, +cx~) x RN); (ii) if q~(lv -- v.[) --Iv -- v.[ • where Y + v > -2, and fo has finite entropy, then there exists a weak solution f (t, v) to the Boltzmann equation with initial datum fo, such that MY~2 & (iii) if r - v,I) = Iv - v,I • where Y + v > O, then, without further assumptions on fo there exists a weak solution f (t, v) to the Boltzmann equation with initial datum fo, such that Yt > O, f (t, .) ~ L logL(I~u). Point (i) of this theorem is work in progress by Desvillettes and Wennberg if one assumes that f0 has finite entropy. Then, in the case where one only assumes that f0 has finite mass and energy, work in progress by the author [440] shows that the entropy becomes finite for any positive time (actually, one proves estimates in L~oc(dr; L p (RN)), for arbitray large p). Key tools in these works are the Plancherel-like formula of Section 4.4 and the cancellation lemma of Section 4.2. Related to point (i) are probabilistic works by Fournier [217], Fournier and Mrlrard [219, 220] who prove immediate appearance of an L 1 density if the initial datum is not a Dirac mass, and C ~ smoothness for Maxwellian collision kernel in two dimensions [217]. The results by Fournier and Mrlrard are considerably more restricted because of strong de- cay assumptions on the initial datum, stringent assumptions on the smoothness of the ki- netic collision kernel and restrictions on the strength of the singularity. However, they have the merit to develop on Tanaka's approach [415] and to build a stochastic theory of the Boltzmann equation, whose solution is constructed via a complicated nonlinear stochastic l lBecause a Dirac mass is a stationary solution of the spatially homogeneous Boltzmann equation! so, starting from a Dirac mass does not lead to any regularization.
- 180. 178 c. Villani jump process. These works constitute a bridge between regularization tools stated here, and Malliavin calculus. They also have applications to the study of stochastic particle systems which are used in many numerical simulations [177,221,223,222,256,257]. In particular, they are able to study the numerical error introduced in Monte Carlo simulations when replacing a non-cutoff Boltzmann equation by a Boltzmann equation with small cut-off. 12 As for point (ii), it follows from the entropy dissipation estimates in Alexandre, Desvillettes, Villani and Wennberg [10] and by now standard computations which can be found, for instance, in Villani [446]. One would expect that when y > 0, C ~ smoothness still holds; current techniques should suffice to prove this, but it remains to be done. Point (iii) is from [440]. Uniqueness is still an open problem in this setting, on which the author is currently work- ing. This question is related to smoothing: if one wants to use a classical Gronwall strategy, like in the proof of uniqueness for the spatially homogeneous Landau equation [182], then one sees that the key property to prove is that (essentially) the non-cutoff bilinear Boltz- mann operator is not only "at least" as singular as the fractional Laplace operator of order v, but also "at most" as singular as this one, in the sense that it maps L 2 into H -v/2 (locally). We do hope for rapid progress in this direction! In the case of the spatially homogeneous Landau equation, then the same regularization results hold true, and are easier to get because the Landau equation already looks like a nonlinear parabolic equation. Hence the smoothing effect can be recovered by standard estimates (only complicated), bootstrap and interpolation lemmas between weighted Sobolev and Lebesgue spaces. It is possible to go all the way to C ~ smoothness even in cases where q/is not so smooth: for instance, qJ(Iv - v.I) = KIv - v.I z+2, y > 0. This study was performed in Desvillettes and Villani [182]. For this case the authors proved immediate regularization in Schwarz space, and uniqueness of the weak solution, in the class of solutions whose energy is nonincreasing, as soon as the initial datum satisfies f f2(v)(1 + Ivl2s) dv < +c~, 2s > 5y + 12 + s (N = 3). By the way, this uniqueness theorem of a weak solution, building on ideas by Arsen' ev and Buryak [41], required some precise Schauder-type estimates for a linear parabolic equation whose diffusion matrix is not uniformly elliptic in the usual sense, and our work has motivated further research in this area [11]. We emphasize that the picture is much less complete in the case y < 0. In particular, for the Landau equation with Coulomb potential (y = -3 in dimension N = 3), nothing is known beyond existence of weak solutions (see Villani [446] or the remarks in [10]). We now turn to the spatially inhomogeneous setting. It is a striking fact that no theorem of existence of classical small solutions of the Boltzmann equation without cut-off has ever been proven to this day, except maybe for the isolated results in [9] which still need further clarification. So we only discuss renormalized solutions. 12Apparently, Monte Carlo methods cannot be directly applied to the study of the non-cutoffBoltzmann equation. The onlymethodwhichseemsableto directlydealwithnon-cutoffcollisionkernels,withoutmaking somea prioritruncation,is the Fourier-baseddeterministicschemedescribedin Section4.9.
- 181. A reviewofmathematicaltopicsincollisionalkinetictheory 179 THEOREM 10. Assume that the collision kernel B is given by (133)-(134), and ~([v - v,I) = Iv - v,I • with 0 ~<v < 2, g ~> -N, y + v < 2. (135) Let (fn) be a sequence of solutions 13 of the Boltzmann equation, satisfying uniform estimates of mass, energy, entropy and entropy dissipation: sup sup fR n~N tc[0,T] N • fn(t,x, v)[1 + Ixl2 + Ivl2 + logfn(t,x, v)] dx dv < -+-~. (136) fo T sup D( fn (t, x, .)) dx dt < +{x). (137) n6N Without loss of generality, assume that fn __+f weakly in L p ([0, T]; L I(RN • INN)). Then, (i) f is a renormalized solution of the Boltzmann equation with a defect measure; (ii) automatically, fn __+f strongly in L 1. COROLLARY 10.1. Let fo be an initial datum with finite mass, energy and entropy: f• Co(x,v)[1 -t-Ivl 2 N • + Ix 12+ log f0 (x, v)] dx dv < +o~. Then there exists a renormalized solution with a defect measure, f (t,x, v), of the Boltzmann equation, with f (0,., .) -- Co. This theorem is proven in Alexandre and Villani [12], answering positively a conjecture by Lions [308]. The result holds in much more generality, for instance, it suffices that the angular collision kernel be nonintegrable (no need for a power-law singularity), and the kinetic collision kernel need not either take the particular form of a power-law, if it satisfies some very weak regularity assumption with respect to the relative velocity variable. And also, it is not necessary that the collision kernel split into the product of a kinetic and an angular collision kernel. We mention all these extensions because they are compulsory when one wants to include realistic approximations of the Debye collision kernel, which is not cut-off, but not in product form .... The strategy of proof is the following. First, by Dunford-Pettis criterion, the sequence (fn)ncr~ is weakly (relatively) compact in L 1. Then, by the renormalized formulation, and the averaging lemmas, one shows that velocity-averages of the fn's are strongly compact. Then the entropy dissipation regularity estimates yield bounds of regularity in the v variable, outside of a small set and outside of a set where the fn's are very small. As 13Eitherrenormalizedsolutions,or renormalizedsolutionswitha defectmeasure,or approximatesolutions,as in Theorem5.
- 182. 180 C. Villani a consequence, the sequence (fn)n6N can be very well approximated by velocity-averages, and therefore it lies in a strongly compact set (as in [311 ]). Let us comment on the range of parameters in (135). The assumption y + v < 2 is just a growth condition on the kinetic collision kernel, and is a natural generalization of the assumption y < 2 in the DiPerna-Lions theorem; by the way, for inverse s-powers in three dimensions, the inequality y + v < 1 always holds true. But now, we see that there are two extensions: first, the possibility to choose v 6 [0, 2) (which is the optimal range), secondly, the possibility to have a nonintegrable kinetic collision kernel, provided that the singularity be homogeneous of degree -N. This feature allows to deal with Coulomb-like cross-sections in dimension 3. By the way, a problem which is left open is whether the theorem applies when the collision kernel presents a nonintegrable kinetic singularity of order -N but no angular singularity. Such collision kernels are unrealistic, but sometimes suggested as approximations of Debye collision kernels [162]. The renormalized formulation above is able to handle this case (contrary to the DiPerna-Lions renormalization), but without angular singularity the regularizing effect may be lost- or is it implied by the nonintegrable singularity, as some heuristic considerations [12] may suggest? A result quite similar to Theorem 10 (actually simpler) holds for the Landau equation, see Lions [311 ], and also Alexandre and Villani [13]. To this day, no clean implementation of a regularization effect has been done in the framework of spatially inhomogeneous small solutions. Desvillettes and Golse [176] have worked on an oversimplified model of the Boltzmann equation without cut-off, for which L ~ solutions can be constructed for free. For this model equation they prove immediate H a regularization for some ot which is about 1/30. In fact, regularization for the spatially inhomogeneous Boltzmann equation without cut- off may be understood as a hypoellipticity problem- with the main problem that the diffusive operator is of nonlocal, nonlinear nature. F. Bouchut has recently communicated to us some very general methods to tackle hypoelliptic transport equations in a Sobolev space setting, via energy-type methods; certainly that kind of tools will be important in the future. 5. The Landau approximation In this section, we address the questions formulated in Section 2.7. In short, how to justify the replacement of Boltzmann's operator by Landau's operator in the case of Debye (= screened Coulomb) potential when the Debye length is very large compared to the Landau length? 5.1. Structureof the Landauequation We recall here the structure of the Landau operator, in asymmetric form: QL(g' f)= Vv "(f RN dv, a(v- v,)[g,(V f) - f (Vg),]), (138)
- 183. A reviewof mathematical topics in collisional kinetic theory 181 ZiZj] aij(Z) = q/(Izl) ~ij iz[2 . (139) The Landau operator can also be rewritten as a nonlinear diffusion operator, Q L(g, f)= V . ({tV f -- [~f ) - Z {tijOijf -- ~" f , ij (140) where b = V .a, c = V. b, or more explicitly bj = Z Oiaij, c- ~-~ Ojbj, i j and ~=a.g, /~=b.g, ?=c.g. There is a weak formulation, very similar to Boltzmann's, for instance, fR QL(g, f)~o=fR g, f TLqgdvdv,, N 2N where [7-s v,) = -2b(v- v,) . Vgo(v) -+-a(v- v,) : D2qg(v). (141) Compare this with the following rewriting of Maxwell's weak formulation of the Boltzmann equation: f~ QB(g, f)go= fR g*f 7"q9dv dr,, N 2N where [7"~0](V, V#) -- fsN-1 B(v - v,, a)(qg' - qg)da. (142) 5.2. Reformulation of the asymptotics of grazing collisions As we explained in Section 3.5, one expects that the Boltzmann operator reduce to the Landau operator when the angular collision kernel concentrates on grazing collisions, the total cross-section for momentum transfer being kept finite. The first rigorous proofs concerned the spatially homogeneous situation: Arsen'ev and Buryak [41] for a smooth kinetic collision kernel, Goudon [248] for a kinetic singularity
- 184. 182 C. Villani of order less than 2, Villani [446] for a kinetic singularity of order less than 4. All proofs were based on variants of the weak formulations above, and used the symmetry v +-~ v.. In order to extend these results to the spatially inhomogeneous setting, there was need for a renormalized formulation which would encompass at the same time the Boltzmann and Landau collision operators. This was accomplished with the results about the Boltzmann equation without cut-off in [12]. Here is the renormalized formulation of the Landau equation: fl'(f)QL(g, f) = -~[ffl'(f) - fl(f)] + V. [V. (~fl(f)) - 2/~fl(f)] fl"(f) aVfl(f) Vfl(f) (143) fl,(f)2 Again, fl stands for a concave nonlinearity, typically fl (f) = f~ (1 + 8f). If one notes that the second term in the fight-hand side of (143) can be rewritten as Qc (g, fl(f)), there is an excellent analogy between this renormalization and the renormalization of the Boltzmann operator which was presented in Section 4.6. This is what makes it possible to pass to the limit. The convergence of the first and second terms in the renormalized formulation can be expressed in terms of the kernels S (appearing in the cancellation lemma) and T. This allows one to cover very general conditions for the asymptotics of grazing collisions, and this generality is welcome to treat such cases as the Debye approximation. Here we only consider a nonrealistic model case. Let (Bn)n~N be a sequence of collision kernels Bn(v - v., or) -- clg(Iv - v.I) bn(cosO), (144) where the kinetic collision kernel ~/, satisfies 9 (Izl) >0, (145) Izl~c~ [ ~(~lzl) - ~(Izl) ] q'(lzl), sup 6 L~oc(RN) (146) 1<z~<J~L )~-- 1 and the family of angular collision kernels (bn)n6Nconcentrates on grazing collisions, in the sense V00 > 0, sup bn (cos 0) ~ 0, 0/>00 n--+c~ L bn(k" or)(1 - k. or) dtr ~/z > 0, Ikl = 1. N-1 n---+ Oo (147) Let Sn be the kernel associated to Bn as in Section 4.2, and Tn be the linear operator associated to Bn as in formula (142). Moreover, let Izl2~(Izl) (Izl) = 4(N - 1)
- 185. A reviewofmathematicaltopicsin collisionalkinetictheory 183 and let QL, TL be the associated quantities entering the Landau operator. Then, (z ) Sn(IZl) n__,~(N-1)V" ~~(IZl) in weak-measure sense, and % >7c I"/----->oo in distributional sense. In this sense one can say that the sequence of Boltzmann kernels Qn approaches QL. These lemmas are not enough to pass to the limit. It still remains (1) to gain strong compactness in the sequence of solutions to the Boltzmann equation, (2) to pass to the limit in the last term of the renormalized solution. Task (2) is a very technical job, based on auxiliary entropy dissipation estimates and quite intricate computations, from which the reader is unlikely to learn anything interesting. On the other hand, we explain a little bit about the strong compactness. 5.3. Damping of oscillations in the Landau approximation As we have seen earlier, entropy dissipation bounds for singular Boltzmann kernels entail the appearance of strong compactness, or immediate damping of oscillations. In the case of the Landau equation, this is the same. It turns out that it is also the same if one considers a sequence of solutions of Boltzmann equations in which the collision kernel concentrates on grazing collisions, in the sense of (147). This is a consequence of the following variant of our joint results in [10]: PROPOSITION 11. Assume that Bn(v - v,, a) ~ 450(Iv- v,I)bo,n(cosO), where 49o is continuous, q~(lzl) > 0 for Izl > 0, and bo,n concentrates on grazing collisions, in the sense of (147). Then there exists #: > 0 and a sequence or(n) --+ 0 such that f0 a(n) sinN-2 0 bn(cosO) (1 - cos0) dO >/z' > 0, n---+ oo f Jr bn(cosO) = ~(n) sinN-2 0 dO (n) > _t_oo, n---+ oo (148) and there exists K > 0 such that f0 rr sinN-2 0 bn(cosO) (021~12A 1)dO /> K min[ap(n), 1~12]. (149)
- 186. 184 C Villani In particular, for any distribution function f, let F = X~ be obtained by multiplication of q/-f with a smooth cut-offfunction X, then I~>R 1 1 )[Dn(f) -+-IlfllL2~ ] d~: ~<C max ap(n)' R2 where Dn is the entropy dissipation functional associated with Bn, and C depends on f only via a lower boundfor f f dv and an upper bound for f f(1 + Ivl2 + Ilog f[) dr. As a consequence of this proposition, strong compactness is automatically gained in the asymptotics of grazing collisions. By the way, this simplifies already existing proofs [446] even in the spatially homogeneous setting. 5.4. Summary Here we give a precise statement from [13]. THEOREM 12. Let Bn be a sequence of collision kernels concentrating on grazing collisions, in the sense of (144)-(147). Further assume that ~(Izl) > 0 as Izl > 0. Let (fn)nrN be a sequence of renormalized solutions of the Boltzmann equation (with a defect measure) ofn Ot -~-1). 7xf n = Qn(f n, fn), satisfying uniform bounds of mass, energy, entropy, entropy dissipation. Without loss of generality, assume that fn __+f in weak L 1. Then, the convergence is automatically strong, and f is a renormalized solution (with a defect measure) of the Landau equation with !P(lzl) = 4(N- 1)Izl2q~(Izl)" REMARK. Theorem 12 allows for kinetic collision kernels with a strong singularity at the origin, but does not allow collision kernels which are unbounded at large relative velocities. This theorem includes all preceding results in the field, however in a spatially homogeneous situation one could reasonably hope that present-day techniques would yield an explicit rate of convergence (as n --+ o0) when q~is not too singular. On the other hand, when ~(Izl) = 1/Izl 3, an improvement of this theorem even in the spatially homogeneous setting would require a much deeper understanding of the Cauchy problem for the Landau equation for Coulomb interaction. 14 14Seethe discussionin Section1.3ofChapter2E.
- 187. A review of mathematical topics in collisional kinetic theory 185 6. Lower bounds We conclude this chapter with estimates on the strict positivity of the solution to the Boltzmann equation. Such results are as old as the mathematical theory of the Boltzmann equation, since Carleman himself proved one of them. At the present time, these estimates are limited to the spatially homogeneous setting, and it is a major open problem to get similar bounds in the full, x-dependent framework in satisfactory generality. Therefore, we restrict the ongoing discussion to spatially homogeneous solutions. Even in this situation, more work remains to be done in the non-cutoff case. 6.1. Mixing effects First consider the case when Grad's angular cut-off is satisfied, and Duhamel's for- mula (100) applies. Then one is allowed to write f0t f (t, v) >~ e-f;' Lf(r'v)dr Q+(f, f)(s, v)ds, (150) f(t, v) ~ e-fo Lf(r,v)dr fo(U), (151) where Lf = A 9 f, A(z) -- f B(Z, a)da. As a trivial consequence of (151), if f0 is strictly positive (resp. bounded below by a Maxwellian), then the same property will be true for f (t, .). But a much stronger effect holds true: whatever the initial datum, the solution will be strictly positive at later times. Just to get an idea of this effect, assume that A is bounded from above and below, so that u t 6 [0, T], e-f;' Lf(r,v)dr ~ KT > 0 for some constant KT depending on T. Then, as a consequence, f0t f (t, v) ~ KT Q+ (f, f)(s, v) ds, 0 ~<t ~<T. (152) Further assume that f0 >~otls, c~>0, (153) where 18 is the characteristic function of some ball B in velocity space, without loss of generality B is centered on 0. From (151) it follows that f (t, .) >/OtKT1B, O~ t <<.T.
- 188. 186 C. Villani Now, plug this inside (152), to find that fO t f(t, v) >~ot2K3T Q+(1B, 1B)(S, v)ds, O<~t<.T. But Q+(1 B, 1B) is positive and bounded below in all the interior of the ball (1 + 6)B for 6 small enough. In particular, there is a positive constant 13 such that f (t, v) >~ot2K3Tfl l(l+a)B. By an immediate induction, Yt > 0, Yv ~ ]1~ N, f (t, v) > O. Precise estimates of this type have enabled A. Pulvirenti and Wennberg [392,393] to prove optimal (Gaussian-type) bounds from below on f, for the spatially homogeneous Boltzmann equation with Maxwellian or hard potentials. In this respect they have improved on the old results by Carleman [119], who obtained a lower bound like e-Ivl2+e (e > 0) in the case of hard spheres. Assumption (153) can also be dispended with, by use of the Q+ regularity. Also the proofs in [393] are sharp enough to prove existence of a uniform (in time) Maxwellian lower bound. 6.2. Maximumprinciple The author suggests another explanation for the immediate appearance of strict positivity, which is the maximumprinciple for the Boltzmann equation. The study of this principle is still under progress, so we cannot yet display explicit lower bounds obtained with this method; the most important feature is that it applies in the non-cutoff case. Let us just give an idea of it. Rewrite the spatially homogeneous Boltzmann equation as -- B f, (f - f) Ot N• + f(fRN• dr, dcr B(f~- f,)). (154) We assume that we deal with a C ~ solution, which is reasonable when the kinetic collision kernel is nice and when there is a nonintegrable angular singularity. The good point about the decomposition (154) is that it is well-defined 15 even in the non-cutoff case. Assume now, by contradiction, that there is some point (to, v0) (to > 0) such that f(to, v0) = 0. Obviously, 8f/St = 0 at (to, v0). Thus the left-hand side of (154), and also 15By cancellation lemma, for instance, see Section 4.2.
- 189. A reviewof mathematical topics in collisionalkinetic theory 187 the second term on the right-hand side vanish at (to, v0). But, when v = v0, f' - f ~>0, for all vf. Thus the integrand in the first term on the left-hand side of (154) is nonnegative, but the integral vanishes, so f' = f = 0, for all vt. This entails that f is identically 0, which is impossible. In other words, we have recovered the weak result that f(t, .) is strictly positive on the whole of ~U for t > 0. 6.3. Summary THEOREM 13. Let B be a collision kernel of the form B(v - v., or) = Iv - v.l• where y ~ O. Let fo be an initial datum with finite mass and energy, and f (t,-) be a solution of the spatially homogeneous Boltzmann equation. Then, (i) if Grad's angular cut-off condition holds, then for any to > O, there exists a Maxwellian distribution M(v) such that for all t ~ to, f (t, v) >1M(v); (ii) if Grad's angular cut-off condition does not hold, and f (t, v) is a C ~ function on (0, +oo) x IRy, then for any t > O, v ~ It~N, f (t, v) > O. Point (i) is due to A. Pulvirenti and Wennberg [392,393]. Point (ii) was first proven by Fournier, using delicate probabilistic methods, in the special case of the Kac equation without cut-off [218], then also for the two-dimensional Boltzmann equation under technical restrictions [218]. Then it was proven in a much simpler way by the author, with the analytical method sketched above. Current work is aiming at transforming this estimate into a quantitative one. We note that in the case of the Landau equation with Maxwellian or hard potential [182], one can prove a theorem similar to that of A. Pulvirenti and Wennberg by means of the standard maximum principle for parabolic equations. 16 16Actually,in [182] the statedresult is not uniform in time, but, as suggested to us by E. Carlen,a uniform bound is easily obtainedby tracing back all the constants: sincethey are uniform for t ~ (e, 2e) and do not depend on the initial datum, it follows that they are uniform in t > e.
- 190. CHAPTER 2C H Theorem and Trend to Equilibrium Contents 1. A gallery of entropy-dissipating kinetic models ............................... 191 1.1. Spatially homogeneous models ..................................... 192 1.2. Spatially inhomogeneous models .................................... 195 1.3. Related models .............................................. 198 1.4. General comments ............................................ 199 2. Nonconstructive methods ........................................... 200 2.1. Classical strategy ............................................. 200 2.2. Why ask for more? ............................................ 202 2.3. Digression ................................................ 203 3. Entropy dissipation methods ......................................... 203 3.1. General principles ............................................ 203 3.2. Entropy-entropy dissipation inequalities ................................ 205 3.3. Logarithmic Sobolev inequalities and entropy dissipation ....................... 206 4. Entropy dissipation functionals of Boltzmann and Landau ......................... 208 4.1. Landau's entropy dissipation ...................................... 208 4.2. Boltzmann's entropy dissipation: Cercignani's conjecture ....................... 210 4.3. Desvillettes' lower bound ........................................ 212 4.4. The Carlen-Carvalho theorem ...................................... 213 4.5. Cercignani's conjecture is almost true ................................. 215 4.6. A sloppy sketch of proof ......................................... 217 4.7. Remarks .................................................. 222 5. Trend to equilibrium, spatially homogeneous Boltzmann and Landau ................... 224 5.1. The Landau equation ........................................... 224 5.2. A remark on the multiple roles of the entropy dissipation ....................... 225 5.3. The Boltzmann equation ......................................... 226 5.4. Infinite entropy .............................................. 227 6. Gradient flows ................................................. 228 6.1. Metric tensors ............................................... 228 6.2. Convergence to equilibrium ....................................... 229 6.3. A survey of results ............................................ 232 7. Trend to equilibrium, spatially inhomogeneous systems .......................... 235 7.1. Local versus global equilibrium ..................................... 235 7.2. Local versus global entropy: discussion on a model case ....................... 237 7.3. Remarks on the nature of convergence ................................. 240 7.4. Summary and informal discussion of the Boltzmann case ....................... 241 189
- 191. Preface The motion of fluids has intrigued scientists since antiquity and we may say that the field of mathematical fluid dynamics originated more than two centuries ago. In 1755 Euler [2] gave a mathematical formulation of the principle of conservation of mass in terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear and angular momentum by PDEs. Material symmetry and frame invariance were used by Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint- Venant [11] and Stokes [12]. In the 19th century no sharp distinction was drawn between mathematicians and physicists as we sometime see in more recent times. The formulation of the equations of fluid motion could be considered as either mathematics or physics. The first work in fluid dynamics that has a "modern" mathematical flavor may have been done by Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous problem. Riemann recognized the mathematical nature of the entropy. This notion led him to his duality method for solving the non-characteristic Cauchy problem for linear hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his contemporaries. What we now call the Cauchy problem for a PDE and the search for its solution did not have the significance that it is accorded nowadays. Even Poincar6 did not raise that kind of question in his Th~orie des tourbillons. For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28. Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much as Riemann, Leray developed new mathematical tools which proved to have independent interest: e.g., weak solutions (that we now call Leray's solutions in this context) and topological degree (a joint work with Schauder [4]). Since the 1930s, the interest that mathematicians devote to fluid dynamics has unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic displays such a variety of models and questions that thousands of scientists, among them many mathematicians, focus their research on fluid dynamics.
- 192. A review of mathematical topics in collisional kinetic theory 191 In Chapter 2A we have discussed Boltzmann's H theorem, and the natural conjecture that the solution of Boltzmann's equation converges towards statistical equilibrium, which is a global Maxwellian distribution. In this chapter we shall study this problem of trend to equilibrium, and also enlarge a little bit the discussion to models of collisional kinetic theory which are variants of the Boltzmann equation: for instance, Fokker-Planck-type equations, or simple models for granular media. The Cauchy problem for these equations is usually not so challenging as for the Boltzmann equation, but the study of trend to equilibrium for these models may be very interesting (both in itself, and to enlighten the Boltzmann case). As a general fact, one of the main features of many collisional kinetic systems is their tendency to converge to an equilibrium distribution as time becomes large, and very often a thermodynamical principle underlies this property: there is a distinguished Lyapunov functional, or entropy, and the equilibrium distribution achieves the minimum of this functional under constraints imposed by the conservation laws. In Section 1 we shall review some of these models. For each example, we shall be interested in the functional of entropy dissipation, defined by the equation d] e[:<t)], D(fo) =-~--~ t=0 where E is the Lyapunov functional, and (f(t))t>~o the solution to the equation under study, f (0) = f0. We shall use the denomination "entropy dissipation" even when E is not the usual Boltzmann entropy. Traditional approaches for the study of trend to equilibrium rely on soft methods, like compactness arguments, or linearization techniques, which ideally yield rates of convergence. In Section 2 we briefly review both methods and explain why they cannot yield definitive answers, and should be complemented with other, more constructive methods. This will lead us to discuss entropy dissipation methods, starting from Section 3. In Section 4, we expose quantitative versions of the H theorem for the Boltzmann and Landau operators, in the form of some functional inequalities. Then in Section 5 we show how these inequalities can be used for the study of the trend to equilibrium for the spatially homogeneous Boltzmann and Landau equations. Section 6 is devoted to a class of collision models which exhibit a particular gradient structure. Specific tools have been devised to establish variants of the H theorem in this case. Finally, Section 7 deals with the subtle role of the position variable for spatially inhomogeneous models. The construction of this area is only beginning. 1. A gallery of entropy-dissipating kinetic models Let us first review some of the basic models and the associated entropy functionals, equilibria, entropy dissipation functionals. We shall not hesitate to copy-cut some of the formulas already written in our introductory chapter.
- 193. 192 C. Villani 1.1. Spatially homogeneous models These models read of = Q(f), t ~ O, 1) e ]I~N , Ot where the collision operator Q, linear or not, may be (1) the Boltzmann operator, " Q(f) = QB(f, f)= dv, da B(v - v,, a)(f f, - ff,); N N-1 (155) then there are three conservation laws: mass, momentum and energy. Moreover, the natural Lyapunov functional is the H-functional, H (f) = fRN f log f, and its dissipation is given by the by now familiar functional if ,, D(f) -- -~ dv dr, dcr B(v - v,, a)(f'f~, - ff,) log f f* ~>O. (156) ff, Define p, u, T by the usual formulas (1), then the equilibrium is the Maxwellian M(v)--Mf(v)= Iv-ul2 e 2T (2Jr T)N/2 " IMPORTANT REMARK. We shall only consider here the case of the Boltzmann equation with finite temperature. In the case of infinite temperature, almost nothing is known, except for the very interesting recent contribution by Bobylev and Cercignani [81]. It should be noted that, since M has the same moments as f up to order 2, H (f) - H (M) = fR f log f N M' which is nothing but the Kullback relative entropy of f with respect to M, and that we shall denote by H (f[ M). Generally speaking, the Kullback relative entropy between two probability densities (or more generally two nonnegative distributions) f and g is given by the formula H(flg) = f f log f. g (157)
- 194. A reviewof mathematical topics in collisional kinetictheory 193 It is well-known 1that H(flg) ~ 0 as soon as f and g have the same mass; (2) the Landau operator, Q(f)=QL(f,f)=Vv.(s dv, a(v-v,)[f,(Vf)- f(Vf),]), aij (z) -- O(Izl) aij izi2 , (158) (159) in this case there are also three conservation laws, and the natural Lyapunov functional is also the H-functional. Now the entropy dissipation is 1s DL(f ) -- -~ NxRUff, lP(Iv- v,l)lFI(v- v,)(V(log f)- [V(log f)],) 12, (160) where H (z) stands for the orthogonal projector onto z• As for the equilibrium state, it is still the same as for the Boltzmann equation; (3) the linear Fokker-Planck operator, Q (f) = Q FP (f) -- Vv" (gv f + f v). (161) In this case there is only one conservation law, the mass (p = f f dr), and the natural Lyapunov functional is the free energy:this is the sum of the H-functional and the kinetic energy, E(f) fir f log f + fR fly]2 = dr. (162) N N 2 Moreover, the entropy dissipation is DFP(f) = fRu f -+v 2 dr, which can be rewritten as the so-called relative Fisher information of f with respect to M, thereafter denoted by I (flM). More generally, I (fig) -- ~N f f V log g (163) 1The classical proof is to rewrite (157) as f f[- log(g/f) + g/f - 1] (or as f g[(f/g) log(f/g) - (f/g) + 1]) and to use the inequality log X ~<X - 1 (or X log X ~>X - 1). Compare with the Cercignani-Lampis trick of Equation (50).
- 195. 194 C. Villani Compare with the definition of the relative Kullback entropy (157); (4) a coupled Fokker-Planck operator, like p"v~. [rv~f + f(v - u)], where 0 ~<a ~< 1 and p, u, T are coupled to f by the usual formulas (1). In this case there are three conservation laws, the natural Lyapunov functional is the H-functional, and the entropy dissipation is P~fR~u f 12__pOt f) Vv log ~--f I(flM . The equilibrium is the same as for the Boltzmann operator. Other couplings are possible: one may decide to couple only T, or only u ... ; (5) some entropy-dissipating model for granular flow, like the one-dimensional model proposed in [70], Q(f) = Vv . (f Vv(f 9 u)), (164) where U (z) = Izl3/3. Then there are two conservation laws, mass and momentum; and the natural Lyapunov functional is 1s f(v)f(w)U(v-w) dvdw, E(f) = -~ 2N (165) while its dissipation is D(f) = s flVU . fl 2. Moreover the equilibrium is p6u, i.e., a multiple of the Dirac mass located at the mean velocity. A particular feature of this model is its gradient flow structure. Generally speaking, models of the form Ofat-- V " (f V6-~)' (166) where E is some energy functional and 6E/6f stands for its gradient with respect to the usual L 2 structure, can be considered as gradient flows [364,365], via geometric and analytical considerations which are strongly linked with the Wasserstein distance. 2 An 2TheWassersteindistanceis definedby Equation(244).The gradientstructureis explainedin Section6.1.
- 196. A review of mathematical topics in collisional kinetic theory 195 integration by parts shows that solutions of (166) admit E as a Lyapunov functional, and the dissipation is given by D(f) = fIRNf 6E Falling into this category is in particular the model for granular flow discussed in [68], in which one adds up the collision operators (164) and (161). 1.2. Spatially inhomogeneous models These models can be written in the general form 0f + v. Vxf + F(x). Vvf -- Q(f), t ~ O, X E •N, V E R N, (167) Ot where F is the sum of all macroscopic forces acting on the system, and Q is one of the collision operators described in the previous paragraph (acting only on the velocity variable!). In the sequel we shall only consider the situation when the total mass of the gas is finite; without loss of generality it will be normalized to 1. We mention however that the case of infinite mass deserves interest and may be studied in the spirit of [310]. If the total mass is finite, then among the forces must be a confinement which prevents the system from escaping at infinity, and ensures the existence of a relevant equilibrium state. There are several possibilities: Potential confinement. Assume that the particles interact with the background environ- ment via some fixed potential, V(x). Then the force is just F(x) = -VV(x). The minimum requirement for V to be confining is e-v 6 L 1. Since V is defined up to an additive constant, one can assume without loss of generality that fR e-V(x) dx - 1. N The presence of the confining potential does not harm the conservation of mass, of course; on the other hand, when Q is a Boltzmann-type collision operator (with three conservation laws), it usually destroys the conservations of momentum and energy. Instead, there is conservation of the total mechanical energy, E v2] f (x, v) V (x) + dx dv N • --2 "
- 197. 196 C. Villani And as far as the entropy is concerned, it is not changed for Boltzmann-type models: this is still the usual H-functional, the only difference being that now the phase space is f H (f) -- / f log f. JRN• This similarity is a consequence of the physical assumption that collisions are localized in space. For the linear Fokker-Planck equation, the situation is different: to the free energy one has to add the potential energy, so the natural Lyapunov functional is E(f) = LN xRN f log f + f~N > < ~ N f (x v)[V(x) + I~ 2] , ~ dxdv. Box confinement. Another possible confinement is when the system is enclosed in a box X C NN, with suitable boundary conditions. The most standard case, namely specular reflection, is a limit case of the preceding one: choose V = +cx~ outside of X, V -- const. within X. When specular reflection is imposed, then the energy conservation is restored (not momentum conservation), and the Lyapunov functional is the same as in the spatially homogeneous case, only integrated with respect to the x variable. For other boundary conditions such as diffusive, the entropy functional should be modified [141,143]. Torus confinement. This is the most convenient case from the mathematical point of view: set the system in the toms TN, so that there are no boundaries. Physicists also use such models for discussing theoretical questions, and numerical analysts sometimes find them convenient. Additional force terms. Many models include other force terms, in particular self- consistent effects described by mean-field interactions: typically, F(x) = -74~(x), I~ =~,p, P = f]t{N f d v , where r is an interaction potential between particles. As we already mentioned, from the physical point of view it is not always clear whether interactions should be modelled via collisions, or mean-field forces, or both .... A very popular model is the Vlasov-Fokker- Planck equation, in which the collision operator is the Fokker-Planck operator and the forces include both confinement and self-consistent interaction. Let us rewrite the model explicitly: Of -57 + v . Vxf + F(x) . Vvf = Vv . (Vvf + fv), F -- -V(V + r 9 p), p = fRN f dr.
- 198. A review of mathematical topics in collisional kinetic theory 197 If the interaction is Coulomb, then one speaks of Vlasov-Poisson-Fokker-Planck model; this case is very singular, but it has a lot of additional structure because (by definition) the potential 4~is the fundamental solution of the Laplace operator. In the self-consistent case, one has to add a term of interaction energy to the free energy: 2 N• p(x)p(y)ck(x - y) dx dy. Then the entropy dissipation is unchanged. Let us now turn to equilibrium states. Their classification in a spatially inhomogeneous context is quite a tedious task. Many subcases have to be considered, the dimension of the space comes into play, and also the symmetries of the problem. We are not aware of any systematic treatment; we shall only consider the most typical situations. 9 The Boltzmann (or Landau) equation in a box. Then, in dimension N = 2, 3 there is a unique steady state which takes the form of a global Maxwellian: f(x, v) = M(v). The mass and temperature of the Maxwellian are determined by the conservation laws, while the mean velocity is 0. This result holds true on the condition that the box be not circular in dimension N = 2, or cylindric in dimension N = 3 (i.e., with an axis of symmetry). For this one can consult [254,167,143]. 9 The Boltzmann (or Landau) equation in a confining potential. Then, the unique steady state has the form f(x, v) = e-V(X)M(v). Again, the mass and temperature of M are determined by the conservation laws, and the mean velocity is 0. This result holds true if the potential V is not quadratic; if it is, then there exist periodic (in time) solutions, which can be considered as stationary even if they are time-dependent. This was already noticed by Boltzmann (see, for instance, [143]). 9 The Boltzmann (or Landau) equation in a toms. Then, the unique steady state has the form f (x, v) = M (v) where M is an absolute Maxwellian. The mass and temperature, but also the mean velocity of M are determined by the conservation laws. 9 The Fokker-Planck equation in a confining potential. Then, the unique steady state is f(x, v) -- e-V(x)M(v), where M is the Maxwellian with unit temperature and zero mean. The mass is of course determined by the conservation law. 9 The Vlasov-Fokker-Planck equation in a confining potential. In general there is a unique steady state in this situation, and it takes the form f(x, v) = p~(x)M(v), where M is the Maxwellian with unit temperature and zero mean. The density p~ is nonexplicit, but solves a nonlinear equation of the form e-(V+4~,p~) fRN e-(V+g'*P~ dx There are also variational formulations of this problem. In the case of the Vlasov- Poisson-Fokker-Planck equation, a detailed survey of the situation is given by Dolbeault [197]. In all the preceding discussion, we have avoided the models for granular collisions, Equation (164). A naive guess would be that the natural Lyapunov functional, in the spatially inhomogeneous case, is obtained by integrating its spatially homogeneous
- 199. 198 C. Villani counterpart, Equation (165), in the x variable. This is false! Because the transport operator -v. Vx may have an influence on the evolution of this functional. 1.3. Related models The following models are not kinetic models, but have come to be studied by members of the kinetic community because of the unity of methods and problematics. - The spatial Fokker-Planck equation, or Smoluchowski 3 equation [399] Op -- Vx " (Txp -Jr-RVV(x)), t ~ 0, x E ~N. (168) 0t One always assume e-v E L 1, and without loss of generality e-v should be a probability measure, just as p. For this equation the natural Lyapunov functional is the free energy, or relative entropy, H (p le- v), and the entropy dissipation coincides with the relative Fisher information, I (pie -v). For a summary of recent studies concerning the trend to equilibrium for (168), the reader may consult Arnold et al. [39], or Markowich and Villani [330]. - Equations modelling porous medium with confinement: Op = 7x. (Vx P(p) + pVV(x)), t ~ O, X E ]I~N, (169) Ot where P is a nonlinearity, P(p) standing for a pressure term, for instance P(p) = p• In this last case (power law), equations like (169), with a quadratic confinement po- tential, naturally arise as rescaled versions of their counterparts without confinement. The natural Lyapunov functional for (169) is f A(p)dx + fpvr where P (p) = p A' (p) - A (p). The trend to equilibrium for (169) has been studied independently by Carrillo and Toscani [131], Dolbeault and Del Pino [163], Otto [364] for the power law case, then more generally by Carrillo et al. [129]. One of the most remarkable features of Equations (168) and (169) is that they have the form of a gradient flow, Ot For a general discussion of the implications, see, for instance, Otto and Villani [365], Markowich and Villani [330]. 3There are several types of equations which are called after Smoluchowski!
- 200. A review of mathematical topics in collisional kinetic theory 199 1.4. General comments In this section we shall informally discuss the features which may help the trend to equilibrium, or on the contrary make it more difficult- both from the physical and from the mathematical point of view. 9 First of all, the distribution tails are usually at the origin of the worst difficulties. By distribution tails, we mean how fast the distribution function decreases as Ivl ~ ~, or Ix l~ c~. This is not only a technical point; Bobylev has shown that large tails could be a true obstacle to a good trend to equilibrium for the Boltzmann equation, even in the spatially homogeneous case. More precisely, he proved the following result [79]. Consider the spatially homogeneous Boltzmann equation with Maxwell collision kernel (with or without cut-off), and fix the mass, momentum, energy of the initial datum f0. Let M(v) be the corresponding equilibrium state. Then, for any e > 0 one can construct an initial datum f0 = f~ such that the associated solution f~ (t, v) of the Cauchy problem satisfies vt Ilf (t,.)-Mll Kee-et, Ke >0. At this point we should make a remark to be honest: an eye observation of a plot of these particular solutions will show hardly any departure from equilibrium, because most of the discrepancy between fe and M is located at very high velocities - and because the constant Ke is rather small. This illustrates the general fact that precise "experimental" information about rates of convergence to equilibrium is very difficult to have, if one wants to take into account distribution tails. 9 Moreover, recent studies have shown that the Boltzmann equation, due to its nonlocal nature, is more sensitive to this tail problem than diffusive models like Landau or Fokker- Planck equations. For the latter equations, it is not possible to construct "pathological" solutions as Bobylev; the trend to equilibrium is typically exponential, with a rate which is bounded below. We shall come back to this point, which by the way is also folklore in the study of Markov processes: it is known that jump processes have more difficulties in going to equilibrium than diffusion processes. 9 Next, it is clear that the more collisions there are, the more likely convergence is bound to be fast. This is why the size of the collision kernel does matter, in particular difficulties arise in the study of hard potentials because of the vanishing of the collision kernel at zero relative velocities; and also in the study of soft potentials because of the vanishing of the collision kernel for large relative velocities. A common belief is that the problem is worse for soft potentials than for hard. Also note that hard potentials are associated with a good control of the distribution tails, while soft potentials are not. Studies of the linearized operator show that in principle, one could expect an exponential decay to equilibrium for the spatially homogeneous Boltzmann equation with hard or Maxwellian potentials (under strong control of the distribution tails), while for soft potentials the best that one could hope is decay like O(e -t~) for some ot E (0, 1) (see Caflisch [111 ]). This is of course related to the fact that there is a spectral gap in the first case, not in the second one. 9 In the case of Boltzmann or Landau models (or some versions of coupled Fokker- Planck), the collision frequency also depends on the density of particles. This of course can
- 201. 200 C. Villani be seen via the fact that Boltzmann and Landau operators are quadratic, while the linear Fokker-Planck is not. As a consequence, the trend to equilibrium should be extremely slow at places where the density stays low: typically, very large positions. Therefore, the trend to equilibrium is expected to hold on extremely long scales of times when one considers the Boltzmann equation in a confinement potential, as opposed to the Boltzmann equation in a finite box. 1~ In the x-dependent case, a strong mathematical difficulty arises: the existence of local equilibria. These are states which make the entropy dissipation vanish, but are not stationary states. In fact they are in equilibrium with respect to the velocity variable, but not with respect to the position variable; for instance they are local Maxwellians Mx(v), with parameters p, u, T depending on x. Of course the trend to equilibrium is expected to be slowed down whenever the system comes close to such a state. We shall discuss this problem in more detail in Section 7. t~ Finally, a gradient flow structure often brings more tools to study the trend to equilibrium. We shall see this in the study of such models as (161) or (164). As we mentioned in Section 2.4 of Chapter 2A, in the case of the spatially homogeneous Boltzmann equation no gradient flow structure has been identified. Moreover, for all the spatially inhomogeneous equations which are considered here, the existence of the local equilibria rules out the possibility of such a structure. 2. Nonconstructive methods In this section, we briefly review traditional methods for studying the convergence to equilibrium. 2.1. Classical strategy A preliminary step of (almost) all methods is to identify stationary states by searching for solutions of the functional equation D(f) = 0, or more generally ff D(f(t)) dt = O. Once uniqueness of the stationary solution has been shown, then weak convergence of the solution towards equilibrium is often an easy matter by the use of compactness tools. Uniqueness may hold within some subclass of functions which is left invariant by the flow. For instance, in the case of the spatially homogeneous Boltzmann equation, it is easy to prove weak convergence as n --+ cx~ of f(n + t, U)nEN towards the fight Maxwellian distribution in weak-LP ([0, T] x ~U), as soon as /, lim limsup [ f(t, v)lvl 2dv = 0. R--+ cx~ t--~ cx~ ,] lv l>/ R (170) Condition (170), thereafter referred to as "tightness of the energy", ensures that there is no leak of energy at large velocities, and that f (t, .) does converge towards the fight Maxwellian distribution- and not towards a Maxwellian with too low temperature. In all
- 202. A review of mathematical topics in collisional kinetic theory 201 the sequel, we will assume that the moments of f are normalized, so that the equilibrium distribution is the standard Maxwellian M with zero mean and unit temperature. As a typical result, under general conditions Arkeryd [17] proved that the solution to the spatially homogeneous Boltzmann equation with hard potentials does converge to M, weakly in L 1, as t ~ cx~.This result is facilitated by the fact that Equation (170) is very easy to prove for hard potentials, while it is a (seemingly very difficult) open problem for soft potentials. In the framework of the spatially homogeneous Boltzmann equation with Max- wellian collision kernel, other approaches are possible, which do not rely on the entropy dissipation. Truesdell [274] was the first one to use such a method: he proved that all spherical moments satisfy closed differential equations, and converge towards corresponding moments of M. This implies weak convergence of f (t, .) towards M. Also contracting metrics4 can be used for such a purpose along the ideas of Tanaka [414, 415]. A refinement is to prove strong convergence of f(t, .) towards M as t --~ cxz, for instance, as a consequence of some uniform (in time) smoothness estimates. The first result of this kind is due to Carleman [118]: he proved uniform equicontinuity of the family (f(t, "))t>~o when f is the isotropic solution of the spatially homogeneous Boltzmann equation with hard spheres, assuming that the initial datum decays in O(1/Ivl6). As a consequence, he recovered uniform convergence to equilibrium. This method was improved by Gustafsson [270] who proved strong L p convergence for the solution of the spatially homogeneous Boltzmann equation with hard potentials, under an ad hoc LP assumption on the initial datum. In the much more general framework of the spatially inhomogeneous Boltzmann equation, by use of the Q+ regularity, Lions [308] proved strong L 1 compactness as t --+ cx~,say when the system is confined in a torus. This however is not sufficient to prove convergence, because there is no clue of how to prove the spatially-inhomogeneous variant of (170), lim limsupf~, dxfv R--+ oe t--+ o~ N i/>R f(t, v)lvl 2 dv = 0. At this point we have to recognize that there is, to this date, no result of trend to equilibrium in the spatially inhomogeneous context, except in the perturbative framework of close- to-equilibrium5 solutions: see, for instance, [286] (perturbation setting in whole space) or [404] (in a bounded convex domain) - with just one exception: the case of a box with uniform Maxwellian diffuse boundary conditions, which was solved by Arkeryd and Nouri [35] in a non-perturbative setting. On the contrary, it is rather easy to prove convergence to equilibrium for, e.g., the spatially inhomogeneous linear Fokker-Planck equation. Once strong convergence to equilibrium has been established (for instance, in the case of the spatially homogeneous Boltzmann equation with hard potentials), a natural 4SeeSection2 in Chapter2D. 5Of courseit is not a verysatisfactorysituationif one is ableto proveconvergenceto equilibriumonlywhen one startsextremelycloseto equilibrium....
- 203. 202 C.Villani refinement is to ask for a rate of convergence. In the "good" cases, a hard work leads to exponential rates of decay thanks to linearization techniques and the study of the spectral gap of the linearized operator. This strategy was successfully applied by Arkeryd [23], Wennberg [456] to the spatially homogeneous Boltzmann equation with hard, or Maxwellian potentials. In the spatially inhomogeneous context, it was developed by the Japanese school under the assumption that the initial datum is already extremely close to equilibrium. In the case of soft potentials, though there is no spectral gap for the linearized operator, Caflisch [111] was able to prove convergence to equilibrium like e-t~ for some exponent fl 6 (0, 1) - also under the assumption that the initial datum belong to a very small neighborhood of the equilibrium. 2.2. Why askfor more ? The preceding results, as important as they may be, cannot be considered as a definitive answer to the problem of convergence to equilibrium. There are at least two reasons for that: (1) Non-constructiveness. The spectral gap (when it exists, which is not always the case !) is usually nonexplicit: for the Boltzmann equation with hard or soft potentials there is only one exception, the spatially homogeneous operator with Maxwellian collision kernel. What is more problematic, nobody knows how to get estimates on its size: usual arguments for proving its existence rely on Weyl's theorem, which asserts that the essential spectrum is invariant under compact perturbation. But this theorem, which is based on a compactness argument, is nonexplicit .... Another problem arises because the natural space for the linearized operator (the space in which it is self-adjoint) is typically LZ(M-1), endowed with the norm []f]122(M_l) = f fZ/M, which is of course much narrower than the natural spaces for the Cauchy problem (say, Lebesgue or Sobolev spaces with polynomial weights). A new compactness argument is needed [457] to prove the existence of a spectral gap in these much larger spaces. REMARK. This problem of functional space arises even for linear equations! For instance, if one considers the Fokker-Planck equation, then the spectral gap exists in the functional space L2(M -1), but one would like to prove exponential convergence under the sole assumption that the initial datum possess finite entropy and energy. (2) Nature of the linearization procedure. In fact, even if linearization may predict an asymptotic rate of convergence, it is by nature unable to yield explicit results. Indeed, it only shows exponential convergence in a very small neighborhood of the equilibrium: a neighborhood in which nonlinearities are negligible in front of the linear terms. It cannot say anything on the time the solution needs to enter such a neighborhood .... This of course does not mean that linearization is in essence a bad method, but that it is a valuable method onlyfor perturbations of equilibria.
- 204. A review of mathematical topics in collisional kinetic theory 203 Entropy dissipation methods have been developed to remedy these problems, and yield explicit estimates of trend to equilibrium in a fully nonlinear context. We note that these methods are not the only effective methods in kinetic theory: other techniques, which have been developed in the particular framework of Maxwellian collision kernels, will be reviewed in Chapter 2D. Thus, the ideal mathematical situation, combining the power of both entropy methods and linearization techniques, would be the following. From a starting point which is far from equilibrium, an entropy method applies to show that the solution approaches equi- librium, possibly with a non-optimal rate (maybe not exponential...). After some explicit time, the solution enters a small neighborhood of equilibrium in which linearization ap- plies, and a more precise rate of convergence can be stated. For this plan to work out, it would seem necessary to (1) refine linearization techniques to have explicit bounds on the spectral gap, (2) establish very strong a priori estimates, so that convergence in entropy sense imply a much stronger convergence, in a norm well- adapted to linearization- or (2') show that the solution can be decomposed into the sum of an exponentially small part, and a part which is bounded in the sense of this very strong norm. 2.3. Digression At this point the reader may ask why we insist so much on explicit estimates. This of course is a question of personal mathematical taste. We do believe that estimates on the qualitative behavior of solutions should always be explicit, or at least explicitable, and that a compactness-based argument showing trend to equilibrium cannot really be taken seriously. First because it does not ensure that the result is physically realistic, or at least that it is not unrealistic by many orders of magnitude. Secondly because of the risk that the constants involved be so huge as to get out of the mathematical range which is allowed by the model. For instance, what should we think of a theorem predicting trend to equilibrium like e-l~176176176 The corresponding time scale is certainly much larger than the time scale on which the Boltzmann description may be relevant.6 Of course, asking for realistic estimates may be a formidable requirement, and often one may already be very lucky to get just constructive estimates. Only when no such estimates are known, should one take into account nonexplicit bounds, and they should be considered as rough results calling for improvements. This is why, for instance, we have discussed the results of propagation, or appearance, of strong compactness in the context of the Cauchy problem for renormalized solutions .... 3. Entropy dissipation methods 3.1. Generalprinciples The main idea behind entropy dissipation methods is to establish quantitative variants of the mechanism of decreasing of the entropy: in the case of the Boltzmann equation, this 6Seethe discussionatthe end of Section2.4.
- 205. 204 C. Villani is the H theorem. This approach has the merit to stand upon a clear physical basis, and experience has shown its robustness and flexibility. RULE 1. The "discrepancy" between a distribution function f and the equilibrium f~ should not be measured by the L 1 norm, but rather by E[flf~] = E(f) - E(f~), thereafter called relative entropy by abuse of language. Thus, one should not try to prove that f(t) converges to f~ in L 1, but rather show that E(f(t)) --+ E(fe~) as t --+ ~, which will be called "convergence in relative entropy". A separate issue is to understand whether convergence in relative entropy implies convergence in some more traditional sense. RULE 2. One considers as a main object of study the entropy dissipation functional D. Of course, the definition of the entropy dissipation relies on the evolution equation; but it is important to consider D as a functional that can be applied to any function, solution or not of the equation. RULE 3. One tries to quantify the following idea: if at some given time t, f (t) is far from f~, then E (f) will decrease notably at later times. Before turning to less abstract considerations, we comment on the idea to measure the distance in terms of the entropy, rather than, say, in terms of the well-known L 1 distance. A first remark is that there is no physical meaning, in the context of kinetic equations, in L1 distance. Some rather violent words by Truesdell will illustrate this. After proving exponential convergence of all moments in the framework of the spatially homogeneous Boltzmann equation with Maxwell collision kernel, he adds [274, p. 116] "Very likely it can be shown that [the solution] itself approaches Maxwellian form, but there is little interest in this refinement." A justification of this opinion is given on p. 112: "Since apart from the entropy it is only the moments of the distribution function that have physical significance, the result sought is unnecessarily strong". Thus, at the same time that he attacks the relevance of L 1results, Truesdell implicitly supports entropy results .... A second remark is that, very often, convergence of the entropy implies convergence in L 1 sense. In the case of the H-functional, or more generally when E (f) - E (f~) takes the form of a relative Kullback entropy, this is a well-known result. Indeed, the famous (and elementary) Csisz~ir-Kullback-Pinsker inequality states that whenever f and g are two probability distributions, 1 f i 2llf-gll 2 - L' ~< f log -- = H (fig). g In many other instances, especially when a gradient flow structure is present, the quantity E (f) - E (f~) can also be shown to control some power of the Wasserstein distance. 7 For this see in particular Otto and Villani [365]. A basic example is the Talagrand inequality, 1W(f, M) 2 <~H(fIM), 2 7Equation(244)below.
- 206. A review of mathematical topics in collisional kinetic theory 205 where M is the zero-mean, unit-temperature Maxwellian distribution. Usually, one can then obtain control of the L 1 norm via some ad hoc interpolation procedure [130]. As a final remark, we comment on the entropy dissipation equality itself- say in the case of the Boltzmann equation. As we saw, formally, solutions of the spatially homogeneous Boltzmann equation satisfy the identity d --H(f(t .)) -- -D(f) dt ' ' but when is this rigorous? It was actually proven by Lu [326] that this equality always holds for hard potentials with cut-off, under the sole assumptions that the initial datum has finite mass, energy and entropy. In fact, we shall always work under much stronger conditions. Thus, in all the sequel, we shall always consider situations in which the estimates for the Cauchy problem are strong enough, that the entropy dissipation identity can be made rigorous. Such is not the case, for instance, in the framework of the DiPerna-Lions theory of renormalized solutions, s 3.2. Entropy-entropy dissipation inequalities When trying to implement the preceding general principles, one can be lucky enough to prove an entropy-entropy dissipation inequality" this is a functional inequality of the type D(f) >~O(E[flfoc]), (171) where H ~ O (H) is some continuous function, strictly positive when H > 0. The main idea is that "entropy dissipation controls relative entropy". Such an inequality implies an immediate solution to the problem of trend to equi- librium. Indeed, let f(t) be a solution of the evolution equation. Since D(f(t))= -(d/dt)E[f(t)lf~], it follows that the relative entropy H(t) -- E[f(t, ")lfc~] satisfies the differential inequality dt ---H(t) ~ O(H(t)). (172) This implies that H(t) --+ 0 as t --+ +oo, and if the function 69 is known with enough details, one can compute an explicit rate of convergence. For instance, a linear bound like D(f) >i 2~.E[fJfcc] will entail exponential convergence to equilibrium, relative entropy converging to 0 like e-2xt. On the other hand, an exponent bigger than 1, D(f) >1KE[flfool l+a (K > O, ot > O) 8In any case, this theory should be hopelessly excluded from any study of trend to equilibrium until energy conservation, and even local energy conservation, has been proven.
- 207. 206 C. Villani will entail "polynomial" rate of convergence to equilibrium, the entropy going down like O(t-1/a). Situations in which the exponent is lower than 1 are very rare; in such cases the system converges to equilibrium in finite time. This occurs in certain simple model equations for granular media [426]. Very often, one cannot hope for such a strong inequality as (171), but one can prove such an inequality in a restricted class of functions: D(f) ~ Of(E[flf~]), (173) where the explicit form of Of may depend on some features of f such as its size in some (weighted) Lebesgue spaces, its strict positivity, its smoothness, etc.: all kinds of a priori estimates which should be established independently. In collisional kinetic theory, there are many situations in which entropy-entropy dissipation inequalities cannot hold true, in particular for spatially inhomogeneous models when the collisions only involve the velocity variable. As we shall see, in such cases it is sometimes possible to use entropy-entropy dissipation inequalities from spatially homogeneous models. As a final remark, the interest of entropy-entropy dissipation inequalities is not restricted to proving theorems of trend to equilibrium. Entropy-entropy dissipation inequalities may also in principle be applied in problems of hydrodynamic (as opposed to long-time) limits, yielding rather explicit estimates. For this one may consult the work by Carlen et al. [124] on a baby model, the recent paper by Saint-Raymond [400] on the hydrodynamic limit for the BGK model, or the study by Berthelin and Bouchut [74] on a complicated variant of the BGK model. However, to apply this strategy to more realistic hydrodynamic limits, say starting from the Boltzmann equation, we certainly have to wait for very, very important progress in the field. 3.3. Logarithmic Sobolev inequalities and entropy dissipation We illustrate the preceding discussion on the simple case of the spatially homogeneous Fokker-Planck equation, ~f ----Vv. (Vvf + f v). Ot Recall that the entropy functional is the Kullback relative entropy of f with respect to the standard Gaussian M, H (flM) = LN f log fM
- 208. A reviewof mathematical topics in collisional kinetic theory 207 (equivalently, the additive constant in the free energy has been chosen in such a way that the equilibrium state has zero energy). And the entropy dissipation functional is the relative Fisher information, I(flM)--~Nf f Vv log (174) The archetype of (171) is the Stam-Gross logarithmicSobolevinequality[411,261 ]. In an information-theoretical language, this inequality can be written most simply as I(flM) >~2H(fIM). (175) Inequality (175) was first proven, in an equivalent formulation, in a classical paper by Stam9 [411]. The links between the theory of logarithmic Sobolev inequalities and information theory have been pointed out for some time [45,120,165,16]. Of course, inequality (175) immediately implies that the solution to the Fokker-Planck equation with initial datum f0 satisfies H(f (t)lM) <<. e-2tH(folM). This is a complete l~ and satisfactory solution to the problem of trend to equilibrium for the Fokker-Planck equation. Actually, the interplay between functional inequalities and diffusion equations goes in both directions [330]. As was noticed in a famous work by Bakry and Emery [45], some properties of trend to equilibrium for the Fokker-Planck equation can be used to prove inequalities such as (175). We shall discuss their approach in Section 6, together with recent developments. By the way, as a general rule, logarithmic Sobolev inequalities are stronger than spectral gap inequalities [261]. As a typical illustration: if one lets f = M(1 + eh) in (175), where f Mh = 0, and then lets e go to 0, one finds the inequality f Mh--O >f MIVhl2) f Mh2, (176) which is the spectral gap inequality for the Fokker-Planck operator. Inequality (176) implies the following estimate for solutions of the Fokker-Planck equation: fo GL2(M-1) ~ [If(t, .)- MIIL~(M_,) ~<e-t lifo - MIIL2(M-,). 9Stam proved the inequality N'(f)I (f) >~N, which is equivalent to (175) by simple changes of variables, in dimension 1. Here N" is the entropy power functional of Shannon, formula (52). The proof of Stam was not completely rigorous, but has been fixed. 10The assumption that the initial datum possess finite entropy can even be relaxed by parabolic regularization. For instance, one can prove [366] that H(f(t)lM) = O(1/t) as soon as f fo(v)lvl 2dv is finite.
- 209. 208 C. Villani 4. Entropy dissipation functionals of Boltzmann and Landau In this section, we discuss entropy-entropy dissipation inequalities for functionals (156) and (160). A common feature of both functionals is monotonicity: the Boltzmann entropy dissipation is a nondecreasing function of the collision kernel B, while the Landau entropy dissipation is a nondecreasing function of qJ. This property makes it possible to only treat algebraically simplified cases where B (resp. q/) is "small". As a typical application, if we find a lower bound for D when the collision kernel is Maxwellian, then we shall have a lower bound for all collision kernels whose kinetic part is bounded below. This reduction is interesting because Maxwellian collision kernels do have many additional properties. We shall see some of these properties in a moment, and shall dig more deeply into them in Chapter 2D. All known lower bounds for the entropy dissipation functionals of Boltzmann or Landau have been obtained from a preliminary study of the Maxwellian case. As a consequence, it will be natural to define an "over-Maxwellian" collision kernel as a collision kernel B which is bounded below by a Maxwellian collision kernel. In the sequel, we shall assume without loss of generality that the first moments of the distribution function f are normalized: (1)v =(1) fI~ f (v) dv 0 N [1312 N (177) and denote by M the associated Maxwellian. 4.1. Landau's entropy dissipation We start with the case of the Landau equation, because its diffusive nature entails better properties of the entropy dissipation functional. Let us first state the main results, then we shall comment on them. THEOREM 14. Let f be a probability distribution satisfying (177). (i) "Over-Maxwellian case": Let qJ(lzl) ~> ]el2, and let DL be the associated entropy dissipation functional, formula (160). Then there exists a constant ~.(f) > O, explicit and depending on f only via an upper boundfor H (f), such that DL(f) >.X(f)I(flM) >~2~,(f)H(flM). (178) More precisely, one can choose f ,k(f)=(N-1) inf I f(v)(v.e) 2dr eES N-1 JRN (179)
- 210. A reviewofmathematicaltopicsincollisionalkinetictheory 209 (ii) "Soft potentials"" Let ~(Izl) ~ Izl2(1 -Jr-Izl) -~, ~ > 0. Then, for all s > O, there exists a constant Cs(f), explicit and depending on f only via an upper bound for H(f), such that DL(f) >~Cs(f)H(flM)l+~7 Fs ~, (180) where Fs = Ms+2(f) Jr-Js+2(f), and Ms+2(f) fR f(v)(1 -+-Iv12)s+2 = N dr, Iv [ (l+lol ) N du. (iii) "Hard potentials"" Let q/(Izl) ~ Izly§ 9/> 0. Then, there exists constants K1 (f), K2 (f), explicit and depending on f only via an upper boundfor H (f), such that DL(f) ~ Kl(f) min[I(flM), l(flM)l+-~] ~>K2(f) min[H(flM), H(fIM)I+-~]. (181) (182) REMARKS. (1) Note that the constant )~ given by (179) has the dimensions of a temperature, and can vanish only if f is concentrated on a line. This is the typical degeneracy of the Landau equation; in particular, the operator in (30) is always strictly elliptic unless f is concentrated on a line. But the finiteness of the entropy prevents such a concentration, and allows one to get a bound from below on ~(f). Of course, other estimates are possible: for instance, by use of some L p, or L c~, or smoothness bound on f. Or, if f is radially symmetric, then automatically )~(f) = 1. (2) Also, as we shall see in the next section, it may sometimes be wiser to estimate from below ~(f) in terms of the entropy dissipation of f! (3) Further note that the inequalities on the fight in (178) and in (182) are nothing but the logarithmic Sobolev inequality (175). In the preceding theorem, point (i) is the starting point for the remaining cases. It was established in Desvillettes and Villani [183] by two different methods. The first one relies on some explicit computations performed in Villani [443], whichare recalled in formula (30). The second strategy is a variant of Desvillettes' techniques, inspired by a method due to Boltzmann himself [93]. It consists in "killing", with a well-chosen operator, the symmetries of the functional DL which correspond to the equilibrium state. 11To be just a little bit more precise, one writes D(f) = f dvdv, ff, lR(v, v,)[ 2 11 See Boltzmann'sargumentin Section4.3.
- 211. 210 C. Villani where R" ]1~2N ---+ ]I~N, and one finds a linear operator T = T(v, v,)" ]l~N ---> 11~N such that T R is identically 0 if and only if D = 0; then DL (f) >~ l/ iiTll2 dvdv, ff, ITR(v, V,)I 2 A careful choice of the operator T enables a very simple computation of the fight-hand side in this inequality. Point (ii) is proven in [429]. The idea is that the vanishing of qJ(lv - v,l)/lv - v,[ 2 as Iv - v,[ --+ cr can be compensated by some good estimates of decay at infinity, in the form of the constant Fs+2 (which involves both moments and smoothness). As for point (iii), it is rather easy to get by "perturbation" from point (i), see Desvillettes and Villani [183]. The idea is that the contribution of small Iv - v,[ is negligible. One writes Iv - v,I • ~> e y Iv - v,[ 2 - 6g+2, then one estimates from below the contribution of e• - V,[2 to the entropy dissipation, and from above the contribution of the small constant function ey+2. A few algebraic tricks [183] lead to the estimate (181) without further bounds on the concentration of f: the constant K in this estimate is essentially )~(f)l+g/2. Theorem 14 gives explicit and satisfactory answers to the quest of entropy-entropy dissipation estimates for the Landau equation; in the next section we shall see that they can be used efficiently for the study of the trend to equilibrium, at least in the spatially homogeneous situation. However, we should avoid triumphalism: it is abnormal that the exponent in the case of hard potentials (which is 1 + g/2) be worse than the exponent in the case of soft potentials (1 + e, with e as small as desired, if f has a very good decay and smoothness at infinity). One would expect that for hard potentials, the inequality DL(f) >~K(f)H(fIM) hold true. 4.2. Boltzmann's entropy dissipation: Cercignani's conjecture Now we turn to the more complicated case of the functional (156). Some parts of the following discussion are copied from [442]. An old conjecture by Cercignani, formulated at the beginning of the eighties, was that the Boltzmann equation would satisfy a linear entropy-entropy dissipation inequality. We state this conjecture here in a slightly more precise form than the original. There are two forms of it, a weak and a strong. CERCIGNANI'S CONJECTURE. Let B ~ 1 be a collision kernel and (156) be the associated entropy dissipation functional. Let f (v) be a probability distribution on R N,
- 212. A reviewofmathematicaltopicsin collisionalkinetictheory 211 with unit temperature, and let M be the associated Maxwellian equilibrium. Then, (strong version) there exists )~ > O, independent of f, such that D(f) >~2~H(fIM); (183) (weak version) there exists )~(f) > O, depending on f only via some estimates of moments, Sobolev regularity, lower bound, such that D(f) >~2~(f)H(f]M). (184) It soon appeared that the strong version of this conjecture had to be false. Indeed, it would have implied a universal exponential rate of convergence for solutions of the spatially homogeneous Boltzmann equation with a collision kernel B ~> 1. But, as we mentioned in Section 1.4, Bobylev [79, p. 224] was able to produce a family of initial data (f~)~>0 with unit mass and temperature, such that the associated solutions of the Cauchy problem (with Maxwellian collision kernel, say B = 1) converge to equilibrium slowly, in the sense Yt ~>0, [[fe(t, .)- M[I ~>Kee -e', Ke > 0. These initial data are constructed more or less explicitly with the help of the Fourier transform apparatus, and hypergeometric functions. Later, Wennberg [461] produced direct counterexamples to (183), covering the case of hard potentials as well. Finally, Bobylev and Cercignani [87] disproved even the weak version of the conjecture. They exhibited a family of distribution functions for which (184) does not hold for a uniform )~, while these distribution functions do have uniformly bounded LP or H k norms (whatever p, k), uniformly bounded moments of order k (whatever k), and are bounded below by a fixed Maxwellian. These counterexamples are obtained by adding a very tiny (but very spread) bump, at very high velocities, to the equilibrium distribution. They again illustrate the principle that distribution tails are the most serious obstacle to a good trend to equilibrium for the Boltzmann equation. Thus, Cercignani's conjecture is false.* It may however be that (184) hold true under more stringent assumptions: - under very strong decay conditions, for instance, f 6 L2(M-1), as in the linearized theory; 12 -or under an assumption of nonintegrable angular singularity, which may help. This conjecture would be supported by the good behavior of the Landau entropy dissipation. *Note added in proof: To my own surprise, after completion of this review, I discovered that Cercignani's conjecture does hold true when B(v- v,, a) ~>1-4-Iv- v,]2. This is not in contradiction with the Bobylev- Cercignani counterexamples,becausetheyassumef Bdo"~<C(1 + iv - v, lY),y < 2! 12Averyrecent,deepresultby Balland Bartheaboutthe centrallimittheoremsuggeststhatthereis somehope if f satisfiesa Poincar6inequality.Thismay be the first step towardsidentifyingsome"reasonable"conditions for Cercignani'sconjectureto be true.
- 213. 212 C. Villani 4.3. Desvillettes' lower bound The first interesting lower bound for Boltzmann's entropy dissipation functional was obtained by Desvillettes [166]. His idea was to go back to Boltzmann's original argument for the identification of cases where the entropy dissipation vanishes. As many proofs of similar results, Boltzmann's proof relies on some well-chosen linear operators which "kill symmetries". Let us sketch this argument (slightly modified) in a nutshell, since it may enlighten a little bit the discussion of the most recent results in the field. BOLTZMANN'S THEOREM. Let N >~2, and let f (v) be a smooth positive solution of the functional equation V ( U, U, , o") E ]1~N >( I[~N X S N-l, f' f~, = f f,. (185) Then f is a Maxwellian distribution; in other words there exist constants )~ ~ R, lZ ~ I[~N such that VV E ~N, V log f (v) = Zv +/z. (186) BOLTZMANN'S ARGUMENT. Average (185) over the parameter o" 6 SN-1 , to find 1 fs (f' f*~)do". ff* = IsN-11 u-, (187) It is easy to convince oneself that the function is ! ! f f, do" = G(v, v,) N-1 depends only on the sphere S(v, v,) with diameter [v, v,]. Actually, up to a Jacobian factor (Iv- v,[/2) N-l, G is just the mean value of the function f(w)f(Co) on this sphere, where t~ stands for the velocity on S which is diametrically symmetric to w. The spheres S(v, v,) are in turn parametrized by only N + 1 parameters, say (v + v,)/2 and Iv - v,I; or, equivalently, by the physical variables m = v + v, [total momentum of colliding particles]; Ivl2 Iv, I2 e = -~ + ~ [total kinetic energy of colliding particles]. (188) Thus we shall abuse notations by writing G(v, v,) = G(m, e). Now, introduce the linear differential operator T = (v - v,)/x (V - V,) (or, which amounts to the same, H(v - v,)(V - V,), where 17(v - v,) is the orthogonal projection
- 214. A review of mathematical topics in collisional kinetic theory 213 on (v - v,)• Its kernel consists precisely of those functions that depend only on m and e. If we apply this operator to the equation log ff, = log G (m, e), we find (v - v.) A [V log f - (g log f),] ----O. In words, for all v, v, there exists a real number •v,v, such that V log f (v) - V log f (v,) = ~,v,v,(v - v,). This functional equation, set in ]~N,N ~>2, implies the conclusion at once. (189) REMARK. The very last part of the proof, starting from (189), is exactly what one needs to identify cases of equality for Landau's entropy dissipation functional. This can make us suspect a deep connection between the entropy dissipations of Boltzmann and Landau. We shall soon see that there is indeed a hidden connection. With the help of the open mapping theorem, Desvillettes was able to produce a "quantitative" version of Boltzmann's argument, leading to the THEOREM 15. Let B >~1, and let D be the associated entropy dissipation functional (47). Let f be a nonnegative density on IRN, with finite mass and energy. Without loss of generality, assume that the first moments of f are normalized by (177). Then, for all R > 0 there is a constant KR > O, depending only on R, such that f D(f) >~Kit inf 1 m~.Ad Jlv [<~R llog f - logmldv, where M is the space of all Maxwellian distributions. Note that the quantity on the right is always positive for some R > 0 if f is not Maxwellian. Several variants were obtained, with better estimates and simpler proofs, and recently Desvillettes [175] found a way to avoid the use of the open mapping theorem, and get explicit constants. Also Wennberg [455] extended the result to hard potentials. Although Desvillettes' result is rather weak, it was important as the very first of its kind. Subsequent developments were partly motivated by the search for stronger estimates. 4.4. The Carlen-Carvalho theorem At the beginning of the nineties, Carlen and Carvalho [121,122] made a crucial contribution to the subject by using the tools of information theory and logarithmic Sobolev inequalities.
- 215. 214 C. Villani They proved that there always exists an entropy-entropy dissipation inequality for Boltzmann's collision operator as soon as one has some (very weak) control on the decay at infinity and smoothness of the distribution function. In their general result, decay at infinity of a distribution function f is measured by the decay of x f "R ~ fv f (v)lvl 2 dv as R 1"c~, while the smoothness is measured by the decay of 7rf "X w-~ H(f) - H(Sz f) as )~$ 0. Here (St)t>>ois as usual the semigroup generated by the Fokker-Planck operator; sometimes it is called the adjoint Ornstein-Uhlenbeck semigroup. Carlen and Carvalho's general theorem [121] can be stated as follows: THEOREM 16. Let B(v - v,, or)/> 1 be a collision kernel. Let Xo, #/o be two continuous functions, decreasing to 0 as R t +cx~ and X $ 0 respectively. Let then f be a probability distribution function with unit mass and temperature, and let M be the associated Maxwellian distribution. Assume that Xf ~ Xo, ~f <<.~PO. (190) Then, there exists a continuous function 0 = Ox,~/, strictly increasing from O, depending on f only via Xo and r such that D(f) >~69(H(flM)). REMARKS. (1) This result crucially uses the special properties of Maxwellian collision kernels, explained in Chapter 2D. (2) The result in [121] is stated for a collision kernel which is bounded below in co- representation. 13 Recent works have shown that this assumption can be relaxed (see the references in Chapter 2D). The main ideas behind the proof of Theorem 16 are (1) the reduction to Maxwellian collision kernel by monotonicity, (2) the inequality 14 D(f) >~H(f) - H(Q+(f, f)) ~ O, which holds true for a Maxwellian collision kernel b(cos 0) such that f b(cos 0) sinN-2 0 dO = 1, and (3) show that when f satisfies (190) and H(f) - H(M) >, s, then f lies in a com- pact set of probability measures on which H - H (Q+) attains its minimum value. 13See Section4.6 in Chapter2A. 14See Section3.2 in Chapter2D.
- 216. A review of mathematical topics in collisional kinetic theory 215 One of the key ingredients is a study of the Fisher information functional I (f) -- f IVfl2/f, and the representation formula f0 ec H(Q+(f, f)) - H(M)= [I(Q+(Sxf, Szf)) - I(Sxf)]d)~. (191) This formula and related estimates are explained in Chapter 2D. A crucial point is to bound below the integrand in (191), for )~positive enough, by the method of Carlen [120]. We note that there is no assumption of lower bound on f in the Carlen-Carvalho theo- rem, though they actually use lower bounds in their estimates. There is no contradiction, because Maxwellian lower bounds are automatically produced by the semigroup (Sz). However, these lower bounds are rather bad, and so are the resulting estimates. Better bounds can be obtained if the probability density f is bounded below by some Maxwellian distribution. In a companion paper [122], Carlen and Carvalho showed how to extend their method to physically realistic cases like the hard-spheres kernel, B(v - v,, or) = Iv - v,I, and gave a recipe for computing the function 0. These results were the first entropy dissipation estimates which would find interesting and explicit applications to the Boltzmann equation, see Section 5. More importantly, they set new standards of quality, and introduced new tools in the field. However, the Carlen- Carvalho entropy-entropy dissipation inequalities are not very satisfactory because the function 69 is quite intricate, and usually very flat near the origin. 4.5. Cercignani's conjecture is almost true As we mentioned earlier, the "linear" entropy-entropy dissipation inequality conjectured by Cercignani (O(H) = const.H) is in general false. Nevertheless, it was proven a few years ago by Toscani and Villani [428] that one can choose O(H) = const.H l+e, with e as small as desired. Here is a precise statement from [428]. We use the notation IlfllL~ -- fRN f(v)(1 + IV12)s/2 dv and its natural extension IlfllLl~ogL-- fR~vf(V)log( 1 + f(v)) (1 -+-Ivl2)~/2dv. THEOREM 17.* (i) "Over-Maxwellian case": Let B >~1 be a collision kernel and D be the associated entropy dissipation functional Equation (156). Let f be a probability density on IRN with unit temperature, and let M be the associated Maxwellian equilibrium. Let e > 0 be arbitrary, and assume thatfor some ~ > O, A, K > O, IlfllL~4+2/~+~ Ilfll 1 < -31-oo ' L2+2/E+~log L ' *Noteaddedinproof:All the resultsin thistheoremhavebeenimprovedin recentworkby the author.
- 217. 216 C. Villani f (v) >/Ke -Alvl2. (192) Then, there exists a positive constant Cs(f), depending only on N, s, ~, [JfllL~+2/~+a, IIf II , A and K, such that L2+2/s+alog L' D(f) > Cs(f)H(fIM) l+e. (193) As an example (choosing 8 = 1), the following more explicit constant works" D(f) >~KTf FZSH(flM) l+~, (194) where K is an absolute constant (not depending on f), Tf is the "temperature" given by (179), and ( 1 ) Fe= log~+A Ilfll 2 Ilfll , L~+2/~ L3+2/~log L" (ii) "Soft potentials": Assume now that B(v - v,, ~) > (1 + Iv - v,I) -e, /~ > O. Then, for all s > O, Equation (194) still holds with (1) Fe = log ~ + A IIf II2 tl5+(2+fl)/sIIf IIt~+(2+fl)/elogL" (iii) "Hard potentials"" Assume now that B(v-v,,o')>~lv-v,I • g>O. Assume, moreover, that f ~ L p, for some p > 1, and tc large enough. Then, there exists > 1, C > O, depending on N, y, p, x, IIf IILff, and on A, K in (192), such that D(f) >~CH(flM) ~. Thus Cercignani's conjecture is "almost" true, in the sense that any power of the relative entropy, arbitrarily close to 1, works for point (i), provided that f decays fast enough and satisfies a Gaussian lower bound estimate. This theorem is remindful of some results in probability theory, about modified logarithmic Sobolev inequalities for jump processes, see Miclo [345]. Even if the situation considered in this reference is quite different, and if the methods of proof have nothing in common, the results present a good analogy. From the physical point of view, this is not surprising, because the Boltzmann equation really models a (nonlinear) jump process. Let us briefly comment on the assumptions and conclusions.
- 218. A review of mathematical topics in collisional kinetic theory 217 (1) The main improvement lies in the form of the entropy-entropy dissipation inequality, which is both much simpler and much stronger. (2) The lower bound assumption can be relaxed into f(v) >~ge -Alvlp for some p > 2, provided that more moments are included in the estimate. (3) Strictly speaking, this theorem is not stronger than the Carlen-Carvalho theorem, because the assumptions of decay at infinity are more stringent. On the other hand, it does not require any smoothness condition. As regards the proof, it is completely different from that of the Carlen-Carvalho theorem, and relies strongly on Theorem 14, point (i). Since this is quite unexpected, we shall give a brief explanation in the next paragraph. Once again, the result for hard potentials is not so good as it should be, because the power in point (iii) cannot be chosen arbitrarily close to 1. We have hope to fix this problem by improving the error estimates for small relative velocities which were sketched in [428]. 15 4.6. A sloppy sketch of proof In this survey, we have chosen to skip all proofs, or even sketches of proof. We make an exception for Theorem 17 because of its slightly unconventional character, and also because of its links with Boltzmann's original argument 16 about cases of equality in the entropy dissipation - with ideas of information theory coming into play. Of course, we shall only try to give a flavor of the proof, and not go into technical subtleties, which by the way are extremely cumbersome. Also we only consider point (i), and set B -- 1, or rather B = ISN-11-1, so that f Bdo- = f da = 1. Thus the functional to estimate from below is 1s dodo, f D(f) = ~ 2N sN-1 da (f' f~, - f f,) log f'f', ff, The three main ingredients in Theorem 17 are - a precise study of symmetries for the Boltzmann collision operator, and in particular the fact that the entropy dissipation can be written as a functional of the tensor product f| - a regularization argument ~ la Stam; - our preliminary estimate for the Landau entropy dissipation, Theorem 14. Stam's argument. At the end of the fifties, Stam [411] had the clever idea to prove the so-called Shannon-Stare inequality, conjectured by Shannon: H(v x + ,/1 + (1 (195) 15As this review goes to print, we just managed to prove the desired result, at the expense of very strong smoothness estimates (in all Sobolev spaces). 16See Section 4.3.
- 219. 218 C. Villani actually equivalent to (52), as a consequence of the Blachman-Stam inequality, which he introduced on that occasion: I(x/~X + ~/1 -otY) <~o~I(X) + (1 -a)I(Y). (196) In inequalities (195) and (196), X and Y are arbitrary independent random variables on ~N, and one writes H(X) = H(f) = f f logf, I(X) = I(f) = f [v fle/f whenever f is the law of X. Stam found out that (196) is essentially an infinitesimal version of (195) under heat regularization. Think that I is nothing but the entropy dissipation associated to the heat equation .... A modern presentation of Stam's argument is found in Carlen and Soffer [125]: these authors replace H and I by their relative counterparts with respect to the standard Gaussian M, and obtain (195) by integrating (196) along the adjoint Ornstein- Uhlenbeck semigroup (St)t>o. More explicitly, since I (f IM) is the derivative of H (flM) along regularization by St, and since also Stf -+ M as t --+ oo, one can write f0 ~176 H(fIM) = I(StflM)dt. The strategy in [428] is inspired from this point of view: we would like to start from +~ d f)] dt. D(f)-- fo [--~D(St This identity is formally justified because Stf ~ M as t ~ oo, and D(M)= 0. Then one can hope that for some reason, the derivative -dD/dt will be easier to handle that the entropy dissipation functional D. This is the case in the proof of the Shannon-Stam inequality, and also here in the framework of the Boltzmann entropy dissipation. It actually turns out, rather surprisingly, that Kf +~ D(f) ~ -R~ DL(Stf) dt, (197) with K > 0, and R a typical size for the velocity. In other words, the entropy dissipation for the Landau equation is a kind of differential version of the entropy dissipation for the Boltzmann equation! Admit for a while (197), and combine it with the result of Theorem 14, in the form DL(Stf) >~(N- 1)TstfI(StflM) >~(N - 1)Tf I(St flM). It follows that D(f) >~C(f) fo +~176 l (Stf lM) dt = C(f)H(flM);
- 220. Areviewofmathematicaltopicsincollisionalkinetictheory 219 which is the statement in Cercignani's conjecture. Of course, we know that Cercignani's conjecture is false, which means that (197) cannot rigorously hold true. A precise variant is established in [428]. The technical problem which prevents (197) is the presence of large velocities, as one could expect. Controlling the contribution of large velocities to the entropy dissipation is the most technical point in the proof presented in [428]. It means for instance establishing quantitative bounds on the tails of the entropy dissipation, like fo +~ fix Cs dt ]xIZ(StF - StG) log StF dX <~ I>~R StG Rs for arbitrary probability densities F(X) and G(X) in ~2N, where Cs is a constant depending on s and on suitable estimates on F and G (moments, lower bound...). In the next two pages, we shall skip all these technicalities and present a sketch of proof of (197) under the absurd assumption that all velocities are bounded, just to give the reader an idea of the kernel of the proof. SLOPPY SKETCH OF PROOF FOR (197). First we introduce the adjoint Ornstein- Uhlenbeck semigroup (St), and we try to compute (-d/dt)D(Stf). At first sight this seems an impossible task to perform in practice, due to the number of occurrences of f in the entropy dissipation functional, and the complicated arguments v', v~,. But a first observation will help: D(f) is actually a functional of the tensor product f | f = ff,. And it is easily checked that the following diagram is commutative, with T standing for tensorization, 7 f > F=ff, I St 7" I St St f > St F. (198) (Here we use the same symbol for the semigroups St in L I(R N) and in L1 (R2N).) This enables to replace in computations (St f)(St f), by St(ff,). One could hope that, similarly, F=ff, > f'f', I St I St St F > St (f' ff,) (199)
- 221. 220 C. Villani is commutative. This is false! The point is that the angular variable a is not intrinsic to the problem. To remove this flaw, we integrate with respect to the parameter a. Since x (x, y) w-~ (x - y) log - Y is a jointly convex function of its arguments, by Jensen's inequality s -- 1 dv dr, If, - da f' f2 log f da f' f~, D(f) >i D(f) = -~ 2N (200) Now it is true, even if not immediate at all, 17 that 7- ,,4 f >F f f, > G f '' = = daff, 7- A &f ~ &f > &G (201) with A standing for the averaging operation over the sphere, is an entirely commutative diagram. This actually is a consequence of the fact that (St) is a Gaussian regularization semigroup. This suggests to work with D instead of D, and to write D(Stf) in the form D(St F, St G), with the abuse of notations -- lf• (F-G) logF D(F, G) = -~ 2N -~ dX, X m. (1), 13,) E ~2N. After these preliminaries, it is not hard to compute dD(StF, StG)= lfR (StF+StG) -dt 4 2N V(&F) V(StG) StF StG 2 dX. (202) Here, of course, V = [V~, V~,] is the gradient in ]~N x Rv N . Under suitable assumptions one can also prove that t ~ D(St f) is a continuous function as t ~ 0, and goes to 0 as t ~ +cx~. Then D(f) = -~ dt 2N(St F + St G) V(StF) 7(StG) StF StG 2 dX 17A weaker property, sufficient for the argument, is that St preserves the class of functions which only depend on v + v, and Ivl2 + Iv, I2.
- 222. A review of mathematical topics in collisional kinetic theory 221 lfo+ f ) -~ dt 2N StF V(StF) StF V(ScG) (203) Since StG is a very complicated object, we would like to get rid of it. Recall from Boltzmann's original argument that StG, being an average on spheres with diameter [v, v, ], does not depend on all of the variables v, v,, but only upon the reduced variables m = v + v,, e = Ivl2/2 + Iv,12/2. Accordingly, we shall abuse notations and write StG(v, v,)-- StG(m,e). Now comes the key point: there is a conflict of symmetries between StG, which only depends on a low-dimensional set of variables, and StF, which is a tensor product. In Boltzmann's argument, the Maxwellian distribution pops out because it is the only probability distribution which is compatible with both symmetries. Here these different structures of StF and StG reflect at the level of their respective gradients: V(StF) StF I V(St f) (VSt f), ] = gy ; (204) V(StG) 1 VmStG -k- 1) , VmStG d- 1), 9 (205) StG StG Oe Oe In particular, V(StG) always lies (pointwise) in the kernel of the linear operator P "[A, B] ~ ]~2N~ 17(1) -- v,)[A - B] ~ ]~N, where H(z) is the orthogonal projection upon z• Of course IIPII = 4~ as a linear operator,18 and so V(StF) V(StG) StF &G 2 2 1 IIPII2 p(VStF _1 2 17(v - v,) Stf (gstf)_*]12" (Stf), (206) By combining (200), (203) and (206), if +~ D(f) >~-~ dt fR2u(St f)(St f), VStf (VStf), ] 2 dv dr,. The reader may have recognized a familiar object in the integrand of the right-hand side. Actually, apart from a factor Iv - v, 12, it is precisely the integrand in the Landau entropy dissipation, computed for Stf! If we now use our absurd assumption of boundedness of 18In contrast with the linear operator appearing in Boltzmann's proof, which was unbounded.
- 223. 222 C. Villani all relative velocities, in the form Iv - v, I ~<R, we get D(f) >~-~ dt 2u(St f)(St f)*lv -- v,I 2 VStf x H(v-v,) Stf 1 fo+~ 8R2 dt DE (St f). (VSt f), ] (St f), 2 dvdv, I-1 4.7. Remarks We shall point out a few remarks about the preceding argument. First of all, in the course of the rigorous implementation, it is quite technical to take into account error terms due to large velocities. One has to study the time-evolution of expressions like f~o(X)(StF- StG)log(StF/StG). But the calculations are considerably simplified by a striking "algebraic" property: a local (not integrated) version of (202) holds true. Let F h (F, G) -- (F - G) log G' j(F,G)=(F+G) VFF VG] 2"G Then, one can check that d dt t=O [St, h]-j, in the sense that for all (smooth) probability distributions F and G, d dt t=O (Sth(F, G) - h(St F, StG)) - j (F, G). This property is somewhat reminiscent of the F calculus used for instance in Bakry and Emery [45] and Ledoux [294]. It yields another bridge between entropy dissipation inequalities and the theory of logarithmic Sobolev inequalities. Our second remark concerns the use of the Fokker-Planck semigroup regularization. As we have seen, the main point above was to estimate from below the negative of the time- derivative of D(f) along the semigroup (St)t 90. As was already understood by Carlen and Carvalho, and even a long time ago by McKean [341 ] in the framework of the Kac model, this estimate has to do with the behavior of the Fisher information I (f) -- f IvflZ/f along the Boltzmann semigroup. Note that I (f) is the dissipation of the H-functional along the semigroup (St)t~o. As we shall explain in Chapter 2D, the semigroup (Bt), generated by
- 224. A reviewofmathematicaltopicsin collisionalkinetictheory 223 the spatially homogeneous Boltzmann equation with Maxwell collision kernel, commutes with (St), and it follows that dt t=o d D(& f) -- --~ t=0 I (Bt f). (207) We shall see in Chapter 2D that the right-hand side of (207) is always nonnegative; this could be considered as an a priori indication that the functional D behaves well under Fokker-Planck regularization. Actually, in the simpler case of the Kac model, 19 McKean [341, Section 7, Lemma d)] used relation (207) the other way round! He proved directly, with a very simple argument based on Jensen's inequality, that the left-hand side of (207) is nonnegative for the Kac model. His argument can be transposed to the Boltzmann equation with Maxwell collision kernel in dimension 2, and also to the case where the collision kernel is constant in co- representation: see [428, Section 8]. As a third remark, we insist that the above argument, besides being rather intricate, is certainly not a final answer to the problem. The use of the average over a seems crucial to its implementation, while for some applications it would be desirable to have a method which works directly for arbitrary Maxwellian collision kernels b(cos0). There is no clue of how to modify the argument in order to tackle the problem of Cercignani conjecture (with exponent 1) for very strongly decaying distribution functions. It also does not manage to recover spectral gap inequalities for Maxwellian collision kernels, which are known to be true. Applied to simpler models than Boltzmann's equation, it yield results which are somewhat worse than what one can prove by other, elementary means! However, in terms of lower bounds for Boltzmann's entropy dissipation, at the moment this is by far the best that we have. Our final remark concerns the problem of solving (53). As mentioned in Section 2.5, many authors have worked to prove, under increasing generality, that these solutions are Maxwellian distributions. The problem with Boltzmann's proof was that it needed C 1 smoothness. However, as suggested by Desvillettes, the use of the Gaussian semigroup (St) (or just the simple heat regularization) allows one to save Boltzmann's argument: let f be a L 1 solution of (53) with finite energy; without loss of generality f has unit mass, zero mean and unit temperature. Average (53) over a to get f f. -" G(m, e) as in formula (187). Then apply the semigroup (St) to find (St f)(St f), = StG(m, e). Since St f is C ~ for t > 0, Boltzmann's proof applies and St f is a Gaussian, which has to be M by identification of first moments. Since this holds true for any t > 0, by weak continuity f = M. 19Equation(21).
- 225. 224 C. Villani 5. Trend to equilibrium, spatially homogeneous Boltzmann and Landau As we already explained, in principle the trend to equilibrium is an immediate consequence of an entropy-entropy dissipation inequality and of suitable a priori estimates. However, there are some interesting remarks to make about the implementation. 5.1. The Landau equation By Theorem 14, one obtains at once convergence to equilibrium for the spatially homogeneous Landau equation - with explicit exponential rate if q'(lz[)/> K[zl2; - with explicit polynomial rate if ~(Izl)/> Klzl • Y > 0. These results hold in the sense of relative entropy, but also in any Sobolev space, thanks to the regularization results which we discussed in Chapter 2B and standard interpolation inequalities. An interesting feature is that the rate of convergence given by the entropy-entropy dissipation inequality is likely to improve as time becomes large, by a "feedback" effect. Indeed, when f approaches equilibrium, then the constant Tf in (179) will approach the equilibrium value TM = 1. In the case q'(Izl) = Izl2, this enables one to recover an asymptotically optimal rate of convergence [183]. The case of soft potentials (y < 0) is more problematic, because the moment estimates are not uniform in time - and neither are the smoothness estimates which enter the constant Fs+2 in Theorem 14. The fact that we do not have any uniform moment estimate for some moment of order s > 2 may seem very serious. It is not clear that condition (170) should be satisfied. Compactness-based methods spectacularly fail in such a situation. However, and this is one of the greatest strengths of the entropy method, it is not necessary that the constant Fs+2 be uniformly bounded. Instead, it is sufficient to have some estimate showing that it does not grow too fast, say in O(t ~) for c~ small enough. With this idea in mind, Toscani and Villani [429] prove the following theorem: THEOREM 18. Let tP(lzl) --Izl2t~D([z[), where r is smooth, positive and decays like [z[-~ at infinity, 0 < fl < 3. Let fo be an initial datum with unit mass and temperature, and let M be the associated Maxwellian distribution. Assume that fo is rapidly decreasing, in the sense that Vs > 0, IlfollL~ < -+-~. Then, for all e > 0 there exists so > 0 and a constant Ce(fo), depending only on e, N, and [[f0[lL2o, such that the unique smooth solution of the spatially homogeneous Landau equation with initial datum f o satisfies H(f (t, ")IM) ~<CE(fo) t -1/e.
- 226. A review of mathematical topics in collisional kinetic theory 225 We note that this theorem does not cover the interesting case fl = 3 (Coulomb potential in dimension 3): the proof in [429] just fails for this limit exponent. Including this case would be a significant improvement. We also note that this theorem deals with a smooth qJ, while realistic q/'s would present a singularity at the origin. This singularity cannot harm the entropy-entropy dissipation inequality, but may entail serious additional difficulties in getting the right a priori estimates, z~ 5.2. A remark on the multiple roles of the entropy dissipation Numerical applications for the constant TU appearing in (179) are very disappointing (say, 10-2~ .... ) This is because the entropy is quite bad at preventing concentration. Much better estimates are obtained via L ~ bounds for instance (which can be derived from regularization). Another possibility is to use the entropy dissipation as a control of concentration for f. The idea is the following: if the entropy dissipation is low (which is the bad situation for trend to equilibrium), then the distribution function cannot be concentrated too much close to a hyperplane, because the entropy dissipation measures some smoothness. As a consequence, Tf cannot be too small. More explicitly, say if q/(lzl) ~> Izl2, then [183, Section 5] TS> (N - 1)2 N + DL (f) " N By re-injecting this inequality in the proof of Theorem 14, one finds the following improvement (still under the assumption ~(Izl)/> Izl2) DL(f) ~ ~ZN(N- 1)2H(fIM)-~ N 4 N 2 4 2 This in turns implies exponential convergence to equilibrium with realistic bounds, which we give explicitly as an illustration. THEOREM 19. Let !It(Izl) ~ Izl2, and let fo be a probability distribution on ]1~N, with zero mean velocity and unit temperature. Let M be the associated Maxwellian distribution. Let f (t, .) be a classical solution of the Landau equation with initial datum fo. Then, for all time t >~O, IITCt,')-MII , ~/N C~e-~ e N < < " N-1 20Seethe discussionin Section 1.3of Chapter2E. CO 2 t 2(N-1)2t + ~/-N(N- 1) e~c~ e----w--, (208)
- 227. 226 C. Villani where Co = ~2N (N - 1)2H (folM) -k N 4 N 2 4 2 This estimate shows that satisfactory bounds can sometimes be obtained by cleverly combining all elements at our disposal! 5.3. The Boltzmann equation Once again, one has to separate between over-Maxwellian collision kernel, hard potentials or soft potentials. In order to apply Theorem 17, we need Moment estimates. They hold true for hard potentials without any assumption on the initial datum, and for Maxwellian collision kernels if a sufficient number of moments are finite at the initial time; we have discussed all this in Chapter 2B. In the case of soft potentials, these estimates are only established locally in time, but in some situations one can control the growth well enough. Lower bound estimates. Uniform such bounds were proven by A. Pulvirenti and Wennberg for Maxwellian collision kernels or hard potentials with cut-off In the case of soft potentials, uniform bounds are an open problem. But, still under the cut-off assumption, local (in time) bounds are very easy to obtain as a consequence of Duhamel's formula (100), if the initial datum satisfies a lower bound assumption. Such a crude bound as f (t, v) ~ Ke -Atlvl2 At- (1 + t) is sufficient in many situations [428, Section 4]. The case of non-cutoff collision kernels is still open. LP estimates. In the model case of hard potentials with Grad's cut-off assumption, such estimates are a consequence of the studies of Arkeryd and Gustafsson, as discussed in Section 3 of Chapter 2B. For instance, if the initial datum lies in L ~ with suitable polynomial decay, then the solution will be bounded, uniformly in time. Also the case of Maxwell collision kernel can be treated in the same way. However, when the collision kernel decays at infinity, things become more intricate. The search for robust estimates led the authors in [428] to a new way of controlling L p norms by the Q+ smoothness,21 moment estimates, and a lot of interpolation. 21For simplicity,kinetic collisionkernels q~(Iv- v,I) consideredin [429] were smooth and bounded from above and below.The authors had forgottenthat in such a case the Q+ smoothnesscould not apply directly, because q~(0) > 0. The proofis howevereasyto fixby treatingseparatelyrelativevelocitieswhichare closeto 0; this has beendonerecentlyby Mouhot.
- 228. A review of mathematical topics in collisional kinetic theory 227 On the other hand, in the case of non-cutoff collision kernels, LP estimates are obtained via Sobolev estimates and regularizing effect. Here we see that many of the estimates which we discussed in Chapter 2B can be combined to yield a qualitative theorem for solutions of the Boltzmann equation: trend to equilibrium with some explicit rate. Since the general panorama of a priori estimates for the spatially homogeneous Boltzmann equation is not completely settled yet, we do not have a general theorem. Let us give one which encompasses the few cases that can be treated completely. We put very strong conditions on the initial datum so that a unified result can be given for different kinds of collision kernels. THEOREM 20. Let B(v - v,, o') --~(Iv - v,l)b(cosO) be a collision kernel satisfying Grad's angular cut-off let fo be an initial datum with unit mass and temperature, and let f (t, .) be a strong solution of the Boltzmann equation with initial datum fo. Assume that fo lies in L ~ and decays at infinity like O(]v[ -k) for any k >10. Assume moreover that fo(v) ~ Ke -alvl2 for some A, K > O. Then (i) if ~ =_ 1, then H(fIM)- O(t-~); (ii) if~(lv - v,I) --Iv - v,[ • V > 0, then H(fIM) -- O(t-K) for some tc > 0; (iii) if q~(lv - v,[) is bounded, strictly positive and decays at infinity like Iv - v,I -r with 0 < fl < 2, then H(fIM) --O(t-~). Moreover, all the constants in these estimates are explicitly computable. Of course, O(t -~) means O(t -K) for any tc > 0. For parts (i) and (ii), see [428]; for part (iii), see [429]. Also we insist that result (i) also holds when q> is bounded from above and below. In Chapter 2D, we shall see that the structure of the particular case q~ = 1 allows a better result, in the form of an explicit exponential rate of convergence. 5.4. Infinite entropy We conclude this section with an interesting remark due to Abrahamsson [1]. Of course, it seems intuitive that entropy dissipation methods require an assumption of finiteness of the entropy. This is not true! In some situations one can decompose the solution of the spatially homogeneous Boltzmann equation into a part with infinite entropy, but going to zero in L 1 sense, and a part with finite entropy, on which the entropy dissipation methods can be applied. If the estimates are done with enough care, this results in a theorem of convergence with explicit rate, even when the initial datum has infinite entropy. It is important here to have a good control of the entropy-entropy dissipation inequality which is used, in terms of the initial datum. With this technique, Abrahamsson [1] was able to prove convergence to equilibrium in L 1 for the spatially homogeneous Boltzmann equation with hard spheres, assuming only that the initial datum has finite mass and energy. On this occasion he used the Carlen- Carvalho theorem, and also some iterated Duhamel formulas in the a priori estimates. Note that this problem mainly arises for cut-off collision kernels, because for most kernels with an angular singularity, entropy becomes finite for any positive time [440] by regularization effects.
- 229. 228 C.Villani This ends our review of applications of entropy methods in the spatially homogeneous Boltzmann and Landau equations. Before discussing spatially inhomogeneous models, we shall briefly consider another class of spatially homogeneous systems, characterized by their gradient flow structure. 6. Gradient flows This section is a little bit outside the main stream of our review, but reflects active trends of research in kinetic theory, and may enlighten some of the considerations appearing here and there in this chapter. The main application to Boltzmann-like equations is Theorem 21 below, for simple models of granular flows. 6.1. Metric tensors As we explained before, several equations in kinetic theory have a gradient flow structure: they can be written Of=vv.(fVv6E ) at -~- (209) for some energy functional f ~ E(f), which we shall always call the entropy for consistency. Typical examples are the Fokker-Planck equation, for which E(f) = flog f + flvl 2dr; N 2 N or the model from [68] for granular flow, af = Vv. (f(f * VU)) +crAvf +OVv. (fv), (210) Ot with U(v) - Ivl3/3, and tr, 0 > 0; then 1 ~ f(v)f(w)U(v- w)dvdw E(f) = -~ 2 s + (r f log f + -~ flvl 2dr. (211) Among examples outside kinetic theory, we have also mentioned the heat equation, the spatial Fokker-Planck equation, the porous medium equation .... Generally speaking, a gradient flow is an equation of the form dX n dt - - grad E(X(t)). (212)
- 230. A review of mathematical topics in collisional kinetic theory 229 Underlying the definition of the gradient operator is that of a Riemannian metric tensor on some "manifold" in which the unknown X lives. Thus, to explain why (209) is a gradient flow, we first have to explain how to define a meaningful metric tensor on the "manifold" of all probability measures. Of course this is a formal point of view, because infinite-dimensional Riemannian geometry usually does not make much sense, even if it is sometimes enlightening, as the well-known works by Arnold [40] in fluid mechanics illustrate. In our context, the relevant metric tensor is defined as follows. Let f be a probability density (assume that f is smooth and positive, since this is a formal definition). Let Of/Os be a "tangent vector": formally, this just means some function with vanishing integral. Then define Of 2 = inf, f } flu] 2dr; -~s + Vv" (fu) = 0 . (213) The infimum in (213) is taken over all vector fields u on ]t~N such that the linear transport equation Of/Os + Vv. (fu) = 0 is satisfied. By polarization, formula (213) defines a metric tensor, and then one is allowed to all the apparatus of Riemannian geometry (gradients, Hessians, geodesics, etc.), at least from the formal point of view. Then, an easy computation shows that (209) is the gradient flow associated with the energy E, on the "manifold" of all (smooth, positive) probability measures endowed with this Riemannian structure. The metric tensor defined by (213) has been introduced and studied extensively by Otto [364]. One of its important features is that the associated geodesic distance is nothing but the Wasserstein distance on probability measures. 22 This is part of the whole area of mass transportation, whose connections with partial differential equations are reviewed in Villani [452]. 6.2. Convergence to equilibrium A general property of gradient flows is that they make the entropy decrease. From formula (212) one sees that dE(x(t)) =- IIgrad E(XCt)) ll dt And the equilibrium positions of (212) are the critical points for E: typically, minima. In all the cases which we consider, there is a unique minimizer for E, which is therefore the only equilibrium state. Now, it is a general, well-known fact that the rate of convergence to equilibrium for a gradient flow is very much connected to the (uniform, strict ...) convexity of the energy functional. A typical result is the following: assume that the energy E is uniformly convex, 22See formula (244) below.
- 231. 230 C. Villani in the sense that its Hessian is bounded below by some positive multiple of the identity tensor, Hess E ~>)~Id. Then, E admits a unique minimizer X~, and the gradient flow (212) satisfies the linear entropy-entropy dissipation inequality IIgrad Ell ~ 2/.[E(X) - E(X~)]. (214) A possible strategy to prove (214) is to go to the second derivative of the entropy functional with respect to time. From (212) and the definition of the Hessian, d 2 dt [[gradE(X(t))II - 2(Hess(E) 9VE, VE). (215) The functional which just appeared in the right-hand side is the dissipation of entropy dissipation. Therefore, the assumption of uniform positivity of the Hessian implies d 2 dr IIgrad E (X (t))II -2~llgrad E(X<t>)ll 2 Integrating this inequality in time, one easily arrives at (214) if everything is well- behaved. 23 This remark shows that the trend to equilibrium for Equation (209) can in principle be studied via the properties of convexity of the underlying energy E. But the right notion of convexity is no longer the usual one: it should be adapted to the definition (213). This concept is known as displacement convexity, and was first studied by McCann [338,340], later by Otto [364], Otto and Villani [365]. DEFINITION 3. Let f0, fl be two (smooth, positive) probability measures on I[~N. By a classical theorem of Brenier [103,339] and others, there exists a unique gradient of convex function, Vqg, such that v~o#fo = f~, meaning that the image measure24 of fo by the mapping Vq9 is the measure fl. Let us define the interpolation (fs)o<~s<<.lbetween f0 and fl by fs -- [(1 - s)Id + sV~o]#fo. 23There are also other, simpler derivations of (214) based on Taylor formula. The above procedure was chosen because this is precisely a way to understand the famous Bakry-Emery method for logarithmic Sobolev inequalities. 24By definition of the image measure: for all bounded continuous function h, f(h o V~o)f0 = f hfl. If ~ois C2, then for all v, fo(v) = fl (V~0(v)) det(D2~0)(v).
- 232. A reviewofmathematicaltopicsincollisionalkinetictheory 231 Then, the functional E is said to be displacement convex if whatever fo, fl, s w-~E (f~) is convex on [0, 1]. It is furthermore said to be uniformly displacement convex with constant )~ > 0 if whatever fo, f~, d2 ds 2 ~E(fs) ~ )~W(fo, fl) 2 (0~<s ~< 1), where W stands for the Wasserstein distance25 between f0 and fl. REMARKS. (1) To get a feeling of this interpolation procedure, note that the interpolation between (~a and 6b is 6(1-s)a+sb,instead of (1 - S)6a + s3t,. Further note that the preceding definition reduces to the usual definition of convexity if the interpolation (fs)o~<s~<l is replaced by the linear interpolation. (2) Let us give some examples. The functionals f f log f, or f fP (p >/ 1) are displacement convex as one of the main results of McCann [340]. The functional f fV is displacement convex if and only if the potential V is convex. Moreover, if V is uniformly convex with constant )~, the functional f f V is uniformly displacement convex, with the same constant. Another interesting example is the case of functionals like f•2N f(v)f(w)U(v- w) dv dw. Such a functional is never convex in the usual sense, except for some very peculiar potentials (power laws ...). On the other hand, it is displacement convex as soon as U is convex. Among the results in Otto and Villani [365], we mention the following statement. If E is uniformly displacement convex, with constant )~ > 0, then the associated gradient flow system (209) satisfies a linear entropy-entropy dissipation inequality of the form fRNf EI2 7-~- dv >i-2s E(f~)], where f~ is the unique minimizer of the energy. This is not a true theorem, because the proof is formal, but this is a general principle which can be checked on each example of interest. A standard strategy of proof goes via the second derivative of the entropy, as we sketched above. In the context of the linear Fokker-Planck equation (168), this strategy of taking the second derivative of the entropy is known as the Bakry-Emery strategy, and goes back to the mid-eighties. To check the assumption of uniform displacement convexity, it is in principle sufficient to compute the Hessian of the entropy. When this is done, one immediately obtains the dissipation of entropy dissipation via formula (215). This calculation is however very intricate, as one may imagine. This is where Bakry and Emery [45] need their so-called 25 Seeformula(244)below.
- 233. 232 C. Villani F2 calculus, which is a set of formal computation rules involving linear diffusion operators and commutators. On the other hand, the formalism developed by Otto [364], Otto and Villani [365] enables simpler formal computations, and can be adapted to nonlinear cases such as granular flows [130]. Just to give an idea of the complexity of the computations, and why it is desirable to have efficient formal calculus here, let us reproduce below the dissipation of entropy dissipation which is associated to the gradient flow for Equation (210): fo ]2 DD(f) = 2rr f (v) -~v~(V) dv + 20 f(v) I~(v)12dv + f(v)f(w)(D2U(v - w) x - - (216) where a[ Iv12 1 (v)=~ (rlogf(vl+O--~+U,f , U(v) = Ivl3/3. 6.3. A survey of results Let us now review some results of trend to equilibrium which were obtained via the considerations above, or which can be seen as related. A survey paper on this subject is Markowich and Villani [330]. The first partial differential equation to be treated in this way was the spatial Fokker- Planck equation, Op Ot =Vx . (Vxp -t- pVV(x)). The classical paper by Bakry and Emery [45] shows that the solution to this equation converges exponentially fast, in relative entropy sense, to the equilibrium e-v (assuming f e- v _ 1), at least if v is uniformly convex with constant )~.The decrease of the entropy is like e-2~t . Underlying entropy-entropy dissipation inequalities are known under the name of logarithmic Sobolev inequalities, and have become very popular due to their relationship with many other fields of mathematics (concentration of measure, hypercontractivity, information theory, spin systems, particle systems ... see the review in [16]). The Bakry-Emery strategy, and the corresponding proof of the Stam-Gross logarithmic Sobolev inequality, were recently re-discovered by Toscani [422,423] in the case of the kinetic Fokker-Planck equation. Instead of F2 calculus, Toscani generalized a lemma by McKean [341] to compute the second derivative of the entropy functional. Though this
- 234. A review of mathematical topics in collisional kinetic theory 233 work was mainly a re-discovery of already known results, it had several merits. First, it suggested a more physical way of understanding the Bakry-Emery proof, in terms of entropy dissipation and dissipation of entropy dissipation. Also, Toscani directly studied the Fokker-Planck equation Otf -- Af + V. (f v), while previous authors mainly worked on the adjoint form, Oth -- Ah - v 9Vh. Last but not least, his paper made these methods and techniques popular among the kinetic community, which began to work on this subject: see in particular the recent synthesis by Arnold, Markowich, Toscani and Unterreiter [39]. Then, these results were generalized to the porous medium equation with drift, Op -- Axp m -k- Vx . (px). Ot It was found that when m ~> 1 - 1/N, solutions to this equation converge exponentially fast (with relative entropy decreasing like e-2t) towards a probability density known as Barenblatt-Pattle profile. These results were obtained independently by Otto [364], Carrillo and Toscani [131], and Del Pino and Dolbeault [163]. All three papers established nonlinear analogues of the logarithmic Sobolev inequalities. The paper by Otto made the link with mass transportation problem. Various generalizations of all these results can be found in [129]. Let us now come back to kinetic, Boltzmann-like systems and display recent results about the equations for granular media suggested in [70,68]. These results were proven by Carrillo, McCann and Villani [130] by using the ideas above, and in particular those of gradient flows, Wasserstein distance and Bakry-Emery strategy. The theorem which we state is slightly more general: the dimension is arbitrary, and the interaction potential is not necessarily cubic. THEOREM 21. Let U be a convex, symmetric potential on ~N, and or, 0 ~ O. Let E(f)--a f fl~ +Of f(v)lv122dv 1s f (v) f (w)U(v - w) dvdw. Let moreover iol2 ) ~-V crlogf +O-~+U. f . We consider the equation of = V. (f~), (217) at for which E is a Lyapunov functional, whose time-derivative is given by the negative of D(f) - f fill2 dr.
- 235. 234 C. Villani Let fo be an initial datum, and f (t) be the solution 26 to (217). Let moreover fc~ be the unique minimizer of E (if 0 -- O, the unique minimizer of E which has the same mean velocity as fo), and let E(flf~)= E(f)- E(f~). Then, (i) if 0 > O, then D(f) ~ 20E(flf~) for all probability distribution f, and E(f (t)lf~) <<.e-20tE(fOIfoo); (ii) if O = 0 and U is uniformly convex, D2U ~ s then D(f) ~ 2s for all f with the same mean velocity as foo, and E(f (t)lf~) <~e-2xt E(folf~); (iii) if 0 = 0 and U is strictly convex, in the sense D2U(z) ~ g(Izl '~ A 1), g,o~ > O, (218) then D(f) ~ CE(flfc~) # for some positive constants C, r, and for any f with the same mean velocity as f~, and E(f (t)lfc~) converges to 0 at least in O(t -K) for some x > 0; (iv) if 0 = O, U is strictly convex in the sense of (218), and moreover a > O, then for all f with the same mean velocity as f~, one has D(f) ~ s for some s > 0 which only depends on an upper bound for E(f). Moreover, E(f (t)lf~) <. e-X~ for some positive constant )~o which only depends on an upper bound for E(fo). REMARKS. (1) The assumptions on the mean velocity reflect an important physical feature: in the cases in which they are imposed, the entropy is translation-invariant. (2) The motivation to study strictly convex, but not uniformly convex potentials comes from the physical model where U (z) = Izl3/3. Then, lack of uniformity may come from small values of z. This difficulty is of the same type than in the study of hard potentials for the Boltzmann equations; this is not surprising since the model in [70] can be seen as a limit regime for some Boltzmann-type equation with inelastic hard spheres. (3) In general, point (iii) cannot be improved into exponential decay: when a -- 0 --- 0 and U(z) = Izl3/3, then the decay of the energy is O(1/t) and this is optimal. (4) We note that part (iv) of this theorem is the most surprising, because in this case the energy functional is not uniformly displacement convex; yet there is a "linear" entropy- entropy dissipation inequality (not uniform in f). This result raises hope that the entropy- entropy dissipation inequalities described in Section 4 for the Landau equation with hard potentials may be improved into inequalities of linear type. To conclude this section, we mention that the trend to equilibrium for Equation (217) has been studied by Malrieu [329] with the F2 calculus of Bakry and Emery. Even though his results are much more restrictive (only a, 0 > 0) and the constants found by Malrieu are not so good, on this occasion he introduced several interesting ideas about particle 26WeassumethatU is sufficientlywell-behavedthatexistenceofaunique"nice"solutionto(217)is guaranteed. Thisis quitea weakassumption.
- 236. A review of mathematical topics in collisional kinetic theory 235 systems and his work provided yet another connection between the kinetic and probabilistic communities.* 7. Trend to equilibrium, spatially inhomogeneous systems We now turn to the study of spatially inhomogeneous kinetic systems like the ones presented in Section 1.2. We first make several remarks. (1) Once again, we are mainly interested in explicit results, and wish to cover situations which are not necessarily perturbations of the equilibrium. Thus we do not want to use linearization tools, and focus on entropy dissipation methods. (2) For many of the spatially inhomogeneous models which we have introduced, the entropy and the entropy dissipation functionals are just the same as in the spatially homogeneous case, up to integration in x. Also, the transport part does not contribute to the entropy dissipation. Thus, one may think, the same entropy-entropy dissipation inequalities which we already used for the spatially homogeneous case will apply to the spatially inhomogeneous case. This is completely false, as we shall explain! And the obstruction is not a technical subtlety, but stands for a good physical reason. (3) Nevertheless, it is plainly irrelevant to ask for an x-dependent version of the entropy- entropy dissipation inequalities presented in Section 4, since the entropy dissipation does not make the x variable play any role. (4) The boundary conditions, and the global geometry of the spatial domain, are extremely important in this study. In this respect the problem of trend to equilibrium departs notably from the problem of the hydrodynamic limit, which fundamentally is a local problem. (5) To work on the trend to equilibrium, one should deal with well-behaved solutions, satisfying at least global conservation laws. In the sequel, we shall even assume that we deal with very well-behaved solutions, for which all the natural estimates of decay, smoothness and positivity are satisfied. Of course, for such equations as the Boltzmann or Landau equation, nobody knows how to construct such solutions under general assumptions .... Therefore, the results dealing about these equations will be conditional, in the sense that they will depend on some strong, independent regularity results which are not yet proven. It is however likely that such regularity bounds can be obtained with present- day techniques in certain particular situations, like weakly inhomogeneous solutions [32]. We wish to insist that even if we assume extremely good a priori estimates, the problem of convergence to equilibrium remains interesting and delicate, both from the mathematical and from the physical point of view! 7.1. Local versus global equilibrium When studying the trend to equilibrium in a spatially dependent context, a major obstacle to overcome is the existence of local equilibrium states, i.e., distribution functions which are *Note added in proof: In a more recent, quite clever work, Malrieu was able to remove the condition 0 > 0.
- 237. 236 C. Villani in equilibrium with respect to the velocity variable, but not necessarily with respect to the position variable. For instance, for the Boltzmann or Landau equation, a local equilibrium is a local Maxwellian, Iv-u(x)l 2 e 2T(x) f (x, v) -- p(x) tZ" lt )) /'''7r~'x''N/2 . (219) For the linear Fokker-Planck equation, a local equilibrium is a distribution function of the form Ivl2 e 2 f (x, v) -- p(x) (2:rr)N/----------------- ~ = p(x)M(v). (220) Local equilibria are not equilibrium distributions in general, but they make the entropy dissipation vanish. This shows that there is no hope to find an entropy-entropy dissipation inequality for the full x-dependent system. If the system ever happens to be in local equilibrium state at some particular time to, then the entropy dissipation will vanish at to, and it is a priori not clear that the entropy functional could stay (almost) constant for some time, before decreasing again. This may result in a strong slowing-down of the process of trend to equilibrium. This difficulty has been known for a long time (even to Boltzmann! as pointed out to us by C. Cercignani), and is discussed with particular attention by Grad [254], Truesdell [430, pp. 166-172] and, in the different but related context of hydrodynamic limits for particle systems, Olla and Varadhan [362]. On the other hand, whenever the solution happens to coincide with some local equilibrium state Mloc at some time, then the combined effect of transport and confinement will make it go out of local equilibrium again, unless Mloc satisfies some symmetry properties which ensure that it is a stationary state. In fact, in most situations these symmetry properties select uniquely the stationary state among the class of all local equilibria. Note the fundamental difference with the problem of hydrodynamical limit: in the latter, one wishes to prove that the solution stays as much as possible close to local equilibrium states, while here we wish to prove that if the solution ever happens to be very close to local equilibrium, then this property will not be preserved at later times. Thus, one can see the trend to equilibrium for spatially inhomogeneous systems as the result of a negociation between collisions on one hand, transport and confinement on the other: by dissipating entropy, collisions want to push the system close to local equilibrium, but transport and confinement together do not like local equilibria- except one. This is why transport phenomena, even if they do not contribute in entropy dissipation, play a crucial role in selecting the stationary state. Our problem is to understand whether these effects can be quantified.
- 238. A review of mathematical topics in collisional kinetic theory 237 An answer to this question was recently obtained by the author in a series of collaborations with Desvillettes. In the sequel, we shall explain it on a simple case: the linear Fokker-Planck equation, with potential confinement, ~f + v. Vxf - VV(x). Vvf = Vv. (Vvf + fv). (221) at The trend to equilibrium for this model was studied by Bouchut and Dolbeault [100] with the help of compactness tools, so no explicit rate of convergence was given. Also, Talay [413] proved exponential decay with a probabilistic method (based on general theorems about recurring Markov chains) which does not seem to be entirely constructive. Other probabilistic approaches have been proposed to study this model, but they also strongly depend on the possibility to interpret (221) as the evolution equation for the law of the solution of some stochastic differential equation with particular properties.* In the sequel, we shall explain how the entropy method of Desvillettes and Villani [184] leads to polynomial (as opposed to exponential), but fully explicit estimates. This method is robust in the sense that it can be generalized to smooth solutions of nonlinear equations, in particular Boltzmann or Vlasov-Poisson-Fokker-Planck equations. With respect to the Boltzmann equation, the model (221) has several pedagogical advantages. First, one can prove all the a priori estimates which are needed in the implementation of the method. What is more important, the local equilibrium only depends on one parameter (the density), instead of three (density, mean velocity and temperature). This entails significant simplifications in the computations and intermediate steps, which however remain somewhat intricate. 7.2. Local versus global entropy: discussion on a model case To use entropy methods in a spatially dependent context, the main idea is to work at the same time at the level of local and global equilibria; i.e., estimate simultaneously how far f is from being in local equilibrium and how far it is from being in global equilibrium. (1) One first introduces the local equilibrium associated with f, i.e., the one with the same macroscopic parameters as f. For instance, in the case of the linear Fokker-Planck equation, the local equilibrium is just (220), with p(t, x) = f f(t, x, v)dr. In the case of the Boltzmann equation, the local equilibrium is the local Maxwellian (219), with p, u, T given by (1). How close f is from local or global equilibrium will naturally be measured by relative entropies. Thus one defines Hglo to be the relative entropy of f with respect to the global equilibrium, and Hloc to be the relative entropy of f with respect to the associated local equilibrium. In the Fokker-Planck (resp. Boltzmann) case, Hloc is H(flpM) (resp. H(fIMf)); note that this is an integral over Rff • R N now. Then, one looks for a system of differential inequalities satisfied by Hglo and Hloc. *Note added in proof: A muchmorecomplete,verysatisfactorystudywas recentlyperformedby Hrrau and Nier; see alsothe referencesprovidedin theirwork.
- 239. 238 C. Villani (2) The first equation is given by an entropy-entropy dissipation estimate of the same type that the ones we discussed in Section 4. We just have to apply this inequality pointwise in x. For instance, for the linear Fokker-Planck equation, dHglo=s I(flpM)dx -- s p I( f D(f ) -- - dt N N M) dx, (222) and by the logarithmic Sobolev inequality (175), D(f)>2s pH( f ~x M) dx = 2H(fIpM) - - 2Hloc (223) (check the last-but-one equality to be convinced!). Note that the symbol H is used above in two different meanings: relative entropy of two probability distributions of the v variable, relative entropy of two probability distributions of the x and v variables. Similarly, if we have nice uniform a priori bounds for the solution of the Boltzmann equation, it will follow from our discussion in Section 4 that dt -- -- Hglo ) K Hl~c, (224) for some constants K > 0, ot > 1. In a spatially homogeneous context, this inequality would be essentially sufficient to conclude by Gronwall's lemma. Here, we need to keep much more information from the dynamics in order to recover a control on how the positivity of Hglo forces Hloc to go up again if it ever vanishes. (3) To achieve this, we now look for a differential inequality involving the time-behavior of Hloc. We start with a heuristic discussion. At a time when the entropy dissipation would vanish, then both the local relative entropy and its time derivative would vanish, since the relative entropy is always nonnegative. Therefore, one can only hope to control from below the second time derivative of the local relative entropy! Taking into account the first differential inequality about Hglo and Hloc, this more or less resembles to considering the third derivative of the entropy at an inflexion point. It is easy to compute (d2/dr 2)Hloc at a time to when f happens to be in local equilibrium. For instance, in the case of the linear Fokker-Planck equation, we have the remarkably simple formula d21 s H(fIpM) - p dt2 t--to N vp 1: v). § VV dx =_Ix (pie- (225) P Here Ix is the Fisher information, applied to functions of the x variable. We do not describe here the corresponding results for the Boltzmann equation, which are of the same nature but much, much more complicated [181]. Here we shall continue the discussion only for the Fokker-Planck equation, and postpone the Boltzmann case to the end of the next section.
- 240. A review of mathematical topics in collisional kinetic theory 239 If V is well-behaved, the logarithmic Sobolev inequality, applied in the x variable, yields d 2 dt 2 t=to H(flpM) >~KH(ple -V) (226) for some positive constant K depending only on V. This is the piece of information that was lacking! Indeed, for the linear Fokker-Planck equation, Hglo-- Hloc4- H(ple-V); thus Equation (226) turns into d 2 dt 2 t=to H (f IpM) >~KHglo- K Hloc. (227) Note that the use of the logarithmic Sobolev inequality in the x variable is the precise point where the geometry of the boundary conditions (here replaced by a confinement potential) comes into play. The fact that this effect can be quantified by a functional inequality is very important for the method; see the remarks in the end of the chapter for the analogous properties in the Boltzmann case. Of course, the preceding calculations only apply at a time to where f happens to be in local equilibrium- which is a very rare event. Therefore, one establishes a quantitative variant of (227), in the form d2 K dt ----TH(flpM) ~> z---x-(H~I~ Hloc)- J(flpM), (228) where J (flpM) is a complicated functional which vanishes only if f -- pM: 1 fR (pu)2 l fR IVx.(pu)12 - J (flpM) -- ~ dx 4- 4 U p -4 N [9 dx UR IVx" (puQu)] 2 fR 4- dx4- U /9 U IVx[p(T- 1)]12 P fR [Vx . S]2 + ~ dx + Iv(f[pM) U p dx 1 4- -~Iv(flp M) 1/2 Ix(flp M) 1/2. (229) Here p, u, T are the usual macroscopic fields, and S is the matrix defined by the equation p(x)u(x) | u(x) + p(x)T(x)IN 4- S(x) -- f•N f (x, V)V | vdv. (230)
- 241. 240 C. Villani (4) The next step of the program is to control J in terms of Hloc, in order to have a closed system of differential inequalities on Hloc and Hglo. This is done by some ad hoc nonlinear interpolation procedure, which yields d 2 dt 2 K --H(fIpM) >~-~ H(flfc~)- Cs(f) H(flpM) 1-E Here e is an arbitrary positive number in (0, 1) and Ce(f) is a constant depending on f via moment bounds, smoothness bounds, and positivity estimates on f. All these bounds have to be established explicitly and uniformly in time, which turns out to be quite technical but feasible [184] (see also Talay [413]); then the constant Ce = Ce (f) can be taken to be independent of t. In the case of the Boltzmann equation, it is possible to perform a similar interpolation procedure; the only missing step at the moment is establishing the a priori bounds. (5) Summing up, for solutions of the Fokker-Planck equation we have obtained the system of differential inequalities d -- ~-THglo > 2Hloc, d2 K d-~ Hloc/> -~- Hglo- Ce Hlloc e 9 (231) The last, yet not the easiest step, consists in proving that the differential system (231) alone implies that Hglo converges to 0 like O(t-K). Since there is apparently no comparison principle hidden behind this system, one has to work by hand .... The bound established in Desvillettes and Villani is Hglo = O(tl-1/e), which is presumably optimal. Thus, the global entropy converges to 0 with some explicit rate, which was our final goal. 7.3. Remarks on the nature of convergence Solutions to (231) do have a tendency to oscillate, at least for a certain range of parameters. In fact, were it not for the positivity of relative entropies, system (231) would not imply convergence to 0 at all! We expect "typical" solutions of (231) to decrease a lot for small times, and then converge to 0 more slowly as t --+ +c~, with some mild oscillations in the slope. This kind of behavior is completely different from what one can prove in the context of spatially homogeneous kinetic equations. 27 We think that it may reflect the physical nature of approach to equilibrium for spatially inhomogeneous systems. As time becomes large and the system approaches global equilibrium, it is more and more likely to "waste 27The rate of convergence typically improves as t --+ +c~, see Section 5.1.
- 242. A reviewof mathematical topics in collisionalkinetic theory 241 time" fighting against local equilibria .... And this may result in oscillations in the entropy dissipation. But examination of a particular, "integrable" case, suggests that (1) these oscillations may be present only when the confinement potential is strong enough, (2) the decay should be exponential. This case corresponds to the quadratic confinement potential, V(x) = 0921xl2/2 -+-C. For this particular shape of the potential, the Fokker-Planck equation can be solved in semi-explicit form [399, Chapter 10], and the rate of decay is governed by the quantities exp(-Xt), where 1 1 - ~/1 - 4092 092 ~< 4 X= 2 ' -' - 1 X-- 1 4- i~/4092 1 092 > 2 ' 4 (232) (in [399], these equations are established only in dimension 1). Thus the decay is always exponential, the rate being given by the real part of X. When the confinement is very tiny, then the convergence is very slow (think that there is no trend to equilibrium when there is no confinement); when the confinement becomes stronger then the rate increases up to a limit value 1/2. For stronger confinements, the rate does not improve, but complex exponentials appear in the asymptotics of the solution. Note that in the same situation, the rate of convergence for the spatially homogeneous equation would be equal to 1. Another integrable case is when there is no confinement potential, but x 6 TN, the N- dimensional toms. Then the decay is always exponential and the rate depends on the size of the periodic box. In dimension 1 of space, it is maximal (equal to 1) when the side of the box has length ~<2zr [152]. It is yet an open problem to generalize the above considerations to nonintegrable cases, and to translate them at the level of entropy dissipation methods. In our opinion, these examples show that a lot of work remains to be done to get an accurate picture of the convergence, even in very simplified situations.* 7.4. Summary and informal discussion of the Boltzmann case We now sum up the state of the art concerning the application of entropy dissipation methods to spatially inhomogeneous systems. The following theorem is the main result of Desvillettes and Villani [184]. THEOREM 22. Let M(v) denote the standard Maxwellian probability distribution on ]KN with zero mean velocity and unit temperature. Let V be a smooth confining potential on ~N, behaving quadratically at infinity: V(x) = ~o2' ' 'x'2 + ~(x), 2 *Note addedinproof: Forprogressonthesequestions,the recentworkby HrrauandNieris recommended.
- 243. 242 C. Villani where co > 0 and 9 9 ~k>~OHk(RN) 9Assume without loss of generality that f e-v(x) dx -- 1, and let foe(X, v) = e-V(X)M(v) denote the unique global equilibrium of the Fokker-Planck equation af Ot - - + v . Vx f - VV(x). Vv f = Vv . (Vv f + f v). Let fo = fo(x, v) be a probability density such that fo/f~ is bounded from above and below, and let f (t) = f (t, x, v) be the unique solution of the Fokker-Planck equation with initial datum fo. Then, for all e > 0 there exists a constant CE(fo), explicitly computable and depending only on V, fo and e, such that H(f (t)lf~) <<.C~(fo) t-1/e. We already pointed out several shorthands of this result: in particular, the convergence should be exponential. We consider it as a major open problem in the field to compute the optimal rate of decay, in relative entropy, as a function of the confinement potential V. Let us now turn to nonlinear situations. The following result was recently obtained by the author in collaboration with Desvillettes [181]. THEOREM 23. Let B be a smooth collision kernel bounded from above and below. Let fo = fo(x, v) be a smooth probability density on I-2x x N3v,where [2 is a smooth bounded, connected open subset of ]R3 with no axis of symmetry, and let f (t) - f (t,x, v) be a smooth solution of the Boltzmann equation af Ot --+ v. Vx f = Q(f, f), t>~O, xeX2, yen 3, with specular reflection boundary condition. Let moreover f~ (x, v) be the unique global equilibrium compatible with the total mass and kinetic energy of fo. Assume that all the moments of f are uniformly bounded in time, that f is bounded in all Sobolev spaces, uniformly in time, and that it satisfies a lower bound estimate, f (t, x, v) >/poe-A~ for some p >~2, Po > O, Ao > O, uniformly in time. Then, for all e > 0 there is a constant Ce, depending only on (finitely many of) the requested a priori bounds, such that H(f (t)lf~) <. Cet -1/e. We do not display here the system of differential inequalities - much, much more complicated than (231) - which underlies this result, and we refer to [181] for more information. An unexpected feature was revealed by this study: not only is the entropy
- 244. A review of mathematical topics in collisional kinetic theory 243 dissipation process indeed slowed down when the distribution function becomes a local Maxwellian state, but also, it is much more slowed down for some particular local Maxwellian states, in particular those of the form p(x)M(v), i.e., with constant temperature and zero velocity. More precisely, the entropy dissipation vanishes in time up to order 4 (instead of 2) when going through such a Maxwellian state. From the mathematical point of view, this entails a spectacular complication of the arguments, and the need for at least three differential inequalities: apart from the behavior of the global entropy, one studies at the same time the departure of f with respect to Mf and the departure of f with respect to pM. From the physical point of view, this additional degeneracy could be interpreted as an indication that the relaxation of the density typically holds on a longer time-scale, than the relaxation of the temperature and the local velocity- although we should be cautious about this. The proof of Theorem 23 combines the general method of Desvillettes and Villani [184] with the entropy dissipation results of Toscani and Villani [428]. In the computations, the natural functionals H(flMU), H(flpM) were traded for the simpler substitutes Ilf - mf 112L2' Ilf - PMIIZL 2 which enable one to weaken significantly the assumptions of Theorem 23. Of course, these assumptions are still very strong, even though rather natural after our discussions in Chapter 2B. The influence of the shape of the box is quantified by the values of several "geometric" constants related to it. One of these constants is the Poincar6 constant P(s defined in a scalar setting by L2(n) whenever F is a real-valued function on f2, and in a vector setting by IlVxull2 L2(n)/> P(Y2)llull 2 L2 (f2) ' whenever u is a vector field in X2, tangent to 0f2. Another constant which appears in the proof is what we call Grad'snumber,defined in [254]: G(f2)-inf {fs2~ij ( Ovi OvJ)2" coo6SN-1 .. ~ + OXi ~] ' V. v--O, V/x v-coo, v.n--Oon Of2}, n standing for the unit normal on OX2. The number G(f2) is strictly positive if and only if s has no axis of symmetry. This number contributes to a lower bound for the constant K (s in a variant of the Korn inequality which reads VxU+rVxU L2(n) >>. K(~)llVxul122(n). (233)
- 245. 244 C. Villani The Korn inequality, of paramount importance in elasticity theory [201], is naturally needed to establish the system of differential inequalities which we use to quantify the trend to equilibrium. Our proof of (233) is partly inspired by Grad [254]. The whole thing adapts to the case of the toms, or to the bounce-back boundary condition, with significant simplifications. On the other hand, in the case of domains with an axis of symmetry, additional global conservation laws (angular momentum) have to be taken into account, and the case of a spherical domain also has to be separated from the rest. These extensions are discussed by Grad [254], but have not yet been transformed in a quantitative variant along the lines above. REMARK. As we have seen in Chapter 2B, if the initial datum is not very smooth and if the Boltzmann collision kernel satisfies Grad's cut-off assumption, then the solution of the Boltzmann equation is not expected to be very smooth. But in this case, as we discussed in Section 3.5 of Chapter 2B, one can hope for a theorem of propagation of singularities in which a vanishingly small (as t ~ c~) singular part could be isolated from a very smooth remainder, and, as in [1], the entropy dissipation strategy would still apply. Theorem 23 certainly calls for lots of improvement and better understanding. Yet, it already shows that, in theory, entropy dissipation methods are able to reduce the problem of trend to equilibrium for the full Boltzmann equation, to a problem of uniform a priori estimates on the moments, smoothness and strict positivity of its solutions. Moreover, it shows that there is no need for stronger a priori estimates than the ones which are natural in a nonlinear setting" in particular, no estimates in L2(M -1) are needed. We hope that these results will also provide a further motivation for the improvement of known a priori bounds.
- 246. CHAPTER 2D Maxwell Collisions Contents 1. Wild sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 2. Contracting probability metrics ........................................ 249 2.1. The Wasserstein distance ........................................ 249 2.2. Toscani's distance ............................................ 251 2.3. Other Fourier-based metrics ....................................... 252 2.4. The central limit theorem for Maxwell molecules ........................... 253 3. Information theory ............................................... 254 3.1. The Fisher information .......................................... 254 3.2. Stam inequalities for the Boltzmann operator ............................. 255 3.3. Consequence: decreasing of the Carlen-Carvalho 7t functional ................... 257 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 4.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 4.2. A remark on sub-additivity ....................................... 260 4.3. Remark: McKean's conjectures ..................................... 260 245
- 247. Preface The motion of fluids has intrigued scientists since antiquity and we may say that the field of mathematical fluid dynamics originated more than two centuries ago. In 1755 Euler [2] gave a mathematical formulation of the principle of conservation of mass in terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear and angular momentum by PDEs. Material symmetry and frame invariance were used by Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint- Venant [11] and Stokes [12]. In the 19th century no sharp distinction was drawn between mathematicians and physicists as we sometime see in more recent times. The formulation of the equations of fluid motion could be considered as either mathematics or physics. The first work in fluid dynamics that has a "modern" mathematical flavor may have been done by Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous problem. Riemann recognized the mathematical nature of the entropy. This notion led him to his duality method for solving the non-characteristic Cauchy problem for linear hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his contemporaries. What we now call the Cauchy problem for a PDE and the search for its solution did not have the significance that it is accorded nowadays. Even Poincar6 did not raise that kind of question in his Th~orie des tourbillons. For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28. Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much as Riemann, Leray developed new mathematical tools which proved to have independent interest: e.g., weak solutions (that we now call Leray's solutions in this context) and topological degree (a joint work with Schauder [4]). Since the 1930s, the interest that mathematicians devote to fluid dynamics has unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic displays such a variety of models and questions that thousands of scientists, among them many mathematicians, focus their research on fluid dynamics.
- 248. A reviewof mathematicaltopicsin collisionalkinetictheory 247 In this chapter we focus on the Boltzmann collision operator when the collision kernel only depends on the deviation angle: B(v - v,, or) = b(cosO). (234) As recalled in Chapter 2A, the modelling of Maxwell molecules, i.e., more or less fictitious particles interacting via repulsive forces in 1/r 5, in three dimensions of space, leads to a collision kernel B which satisfies (234). By extension, we shall call Maxwellian collision kernel any collision kernel of the form (234). Assumption (234) entails a number of particular properties. The gain part of the Boltzmann collision operator Q+(g,f)=s ao,f N N-I cos0= Iv-v,l'a' dcr b(cosO)g(v',) f (v'), (235) shares many features with the (more symmetric) rescaled convolution operator, g. f ----gl 9 f!, (236) 2 2 where the rescaling operation is defined by 1 (v t fk (1)) -- )~N/2f --~ " (237) Note that if X and Y are independent random variables with respective law f and g, then f * g is the law of (X + Y)/~/2. Therefore, with the analogy between the Q+ and * operations in mind, the theory of the spatially homogeneous Boltzmann equation with Maxwellian collision kernel resembles that of rescaled sums of independent random variables. In the sequel, we shall insist on some peculiar topics which illustrate the originality of Maxwellian collision kernels: in Section 1 the Wild sum representation, which is an appealing semi-explicit representation formula for solutions in terms of iterated Q+ operators; in Section 2, the existence and applications of several contracting probability metrics compatible with the Boltzmann equation. Finally, in Section 3, we describe some interesting connections with information theory. For more standard issues concerning the Cauchy problem or the qualitative behavior of solutions, the best reference is the long synthesis paper by Bobylev [79], entirely based on the use of Fourier transform, which also reviews many contributions by various authors. Before embarking on this study, we recall that besides its specific interest, the study of Maxwellian collision kernels is often an important step in the study of more general properties of the Boltzmann operator. 1 1See,for instance,Section4.4 in Chapter2Bor Sections4.3 to4.6 in Chapter2C.
- 249. 248 C. Villani 1. Wild sums If the collision kernel is Maxwellian and Grad's cut-off assumption is satisfied, then one can assume without loss of generality that for some (and thus any) unit vector k, is fo N-1 b(k. a) da = IsN-2I b(cos0) sinN-2 0 dO = 1. (238) Then the spatially homogeneous Boltzmann equation can be rewritten as ~f = Q+ (f, f)- f. (239) Ot However, when the collision kernel is nonintegrable, then one can only write the general form Ofot= f~tN dr, fSN-1 da b(cosO)[fl f~, - ff,]. (240) As was noticed by Wild [464], given any initial datum f0, Equation (239) can be solved recursively in terms of iterated Q+ operators (this is nothing but a particularly simple iterated Duhamel formula, if one considers (239) as a perturbation of Otf = --f). One finds oo f(t,.)=e-tZ(1-e-t)n-lQ+(fo) ' n=l (241) where the n-linear operator Q+ is defined recursively by o+r io, n-1 1 Qn-k(fO))" - 1 ~ (o; Io), + k=l The sum (241) can also be rewritten f(t, v) -- Ze-t(1 e-t) n-1 + - ot(ylQ• , n=l y ) (242) where F(n) stands for the set of all binary graphs with n leaves, each node having zero or two "children", and Q+ (f0) is naturally defined as follows: if y has two subtrees Yl and Y2 (starting from the root), then Q+ (f0) = Q+ (Q• (f0), Q+ (f0)). Moreover, ct(y) are combinatorial coefficients. Wild sums and their combinatorial contents are discussed with particular attention by McKean [341], and more recently by Carlen, Carvalho and Gabetta [126].
- 250. A review of mathematical topics in collisional kinetic theory 249 It follows from the Wild representation that a solution of the Boltzmann equation (239) can be represented as a convex combination, with time-dependent weights, of terms of the form fo, Q+(fo, fo), Q+(Q+(fo, fo), fo), Q+(Q+(fo, fo), Q+(Q+(fo, fo), Q+(fo, fo))), etc. This representation is rather intuitive because it more or less amounts to count collisions: the f0 term takes into account particles which have undergone no collision since the initial time, the term in Q+ (f0, f0) corresponds to particles which have undergone only one collision with a particle which had never collided before, Q+ (f0, Q+ (f0, f0)) to particles which have twice undergone a collision with some particle having undergone no collision before .... This point of view is also interesting in numerical simulations: in a seemingly crude truncation procedure, one can replace (241) by No f (t, V)- e-t E(1 --e-t) n-1Q+(fo)+ (1- e-t)NOM, n--1 where M is the Maxwellian distribution with same first moments as f0. Later in this chapter, we shall explain why such a truncation is rather natural, how it is related to the problem of trend to equilibrium and how it can be theoretically justified. 2 2. Contracting probability metrics In this section, probability metrics are just metrics defined on a subset of the space of probability measures on IRN. We call a probability metric d nonexpansive along solutions of Equation (240) if, whenever f (t, .) and g(t, .) are two solutions of this equation, then d(f (t), g(t)) <~d(fo, go). (243) We also say that d is contracting if equality in (243) only holds for stationary solutions, i.e. (in the case of finite kinetic energy), when f0, go are Maxwellian distributions. 2.1. The Wassersteindistance In his study of the Boltzmann equation for Maxwellian molecules, Tanaka [414,353,415] had the idea to use the Wasserstein (or Monge-Kantorovich) distance of order 2, W(f, g) = inf{y/ElZ - YI2; law(X) - f, law(Y) - g}. (244) 2Thisis the purposeofTheorem24(v).
- 251. 250 C.Villani Here the infimum is taken over all random variables X, Y with respective law f and g. It is always assumed that f and g have finite moments of order 2, which ensures that W(f, g) < +oo. In analytical terms, W can be rewritten as W (f, g) : inf, ~f]~Nx~NIv - wl2dzr(v, w); Jr E/7(f, g) }, where/7 (f, g) stands for the set of probability distributions Jr on ]1~ N X ]1{N which admit f and g as marginals, More explicitly, zr 6/7 (f, g) if and only if c0(R • c0(R ), I +f The Wasserstein distance and its variants are also known under the names of Frrchet, HOffding, Gini, Hutchinson, Tanaka, Monge or Kantorovich distances. The infimum in (244) is finite as soon as f and g have finite second moments. Moreover, it is well-known that convergence in Wasserstein sense is equivalent to the conjunction of weak convergence in measure sense, and convergence of the second moment: V(tg E C o (]~N ) , f f "~odVn__+ ~>f f ~od. , >0 ~---~ w(fn' f)n----~cx~ " " f fn[ul2dun--+cxz> f flul2du" (245) Tanaka'stheorem [415] states that whenever f, g are two probability measures with the same mean, and b is normalized by (238), then W(Q+(f, f), Q+(g, g)) <~w(f, g). (246) Tanaka's representation (74) entered the proof of this inequality, which is formally similar to a well-known inequality for rescaled convolution, W(f, f, g, g) <<.W(f, g). Thanks to the Wild formula, Tanaka's theorem implies that W is a nonexpansive (in fact contractive) metric, say when restricted to the set of zero-mean probability measures. On this subject, besides Tanaka's papers one may consult [391]. As a main application, Tanaka proved theorems of convergence to equilibrium for Equation (240) without resorting to the H theorem. In fact, convergence to equilibrium follows almost for free from the contractivity property, since the distance to equilibrium, W(f(t), M), has to be decreasing (unless f is stationary). So one can prove that W(f(t),M) >0 l----~O0 as soon as f0 has just finite energy, not necessarily finite entropy.
- 252. A reviewofmathematicaltopicsincollisionalkinetictheory 251 Since that time, Tanaka's theorem has been largely superseded: more convenient metrics have been found, and entropy methods have become so elaborate as to be able to cover cases where the entropy is infinite.3 Yet, Tanaka's theorem reminds us that Boltzmann's H theorem is not the only possible explanation for convergence to equilibrium. The next sections will confirm this. 2.2. Toscani's distance Since Bobylev's work, it was known that the Fourier transform provides a powerful tool for the study of the spatially homogeneous Boltzmann equation with Maxwellian collision kernel. To measure discrepancies in Fourier space, Toscani introduced the distance d2(f, g) -- sup . (247) ~C]~N I~12 The supremum in (247) is finite as soon as f and g have finite second moments, and the same mean velocity: Nf (V)Vdv - - fi[{Ng(V)V dr, fR (f + g)lvl 2dr < +ec. N Also, convergence in d2 sense is equivalent to convergence in Wasserstein sense (245). It turns out [427] that, under the normalization (238), d2(Q+(f, f), Q+(g, g)) ~ d2(f, g), (248) with equality only if f, g are Maxwellian distributions. As a consequence, d2 is a contracting probability metric along solutions of (239) (when one restricts to probability measures with some given mean). As shown by Toscani and the author [427], this contracting property remains true for Equation (240) with a singular collision kernel. As a main consequence, the Cauchy problem associated with (240) admits at most one solution. This uniqueness theorem holds under optimal assumptions: it only requires finiteness of the energy and of the cross-section for momentum transfer (63). As other applications of the d2 distance, we mention - a simple proof of weak convergence to equilibrium under an assumption of finite energy only; - some partial results for the non-existence of nontrivial eternal solutions [450, Annex II, Appendix]; - some explicit estimates of rate of convergence in the central limit theorem [427], by refinement of the inequality d2(f, f, M) ~<d2 (f, M). 3RecallSection5.4 in Chapter2C.
- 253. _5_ C. Villani 2.3. Other Foltrier-based metrics Other useful Fourier metrics are defined by d~(g, f)- sup , s > 2. (249) They are well-defined only when f, g have the same moments up to high enough order. For instance, one cannot directly compare f to the associated Maxwellian distribution Mf in distance d4 unless f fl)iVjVk dv -- f MfvivjVk dl) for all i, j, k. But this drawback is easily fixed by subtracting from f a well-chosen Taylor polynomial. The interest of using exponents s greater than 2 comes from the fact that the distances ds become "more and more contracting" as s is increased, and this entails better properties of decay to equilibrium. As soon as s > 2, one can prove [225] exponential decay to equilibrium in distance ds, if the initial datum has a finite moment of order s. If one only assumes that the initial datum has a finite moment of order 2, then the method also yields exponential decay in some distance of the form d~(L g) - sup ~RN ISe124~(~ e) for some well-chosen function 4) with 4)(0) = 0. By taking larger values of s, one improves the rate of convergence in ds metric; in particular, the choice s = 4 yields the optimal rate of convergence [128], which is the spectral gap of the linearized Boltzmann operator.4 This exponent 4 is related to the fact that the linearized Boltzmann operator admits Hermite polynomials as eigenfunctions, and the lowest eigenvalues are obtained for 4th-degree spherical Hermite polynomials. Of course, this result of optimal convergence is obtained in quite a weak sense; but, by interpolation, it also yields strong convergence in, say, L 1 sense if one has very strong (uniform in time) smoothness and decay bounds at one's disposal. Such bounds were established in [128], thanks to an inequality which can be seen as reminiscent of the Povzner inequalities, but from the point of view of smoothness, i.e., with moments in Fourier space, instead of velocity spaceS: IIa+<f, f 112 1 H m ~ ~llfll2m -+-Cm, m E N. (250) This inequality holds true at least when f is close enough to M f in relative entropy sense. After establishing the optimal decay to equilibrium in d4 distance on one hand, and the uniform smoothness bound in H m on the other hand, Carlen, Gabetta and Toscani [128] 4Recall that when the collision kernel is Maxwellian, then the spectrum of the linearized operator can be computedexplicitly [79,p. 135]. 5It is not rare in the theory of the spatiallyhomogeneousBoltzmannequation that smoothnessestimates and decay estimates bear a formal resemblance, and this may be explainedby the fact that the Fourier transform of the spatiallyhomogeneousBoltzmannequationis a kindof Boltzmannequation,see Equation (77) ....
- 254. A review of mathematical topics in collisional kinetic theory 253 had no difficulty in interpolating between both partial results to prove convergence to equilibrium in L 1 at exponential rate. The interpolation can be made at the price of an arbitarily small deterioration in the rate of convergence if m is very large. A precise theorem will be given in Section 4.1. REMARK. This theorem of exponential trend to equilibrium with explicit rate is at the moment restricted to Maxwellian collision kernels. This is because only in this case are nice contracting probability metrics known to exist. Also note that the spectral gap of the linearized collision operator is known only for Maxwellian collision kernel. Accordingly, a lower bound on the spectral gap is known only when the collision kernel is bounded below by a Maxwellian collision kernel. The preceding problem of trend to equilibrium takes its roots on the very influen- cial 1965 work by McKean [341]. In this paper, he studied Kac's equation (21), and at the same time proved exponential convergence to equilibrium, with rate about 0.016, sug- gested the central limit theorem for Maxwellian molecules6 and established the decrease of the Fisher information.7 The value 0.016 should be compared to the optimal rate 0.25, which is obtained, up to an arbitrarily small error, in [128]. McKean's results have inspired research in the area until very recently, as the rest of this chapter demonstrates. Another related early work was Grtinbaum [262]. 2.4. The central limit theoremfor Maxwell molecules Once again, let us consider the Boltzmann equation with Maxwellian collision kernel, fix an initial datum f0 with unit mass, zero mean and unit temperature, and denote by M the corresponding Maxwellian. Recall from Section 1 that the solution to the Boltzmann equation with initial datum f0 can be written as the sum of a Wild series, which is a convex combination of iterated Q+ operators acting on f0. Let us be interested in the behavior of the terms of the Wild series as t --+ +c~. It is obvious that the terms of low order have less and less importance as t becomes large, and in the limit the only terms which matter are those which take into account a large number of collisions. But the action of Q+ is to decrease the distance to equilibrium; for instance, one has the inequality d2(Q+(f, g), M) ~<max[d2(f, M),d2(g, M)], as a variant of the inequalities discussed in Section 2.2. Therefore we can expect that terms of high order in the Wild series will be very close to M, and this may be quantified into a statement that f approaches equilibrium as time becomes large. This however is not true for all terms of high order in the Wild series, but only for those terms Q+ (f0) such that the corresponding tree y is deep enough (no leaves of small 6See next section. 7See Section 3.1.
- 255. 254 C.Villani height). Intuitively, small depth means that all particles involved have collided sufficiently many times. For instance, one would expect8 Ion, o+ io, ion), o+(o+ io, ZoO,o+ io, Ion)) to be rather close to M, but not Ion, Zo), Io), Io), Io), Yo), Io) (think that even Q+ (f0, M) is not very close to M ...). Thus, to make the argument work, McKean [341] had to perform an exercise in combinatorics of trees, and show that the combined weight of "deep enough" trees approaches 1 as time becomes large. These ideas were implemented in a very clean, and more or less optimal way, by Carlen, Carvalho and Gabetta [126], thanks to Fourier-defined probability metrics. A precise result will be given in Section 4.1. 3. Information theory 3.1. The Fisherinformation Among the most important objects in information theory are the Shannon entropy and the Fisher information. Up to a change of sign, Shannon's entropy9 is nothing but the Boltzmann H-functional. As for the Fisher information, it is defined as I(f)-- fR IVfl2 fR =4 ]V~[ 2 (251) U f U (compare with the relative Fisher information (174)). The Fisher information is always well-defined in [0, +cx~], be it via the L2 square norm of the distribution V~/-f or by the convexity of the function (x, y) w-~Ixl2/y. It is a convex, isotropic functional, lower semi- continuous for weak and strong topologies in distribution sense. Fisher [216] introduced this object as part of his theory of "efficient statistics". The Fisher information measures the localization of a probability distribution function, in the following sense. Let f(v) be a probability density on R N, and (Xn) a family of independent, identically distributed random variables, with law f(.- 0), where 0 is unknown and should be determined by observation. A statistic is a random variable - 0 (X1 ..... Xn), which is intended to give a "best guess" of 0. In particular, 0 should converge towards 0 with probability 1 as n --+ cx~;and also one often imposes (especially when n is not so large) that 0 be unbiased, which means E0 = 0, independently of n. Now, 8Drawthe correspondingtrees! 9Seethe referencesin Section2.4 of Chapter2A.
- 256. A review of mathematical topics in collisional kinetic theory 255 the Fisher information measures the best possible rate of convergence of 0 towards 0 in the sense of mean quadratic error, as n --+ ~. More explicitly, if 0 is unbiased, then N 2 Var(tg) ~> ~ . (252) nI(f) Inequality (252) is called the Cram6r-Rao inequality, but for an analyst it is essentially a variant of the Heisenberg inequality, which is not surprising since a high Fisher information denotes a function which is very much "localized" .... In fact, the standard Heisenberg inequality in R N can be written VI)0 E ~N, (fo) I (f) f(v)lv - vol2dv ~>N 2 N f dv For given mass and energy, the Fisher information takes its minimum value for Maxwellian distributions -just as the entropy. And for given covariance matrix, it takes its minimum value for Gaussian distributions. This makes it plausible that the Fisher information may be used in problems such as the long-time behavior of solutions to the Boltzmann equation, or the central limit theorem. The idea of an information-theoretical proof of the central limit theorem was first implemented by Linnik [305] in a very confuse, but inspiring paper. His ideas were later put in a clean perspective by Barron [59] and others, see the references in [156]. The same paper by Linnik also inspired McKean [341] and led to the introduction of the Fisher information in kinetic theory. 10 3.2. Staminequalitiesfor the Boltzmann operator As one of the key remarks made by McKean [341 ], the Fisher information is a Lyapunov functional for the Kac model (21). We already mentioned in Section 4.7 of Chapter 2C that his argument can be adapted to the two-dimensional Boltzmann equation. Also Toscani [421 ] gave a direct, different proof of this two-dimensional result. A more general result goes via Stam-type inequalities. The famous Blachman-Stam and Shannon-Stam inequalities 11 [411,75,125] admit as particular cases I (f * f) ~<I (f), H (f * f) ~<H (f). These inequalities are central in information theory [156,165]. Their counterparts for the Boltzmann equation are l(Q+(f, f)) <~I(f), H(Q+(f, f)) <~H(f), (253) where Q+ is defined by (235), and the collision kernel has been normalized by (238). 10McKean used the denomination "Linnik functional" for the Fisher information, which is why this terminology was in use in the kinetic community for some time. 11Inequalities (196) and (195), respectively.
- 257. 256 C. Villani Inequalities (253) immediately entail (by Wild sum representation when (238) holds, by approximation in the general case - or by an ad hoc application of the definition of convexity) that the Fisher information and the H-functional are Lyapunov functionals along the semigroup generated by the Boltzmann equation with Maxwellian collision kernel. In particular, this gives a new proof of the H theorem in this very particular situation. This remark is not so stupid as it may seem, because this proof of the H theorem is robust under time-discretization, and also applies for an explicit Euler scheme le - apparently this is the only situation in which the entropy can be shown to be nonincreasing for an explicit Euler scheme. Inequalities (253) were proven in dimension 2 by Bobylev and Toscani [83], and in arbitrary dimension, but for constant collision kernel (in w-representation), by Carlen and Carvalho [121]. Finally, the general case was proven by the author in [445]. Apart from rather classical ingredients, the proof relied on a new representation formula for V Q+ in the Maxwellian case: V Q+ (f, f) -- -~ dv, dab(k.a)(f~,(I + P~k)(VI)' + f'(I - P~k)(VI)~,), (254) where k = (v - v,)/lv - v,I, IN" ]1~N -'-> ]I~N stands for the identity map and P~k" ]~N R u is the linear mapping defined by P~k(x) -- (k . a)x + (a . x)k - (k . x)a. Formula (254) was obtained by an integration by parts on S u-1 , which crucially used the assumption of Maxwellian collision kernel. In the non-Maxwellian case, we could only obtain an inequality weaker by a factor 2: I(Q+(f, f)) <~ 211AIIL~I(f), where A (z) "- fSN-1 B(Z, a) do'. Just as in the well-known Stam proof, the first inequality in (253) implies the second one via adjoint Ornstein-Uhlenbeck regularizafion. To be explicit, if (St)t>1ostands for the semigroup associated to the Fokker-Planck equation, and if f has unit mass, zero mean, unit temperature, then H(Q+(f, f)) - H(M)= fo +~ [I ( Q+ (St f, St f)) - I (M)] dt. Underlying this formula is of course the commutation between St and Q+, St Q+ (f, f) - Q+ (St f, St f), which follows from Bobylev's lemma, 13 for instance. REMARKS. (1) The decreasing property of the Fisher information also holds for solutions of the Landau equation with Maxwell molecules. This can be seen by asymptotics 12Also one may dream a little bit and imagine that this remark could end up with new lower bounds for the entropy dissipation! 13See Section 4.8 in Chapter 2A.
- 258. A review of mathematical topics in collisional kinetic theory 257 of grazing collisions, but a direct proof is also possible, and gives much better quantitative results [451 ]. In particular, one can prove that the Fisher information converges exponentially fast to its equilibrium value. (2) Still for the Landau equation, this decreasing property was compared with results from numerical simulations by Buet and Cordier [105]. In many situations, they observed a decreasing behavior even for non-Maxwell situations, e.g., Coulomb potential. They also suggested that this decreasing phenomenon was associated with entropic properties of the code: enforcing the decrease of the entropy in the numerical scheme would have a stabilizing effect which prevents the Fisher information to fluctuate. 3.3. Consequence: decreasing of the Carlen-Carvalho !l*functional In [121], Carlen and Carvalho introduced the function :)~ ~ H(f) - H(Szf) to measure the smoothness of the distribution function f, in the context of entropy- entropy dissipation inequalities. Here (St)t>~o is as usual the adjoint Ornstein-Uhlenbeck semigroup, i.e., the semigroup generated by the Fokker-Planck operator (161). Note that if the moments of f are normalized by (177), then ~ is a nonnegative function. One of the main points in the Carlen-Carvalho theorems 14 was to obtain a control of 7r for )~ close to 0. We now claim that for Maxwell collision kernel, thefunction ~ is pointwise nonincreas- ing: for each value of )~, H (f) - H (Sz f) is nonincreasing as a function of time. To see this, recall that in the case of Maxwellian collision kernel, the Boltzmann semigroup (Bt) com- mutes with (St), as a consequence of Bobylev's lemma. 15 Therefore, the time-derivative of n(f) - H(S~f) is D(Sz f) - D(f). To prove that D(Sz f) - D(f) is nonpositive, we just have to prove d --D(S~f) <~O. d)~ But, 16 by commutation property, the dissipation along (St), of the dissipation of entropy along (Bt), is also the dissipation along (Bt), of the dissipation of entropy along (St). Hence, d d~. dI D(S~ f) -- -~ I (Bt f) <~O. X=0 t=0 This proves the claim. 14Recall Section 4.4 in Chapter 2C. 15See Section 4.8 in Chapter 2A. 16We already made this remark in Section 4.4 of Chapter 2C.
- 259. 258 C. Villani 4. Conclusions 4.1. Summary We now summarize most of our discussion about specific properties of the Boltzmann equation with Maxwellian collision kernel in a single theorem. As usual, we set IIf IILs~(RN)-- fRN f(v)(1 -+-IVl2)s/2 dr. THEOREM 24. Let b(cos0) be a nonnegative collision kernel, satisfying finiteness of the cross-section for momentum transfer, f0 rr b(cos0)(1 - cos0) sinN-2 0 dO < +o~, and let fo ~ L~ (11~ N) be an initial datum with finite mass and energy. Without loss of generality, assume that fo has unit mass, zero mean velocity and unit temperature. Then, (i) there exists a unique (weak) solution (f(t))t>o to the spatially homogeneous Boltzmann equation with initial datum f0; (ii) the quantities H(f (t)), I(f (t)), d2(f (t), M), W(f (t), M) are nonincreasing as functions of t; (iii) d2(f (t), M) and W(f (t),M) converge to 0 as t --~ +oo, and also H(f (t)) - H(M) if H(fo) < +cx~; (iv) assume that the collision kernel satisfies Grad's angular cut-off assumption, i.e., f0 rr b(cos0) sinN-2 0 dO < +oo. Let )~ be the spectral gap of the linearized Boltzmann operator. Then, for all e > O, there exists s > 0 and k ~ N such that, if fo ~ L~+s fq H~ (RN), then there exists a constant C < +cx~, explicit and depending on f only via IIf0 IIL~+s and IIf0 IIn~, such that vt 0, IIf(t) - M IIL' < < -Ce-(X-s)t; (v) assume that the collision kernel satisfies Grad's angular cut-off assumption, and is normalized by (238). Consider the Wild representation of f (t), and for any No >~1 let f Uo(t) be the truncation of the series at order No (take formula (241) and throw away all terms starting from the one in e-t (1 -e-t)u~ Define gNo(t) - fNo(t) + (1 --e-t)N~
- 260. A review of mathematical topics in collisional kinetic theory 259 Assume that fo ~ L~+~ A H2+~(R N~ for some 6 > O. Then, there exist constants C < +oo and ot > O, depending on f only via IIf0 IILI+~and IIf IIH2+~, such that VNo~>I, Vt~>O, IIf (t) - gNo (t) IIt 1 ~ C (1 --e-t) N~ NO Ol Also, Ilf(t) - MIIc~ converges exponentially fast to O. Part (i) of this theorem is from Toscani and Villani [427]. As for part (ii), the statement about W2 is due to Tanaka [415], the one about d2 is in Toscani and Villani [427], the one about I is from Villani [445]. Point (iii) is due to Tanaka for W, the same proof applies for d2. Actually Tanaka's proof was given only under additional moment assumptions; for the general result one needs tightness of the energy, which is proven in [225]. The statement about the entropy is more delicate: in addition to the tightness of the energy, it requires the monotonicity property of the function ~, as described in Section 3.3. These two estimates make it possible to use the main result in Carlen and Carvalho [121] and conclude that the relative entropy satisfies a closed differential equation which implies its convergence to 0 at a computable rate. This argument is explicitly written in Carlen, Carvalho and Wennberg [127] in the particular case when the collision kernel is constant in co-representation. Next, point (iv) is the main result of Carlen, Gabetta and Toscani [128], while point (v) is the main result of Carlen, Carvalho and Gabetta [126]. Point (v) can be seen as a bound (essentially optimal) on the error which is performed when replacing the solution of the Boltzmann equation by a truncation of the Wild sum. Note that the result of exponential convergence in (v) is much more general than the one in (iv), but the rate of convergence is a priori worse. We note that the proof of point (iv) uses the last part of (iii). Indeed, the convergence to equilibrium is shown to be exponential only in a certain neighborhood of the equilibrium; 17 to make the constant C explicit it remains to estimate the time needed to enter such a neighborhood, which is what entropy methods are able to do. In conclusion, one can say that the theory of spatially homogeneous Maxwell molecules is by now essentially complete. The links between information theory and kinetic theory have been completely clarified in the last years, this being due in large part to the contributions by Carlen and coworkers. Among the few questions still open, we mention the classification of all nontrivial eternal solutions 18 - which certainly can be attacked more efficiently in the Maxwellian case, thanks to the many additional tools available, as demonstrated by the advances made by Bobylev and Cercignani [81] - and the problems which are mentioned in the next two sections. Also, it would be extremely interesting to know how point (iv) above generalizes to a spatially inhomogeneous setting, even from the formal point of view, and even assuming on the solutions all the smoothness one can dream of. 17Inparticular, because the bound (250) is only proven when f is close enough to M. 18See Section 2.9 in Chapter 2A.
- 261. 260 C. Villani 4.2. A remark on sub-additivity An interesting problem is the classification of all Lyapunov functionals for the Boltzmann equation. Recall the following result by McKean 19 [342]: for the Kac model, the entropy, or H-functional is, up to addition of an affine function or multiplication by a constant, the only Lyapunov functional of the form f A(f). McKean also conjectured that the Fisher information would be the only Lyapunov functional of the form f A(f, Vf). A related problem is to classify all functionals J, say convex and isotropic, which satisfy J(Q+) <~J, in a way similar to (253), under the normalization (238). Such functionals are particular Lyapunov functionals for the Boltzmann equation (240). In dimension 2 of velocity space, Bobylev and Toscani [83] have obtained the following sufficient condition: for all probability distributions f and g on R2, and for all )~e [0, 1], J(f~ * gl-~.) ~<)~J(f) + (1 - )OJ(g). (255) Whatever the dimension, this criterion is satisfied by all functionals that we have encountered so far: H, I, d2(., M), W(., M) 2. However, nobody knows if it is sufficient in dimension higher than 2. As a consequence of our remarks on the Landau equation with Maxwellian collision kernel [443] and the asymptotics of grazing collisions, any Lyapunov functional J has to satisfy (255) in the particular case when f is radially symmetric and g is the Maxwellian distribution with zero mean, and same energy as f. 4.3. Remark: McKean's conjectures In his seminal 1965 work [341], McKean also formulated several conjectures. Even though they all seem to be false, they have triggered interesting developments. Let us mention two of these conjectures. The "super-H theorem" postulates that the entropy is a completely monotone function of time: dH/dt <~O, (d2H)/(dt 2) >/0.... (-1)n(dnH)/(dt n) >/O. For some time this was a popular subject among a certain group of physicists. This conjecture is however false, as shown by Lieb [303] with a very simple argument. Strangely, for the particular Bobylev-Krook-Wu explicit solutions, this "theorem" holds true for n ~<101 and breaks downs afterwards [361]. The "McKean conjecture", strictly speaking. Let Ms stand for the Maxwellian distribution with zero mean and 3 temperature, and consider the formal expansion n(f , Ms) -- - Z n! n=O (256) 19This result is somewhat reminiscent of the axiomatic characterization of entropy by Shannon, see, for instance, [156, pp. 42-43] and references therein.
- 262. A review of mathematical topics in collisional kinetic theory 261 so that Io(f) -- -H(f), ll(f) -- I(f), I2(f) -- - ~ij f f[Oij(l~ 2, etc. Knowing that dlo/dt >~0 and dI1/dt <<.O, McKean conjectured the more general inequality (-1)ndln/dt >~O. This conjecture seems to be false in view of the formal study realized by Ledoux [295] for the Fokker-Planck equation. Keeping in mind that the entropy measures volume in infinite dimension, the successive terms in (256) could be seen as infinite-dimensional analogues of the mixed volumes arising in convex geometry. It is not even clear that they have alternate signs for n >~ 1 .... However, this conjecture has inspired a few works in kinetic theory, see, for instance, Gabetta [224], or the discussion of the Kac model in Toscani and Villani [428, Section 7]. These ideas have also been used by Lions and Toscani [323] to establish certain strengthened variants of the central limit theorem.
- 263. CHAPTER 2E Open Problems and New Trends Contents 1. Open problems in classical collisional kinetic theory ............................ 265 1.1. Strong solutions in a spatially inhomogeneous setting ......................... 265 1.2. Derivation issues ............................................. 266 1.3. Role of the kinetic singularity ...................................... 266 1.4. Improved entropy-entropy dissipation estimates ............................ 268 1.5. Approach to equilibrium for Kac's master equation .......................... 269 1.6. Influence of the space variable on the equilibration rate ........................ 272 2. Granular media ................................................. 272 2.1. Derivation issues: problems of separation of scales .......................... 273 2.2. Spatial inhomogeneities ......................................... 276 2.3. Trend to equilibrium ........................................... 277 2.4. Homogeneous Cooling States ...................................... 278 3. Quantum kinetic theory ............................................ 279 3.1. Derivation issues ............................................. 281 3.2. Trend 1o equilibrium ........................................... 283 3.3. Condensation in finite time ....................................... 285 3.4. Spatial inhomogeneities ......................................... 285 263
- 264. Preface The motion of fluids has intrigued scientists since antiquity and we may say that the field of mathematical fluid dynamics originated more than two centuries ago. In 1755 Euler [2] gave a mathematical formulation of the principle of conservation of mass in terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear and angular momentum by PDEs. Material symmetry and frame invariance were used by Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint- Venant [11] and Stokes [12]. In the 19th century no sharp distinction was drawn between mathematicians and physicists as we sometime see in more recent times. The formulation of the equations of fluid motion could be considered as either mathematics or physics. The first work in fluid dynamics that has a "modern" mathematical flavor may have been done by Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous problem. Riemann recognized the mathematical nature of the entropy. This notion led him to his duality method for solving the non-characteristic Cauchy problem for linear hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his contemporaries. What we now call the Cauchy problem for a PDE and the search for its solution did not have the significance that it is accorded nowadays. Even Poincar6 did not raise that kind of question in his Th~orie des tourbillons. For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28. Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much as Riemann, Leray developed new mathematical tools which proved to have independent interest: e.g., weak solutions (that we now call Leray's solutions in this context) and topological degree (a joint work with Schauder [4]). Since the 1930s, the interest that mathematicians devote to fluid dynamics has unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic displays such a variety of models and questions that thousands of scientists, among them many mathematicians, focus their research on fluid dynamics.
- 265. A review of mathematical topics in collisional kinetic theory 265 The goal of this chapter is to present some of the main open problems in collisional kinetic theory, then to discuss some of the new questions arising in two developing branches of the field: the study of granular media on one hand, quantum kinetic theory on the other. Other choices could have included semiconductors (whose modelling is very important for industrial applications), modelling of biological interactions (in which problems have not been very clearly identified up to now), the study of aerosols and sprays (which naturally involve the coupling of kinetic equations with fluid mechanics), etc. Also, we only discuss problems associated with the qualitative behavior of solutions, and do not come back on less traditional issues like those which were presented in Section 2.9 of Chapter 2A. Selecting "important" problems is always dangerous because of subjectivity of the matter, and changes in mathematical trends and fashions. To illustrate this, let us quote Kac himself [283, p. 178, footnote 5]: "Since the master equation 1 is truly descriptive of the physical situation, and since existence and uniqueness of the solutions of the master equation are almost trivial, the preoccupation with existence and uniqueness theorems for the Boltzmann equation appears to be unjustified on grounds of physical interest and importance." 1. Open problems in classical collisional kinetic theory 1.1. Strong solutions in a spatially inhomogeneous setting The theory of the Cauchy problem for Boltzmann-like equations is by now fairly advanced under the assumption of spatial homogeneity. For instance, in the case of hard potentials, it seems reasonable to expect that this theory will soon be completed with the help of already existing tools. On the other hand, essentially nothing is known concerning the general, spatially inhomogeneous case in a non-perturbative context. Progress is badly needed on the following issues: - moment estimates, - regularity estimates (propagation of regularity/singularity, regularization), - strict positivity and lower bounds. This lack of a priori estimates is a limiting factor in many branches of the field. A priori estimates would enable one to 9 prove uniqueness of solutions, and energy conservation; 9 perform a simple treatment of boundary conditions (walls, etc.); 9 give estimates of speed of convergence to equilibrium along the lines presented in Chapter 2C (for this one needs uniform estimates as times goes to +cx~); 9 justify the linearization procedure which is at the basis of so many practical applications of kinetic theory, see, for instance, [148]. Such a priori estimates also would be very useful, even if not conclusive, to 9 prove the validity of the Landau approximation in plasma physics, viewed as a large- time correction to the Vlasov-Poisson equation; 1SeeSection1.5below.
- 266. 266 C. Villani 9 prove the validity of the fluid approximation to the Boltzmann equation. For this, local conservation laws seem to be the minimum one can ask for in order to prove the hydrodynamic limit.2 Even in the more modest framework of solutions in the small, where smooth solutions can often be built, many gaps remain, like the treatment of singular collision kernels, or the derivation of uniform smoothness estimates as t --+ +o0. 1.2. Derivation issues As already mentioned, Lanford's theorem is limited to a perturbative framework (small solutions), and concerns only the hard-sphere interaction. The treatment of more general interactions is almost completely open, and would be of considerable interest. As an oustanding problem in the field is of course the formidable task of rigorously deriving collisional kinetic equations for Coulomb interaction. As for the problem of extending Lanford's theorem to a nonperturbative setting, one of the main difficulties is certainly related to the fact that there is no good theory for the Cauchy problem in the large - but we expect much, much more obstacles to overcome here! 1.3. Role of the kinetic singularity Let us consider a Boltzmann collision kernel, say of the form B(v - v,, tr) = t/,(lv - v,[) b(cosO). In Chapter 2B we have seen how the properties of the Boltzmann equation depend on whether b is integrable or singular. On the other hand, what remains unclear even in the spatially homogeneous case, is the influence of the kinetic collision kernel q~. When is singular, does it induce blow-up effects, and in which sense? Does it help or harm regularizing effects induced by an angular singularity? The most important motivation for this problem comes from the modelling of Coulomb collisions in plasma physics: the collision kernel given by the Rutherford formula presents a singularity like Iv - v.1-3 in dimension N = 3. Here are two questions which arise naturally. (1) Consider the Boltzmann equation with truncated Rutherford collision kernel, of the form Iv- v,l-3b(cosO)10~>~, which is physically unrealistic but used in certain modelling papers [162]. This collision kernel presents a nonintegrable (borderline) kinetic singularity. Does it entail that the equation induces smoothness, or even just compactifying effects? Some formal arguments given in Alexandre and Villani [12] may support a positive answer, but the situation seems very intricate. As explained in [12, Section 5], the geometry of the 2Thisis whatthe authorbelieved,untilveryrecentlysomeproofsofhydrodynamiclimitappeared[54,53,240, 245], coveringsituationsin whichlocalconservationlawsare not knownto holdfor fixedKnudsennumber,but are asymptoticallyrecoveredin the limitwhenthe Knudsennumbergoesto 0!!
- 267. A review of mathematical topics in collisional kinetic theory 267 problem is "dual", in some sense, to the one which appears in the study of nonintegrable, borderline angular singularities .... If the answer is negative, this suggests that such collision kernels should be used with a lot of care! (2) On the other hand, consider the Landau approximation for Coulomb collisions, or Landau-Coulomb equation, which is more realistic from the physical point of view. Does this equation have smooth solutions? In the spatially homogeneous situation, the Landau-Coulomb equation can be rewritten as nrv2j -+- 87rf 2 t >~O, V6 R 3 Of Ot -- ~ {lij OyiOy--~" ' ' ij (257) where 1 I l)iVJ1 {tij --~ -~[ ~ij ivl2 * f. If f is smooth, then the matrix ({lij) is locally positive definite, but bounded, and (257) is reminiscent of the nonlinear heat equation 0f = Af + f2 (258) at which has been the object of a lot of studies [468] and generically blows up in finite time, say in L~ norm. The common view about (258) is that the diffusive effects of the Laplace operator are too weak to compensate for blow-up effects induced by the quadratic source term. And if the diffusion matrix ({lij) is bounded, this suggests that (257) is no more diffusive than (258). Weak solutions to (257) have been built in Villani [446]; they satisfy the a priori estimate v/-f 6 L2 (nvl) locally. This estimate is however, to the best of the knowledge of the author, compatible with known a priori estimates for (258). These considerations may suggest that blow-up in finite time may occur for solutions of (257). If there is blow-up, then other questions will arise: how good is the Landau approximation at a blow-up time? What happens to blow-up if the physical scales are such that the Landau effects should only be felt as t --+ +cx~? However, blow-up has never been reported by numerical analysts. And after seeing some numerical simulations by E Filbet, the author has changed his mind on the subject, to become convinced that blow-up should indeed not occur. All this calls for a wide clarification.
- 268. 268 C. Villani 1.4. Improved entropy-entropy dissipation estimates In this section we only consider very nice distribution functions, say smooth and rapidly decaying, bounded below by a fixed Maxwellian. We saw in Chapter 2C that such probability distributions satisfy entropy-entropy dissipation inequalities of the form D(f) >~K H (flMf) ~, where M f is the Maxwellian equilibrium associated with f, K and ot are positive constants, H is the relative entropy functional, and D is the entropy dissipation for either Boltzmann or Landau's equation. In several places do our results call for improvement: Landau equation with hardpotentials. In the case of the Landau equation, ot = 1 is admissible when qJ (Izl) ~ Izl2" ot -- 1 + e (e arbitrarily small) is admissible when qJ(Izl) ~ Izl2+y, y < 0. It is natural to conjecture that also c~= 1 be admissible for hard potentials (y > 0). More generally, this should be true when !P(Izl) = Izl2~p(Izl) with 7~ continuous and uniformly positive for Izl/> ~ > 0. This conjecture is backed by the spectral analysis of the linearized Landau operator [161], and also by the similar situation appearing in Carrillo, McCann and Villani [130] in the study of entropy-entropy dissipation inequalities for variants of granular media models. At the moment, the best available exponent for hard potentials is ct = 1 -+-2/y, from Desvillettes and Villani [183]. Boltzmann equation with hard potentials. In the case of the Boltzmann equation, c~ = 1 + e is admissible for Maxwellian or soft potentials. It is accordingly natural to think that c~-- 1 + e is also admissible3 for hard potentials. Recall that Cercignani's conjecture (or - 1) is false in most cases according to Bobylev and Cercignani [87]. Cercignani's conjecture revisited? Counterexamples in [87] leave room for Cercignani's conjecture to hold true in two situations of interest: 9 when the collision kernel is noncut-off and presents an angular singularity. This would be plausible since grazing collisions behave better with respect to large velocities, as the example of the Landau equation demonstrates4; 9 when f ~ LP ((M f)- 1) for some p ~> 1. Of special interest are the cases p = 1 (cf. Bobylev's estimate for hard spheres, in Theorem 1(ii)); p = 2 (natural space for linearization) and p --- ~ (when f/M f is bounded from above). Maybe a Maxwellian bound from below is also needed for proving such theorems. As we mentioned when discussing Cercignani's conjecture in Chapter 2C, about this topic one also has to make the connection with the recent Ball and Barthe result about the central limit theorem. 3As this review goes to print, the author just managed to prove precisely this result, under the assumption that the density be bounded in all Sobolev spaces. 4Similar results in the theory of linear Markov processes would also be interesting.
- 269. A review of mathematical topics in collisional kinetic" theory 269 1.5. Approach to equilibrium for Kac's master equation A related topic is the Kac spectral gap problem and its entropy dissipation variant. This subject is a little bit in digression with respect to those which we discussed so far, but we wish to explain it briefly because of its intimate (and not well-known) connections with Cercignani's conjecture. These connections were brought to our attention by E. Carlen. In his famous paper [283], Kac introduced a stochastic model which he believed to be a way of understanding the spatially homogeneous Boltzmann equation. His equation models the behavior of n particles interacting through binary elastic collisions occurring at random Poissonnian times, with collision parameter cr randomly chosen on the sphere. It reads Ot 1 dcr B(vi- 1)j, 0")[,/4/o. j fn -- fn], (259) where the summation runs over all pairs of distinct indices (i, j) in {1..... n}, and fn is a symmetric probability distribution on the manifold (actually a sphere) of codimension N + 1 in (RN) n defined by the relations /7 /7 Z Ivi12 -- 2nE > O, ~ IJi -- ng E R N. i=1 i=1 We use the notation f for the normalized integral on the sphere, ISN-1 [-1 f. Moreover the linear operator ,A~ represents the result of the collision of the spheres with indices i and j, ioJ ! ! .fit fn(Vl ..... Vn)- f (Vl ..... Vi..... Vj ..... Vn), I Ui -'[- l)j Iui -- vjl vi= 2 -+-~cr, t l)i + Vj IIJi -- Uj[ vj -- 2 ~ c r . As explained by Kac, the spatially homogeneous Boltzmann equation can be recovered, at least formally, as the equation governing the evolution of the one-particle marginal of fn in the limit n --+ +oo. In this limit, time has to be sped up by a factor n. See [283,412,256] for a study of this and related subjects. A simplified version, which is commonly called Kac's master equation, is given by Ot - - Lnfn -- -~ o dO [fn o R~ - fn], (260) where fn is a probability distribution on the sphere in ]t{ n , defined by ZvZ=2nE. (261)
- 270. 270 C. Villani Moreover, Rb v--(v, ..... ..... v,), where ! (v~, vj) = RO(1)i, Vj) is obtained from (1)i, Vj) by a rotation of angle 0 in the (i, j) plane. Without loss of generality, we set E = 1/2 in (261), so that the sphere has radius ~/-ff.With this choice, the image measure of the uniform probability measure on the sphere, under projection onto some axis of coordinate, becomes the standard Gaussian measure as n --+ c~ (Poincarr's lemma, actually due to Maxwell). Moreover, we shall use the uniform probability measure on the sphere as reference measure, so that probability distributions are normalized by f fnda- 1. ~/-~Sn-1 Among the problems discussed by Kac is that of establishing an asymptotically sharp lower bound on the spectral gap )~n of Ln as n --+ +cx~. Recently, Diaconis and Saloff- Coste [189] proved )~n1 = O(n3), then Janvresse [281] proved Kac's conjecture that )~n1 = O(n); she used Yau's so-called martingale method. Finally, a complete solution was given very recently by Carlen, Carvalho and Loss [123], who managed to compute the spectral gap by a quite unexpected method (also by induction on the dimension). This work also extends to Equation (259) if the collision kernel B is Maxwellian. Since time should be sped up by a factor n in the limit n ~ ~, the corresponding evolution equation will satisfy estimates like 11A(t,.)- IIfn(0, .)- lllt2(4~sn-1 ) (262) for some )~ > 0, which can be chosen uniform as n --+ cx~according to Janvresse's theorem. Here 1 is the equilibrium state, i.e., the density of the uniform probability measure on the sphere v/-nSn-1 . Inequality (262) conveys a feeling of uniform trend to equilibrium as n --+ cx~,which was Kac's goal. However, it is not very clear in which sense (262) is a uniform estimate. Since all the functions fn'S are defined on different spaces, one should be careful in comparing them. In particular, think that if fn satisfies the chaos property, then Ilfn IlL2 is roughly of order C n for some constant C > 0 (which in general is not related to the L 2 (or L2(M-I)) norm of the limit one-particle marginal f, see [283, Equation (6.44)]). And Ilfn - 11122= Ilfn 1122- 1 is also of order C~. Having this in mind, it would be natural to 1In compare distances in dimension n by the quantity II 9 IIL2 9But if we do so, we find 1/n ~ 1/n , LZ(sn-1),
- 271. A review of mathematical topics in collisional kinetic theory 271 which does not behave well in the limits ! A way to circumvent the difficulty would be to compare all first marginals, which all live in L 1(R), and prove that under some precise conditions on the sequence (fn), ElX > O, Vn ~ 1, IIP, A(t, .)- MIIL= M_,> Ce-xt. Now, a problem which looks more natural and more interesting in this context is the problem of the entropy-entropy dissipation estimate for Kac's master equation. Again, we state this problem assuming without loss of generality that E -- 1/2, so that ~ v2 -- n in (261) and we use the uniform probability measure as reference measure for the definition of the entropy: H(fn) - f s.-1 f nlog f ndo'. Note that H (1) -- 0. PROBLEM. Find Kn optimal such that for all symmetric probability distribution fn on ~/-fiS~-l, logfn ~> KnH(fn). (263) If Kn 1 = O(n), then (263) entails the following entropy estimate for solutions of the Kac equation: H(fn(t, ")) - H(1) ~<e-Ut[H(fn(O, .))- H(1)], for some # > 0. Since H (fn) typically is O(n), this would lead to the satisfactory estimate H(fn(t"))- H(1) <~e-ut I H(fn(O"))- H(1) I n n ' (264) and also one-particle marginals of all fn'S could be compared easily as a consequence an adequate chaos assumption for fn (0, .). But from the counterexamples due to Bobylev and Cercignani [87], one expects that Kn 1 --O(n) is impossible. Indeed, by passing to the limit as n --+ cx) in (263), under a chaos assumption, one would have a proof of Cercignani's entropy dissipation conjecture for Kac's model, which should be false (although this has never been checked explicitly) .... On the other hand, the author [450, Annex III, Appendix B] was able to prove Kn 1 -- O(n 2) by the same method as in Section 4.6 of Chapter 2C. This leads to two open questions: 9 What is the optimal estimate? 5Or shouldthe relevantscalingbe fn ~ (1+h/V/if)| whichwouldmeanthatwe are interestedin fluctuations of the equilibriumstate?This ansatzformallyleads to Ilfn- 11122= O(n) ....
- 272. 272 C. Villani 9 Does an estimate like O(n) hold for a well-chosen sub-class of probability distribu- tions? 1.6. Influence of the space variable on the equilibration rate Let us now consider trend to equilibrium in a spatially inhomogeneous context. Diffusive models. The strategy of Desvillettes and Villani, exposed in Chapter 2C, shows trend to equilibrium like O(t -~) for many entropy-dissipating systems when good smoothness a priori estimates are known. However, an exponential rate of convergence would be expected, at least for the linear Fokker-Planck, or Landau equation. The following issues would be of particular interest: 9 admitting that the solution of the linear Fokker-Planck equation with confinement potential V goes to equilibrium in relative entropy like O(e-at), what is the optimal value of ot and how does it depend on V?* Can one obtain this result by an entropy method? 9 admitting that the solution of the Landau equation in a box (periodic, or with appropriate boundary condition) goes to equilibrium in relative entropy like O(e-~ what is the optimal value of ot and how does it depend on the boundary condition or the size of the box? Can one devise an entropy method to obtain exponential decay? Boltzmann equation. When Boltzmann's equation with a Maxwellian collision kernel is considered, then one can compute the spectral gap of the linearized operator. In the spatially homogeneous case, as we have seen in Chapter 2D, this spectral gap essentially governs the rate of decay to equilibrium, even in a non-linearized setting. At the moment, entropy methods seem unable to predict such a result,6 but a clever use of contracting probability metrics saves the game. Now, how does all this adapt to a spatially inhomogeneous context and how is the rate of convergence affected by the box, boundary conditions, etc.? Of course, in a preliminary investigation one could take for granted all the a priori bounds that one may imagine: smoothness, decay, positivity .... This concludes our survey of open problems for the classical theory of the Boltzmann equation. As the reader has seen, even if the field is about seventy years old, a lot remains to be done! Now we shall turn to many other problems, arising in less classical contexts which have become the object of extensive studies only recently. 2. Granular media Over the last years, due to industrial application and to the evolution of the trends in theoretical physics, a lot of attention was given to the modelling of granular material (sand, powders, heaps of cereals, grains, molecules, snow, or even asteroids...). The literature on *Noteaddedinproof: Onthisproblemseetherecentprogressby HrrauandNier. 6Becauseofthe obstructionto Cercignani'sconjecture....
- 273. A review of mathematical topics in collisional kinetic theory 273 the subject has grown so fast that some journals are now entirely devoted to it! And also the number of involved physicists has become extremely large. Among the main motivations are the understanding of how granular material behaves under shaking, how flows are evolving or how to prevent them, how to facilitate mixing, how to prevent violent blow-up of a silo, or avalanches, how matter aggregates in a newborn solar system, etc. One popular model for these studies is a kinetic description of a system of particles interacting like hard spheres, but with some energy loss due to friction. Friction is a universal feature of granular material because of the roughness of the surface of particles. Many studies have been based on variants of the Boltzmann or Enskog equations which allow energy loss. This subject leads to huge difficulties in the modelling; see, for instance, the nice review done by Cercignani [146] a few years ago. Also some mathematical contributions have started to develop, most notably the works by Pulvirenti and collaborators [70-72,68,69,395]. Here we point out some of the most fundamental mathematical issues in the field. Since the physical literature is considerable, we only give a very restricted choice of physicists' contributions. Thanks are due to E. Caglioti for explaining us a lot about the subject and providing references. 2.1. Derivation issues: problems of separation of scales The separation between microscopic and macroscopic scales in the study of granular media is not at all so clear as in the classical situation, and this results in many problems when it comes to derivation of the relevant equations. To illustrate this, we mention the astonishing numerical experiments described in [237]: a gas of inelastic particles is enclosed in a one-dimensional box with specular reflection (elastic wall) on one end, Maxwellian re-emission (heating wall) on the other. It is found that, basically, just one particle keeps all the energy, while all the others remain slow and stay close to the elastic wall. In other words, the wall is unable to heat the gas, and moreover it is plainly impossible to define meaningful macroscopic quantities! Enskog equation. At the basis of the derivation of the Boltzmann equation, be it formal or more rigorous, is the localization of collisions: the length scale for interaction is much smaller than the length scale for spatial fluctuations of density. The Boltzmann-Grad limit n --+ cx~, r --+ O, nr 2 --+ 1 (n = number of particles, r = radius of particles, dimension = 3) is a way to formalize this for a gas of elastic hard spheres. On the other hand, in the case of granular media, the size of particles is generally not negligible in front of the typical spatial length. This is why many researchers use an Enskog-like equation, with delocalized collisions: for instance, ~f Ot +v. Vxf=r2f•3 dv*fs2 dcr Iv - v*l X [J G(x,x + rcr; p)f(t,x, f))f(t,x - rcr, fJ,) - G(x,x - rcr; p)f(t,x, v)f(t,x - rcr, v,)]. (265)
- 274. 274 C. Villani (See Cercignani [146] and Bobylev et al. [86].) Here r > 0 is the radius of particles, v, v, are post-collisional velocities and ~, fi, are pre-collisional velocities, given by the formulas _ v+ v, 1 -e 1 +e v .... ~2-- --~-e (v- v,) + 4e Iv- v, la, _ v+v, 1-e(v_v,)_ l+e v,= ~ ~- 4e 4e Iv-v, la. (266) Moreover e is an inelasticity parameter: when e- 1 the preceding equations are the usual equations of elastic collisions, while the case e = 0 correspond to sticky particles. In general e may depend on Iv - v,I but we shall take it constant to simplify. Finally, to complete the explanation of (265), J is the Jacobian associated to the transformation (266), 1 Iv- o,I j = _ ~ (267) e2 Iv- v,l' and G is the famous but rather mysterious correlationfunction which appears in the Enskog equation. Roughly speaking, G relates the 2-particle probability density with the 1-particle probability density, as follows: f2(t, x, v, y, w) = G(x, y; p(t, x), p(t, y)) f (t, x, v) f (t, y, w), where p(t, x) -- f f dr. In the Boltzmann case this term did not appear because of the chaos assumption .... Exactly which function G should be used is not clear and a little bit controversial: see Cercignani [146] for a discussion. If the reader finds the complexity of (265) rather frightening, we should add that we did not take into account variables of internal rotation, which may possibly be important in some situations since the particles are not perfectly spherical; see [146] for the corresponding modifications. There are no clear justifications for Equation (265), even at a formal level. In the limit of rarefied gases with a chaos assumption, or in the spatially homogeneous case, one formally recovers an inelastic Boltzmann equation, which is more simple: the collision operator just reads, with obvious notations, s f, -- Qe(f, f) - dr, dcr Iv - v,l(f f, - f f,). 3 2 (268) This collision operator also has a nice weak formulation, f Qe(f, f)q)= s ff,[q)'- q)]dvdv, dcr, (269) 3 x ]]~3x S 2
- 275. A review of mathematical topics in collisional kinetic theory 275 (with respect to v, v, taken as pre-collisional where the post-collisional velocities v~, v, velocities) are V + 1 l+e v' -- v, - e(v _ v,) + Iv - v, la, ------2--+ 4 4 v + v, 1- e.v v,) - l + e v,- 2 ------4-- 4 Iv- v, lcr. (270) As in the elastic case, formula (269) can be symmetrized once more by exchange of v and v,. Note that fi, fi, do not coincide with vf, v,f because collisions are not reversible. Also note that f Qe(f, f) dv - O, f Qe (f, f) v dv - O, but f Qe(f, f)lvl 2dr ~<0. Now let us enumerate some more models. Starting from (268), many variants can be obtained by 9 reducing the dimension of phase space by considering only 2-dimensional, or even 1-dimensional models; 9 adding a little bit of diffusion, which is presumably realistic in most situations (heat bath, shaking...); 9 adding some drift term 0Vv. (f v), to model some linear friction acting on the system; 9 change the "hard-sphere-like" collision kernel Iv - v,I for Iv - v,I • with, say -1 ~< y ~< 1. This provides an equation with some inelastic features, and which may be easier to study .... The dimensional homogeneity of the equation can be preserved by multiplying the collision operator by a suitable power of the temperature. For instance, Bobylev, Carrillo and Gamba [86] have performed a very detailed study of the case y = 0 ("pseudo-Maxwellian collision kernel") along the general lines of the theory developed by Bobylev for elastic Maxwellian collisions; 9 only retain grazing collisions by an asymptotic procedure similar to the Landau approximation7 [424,426]. Actually, some physicists mention that grazing collisions do occur very frequently in granular material [236] .... The operator which pops out of this limit procedure is of the form QL(f, f) + Vv. [fVv(f * U)], where U(z) is proportional to Izl3 (or more generally to Izly+2), and QL is an elastic Landau operator with ~(Izl) proportional to Izl3 (or Izl• In particular, 7 See Section2.7 in Chapter2A.
- 276. 276 C. Villani in dimension 1 of phase space, this elastic Landau operator vanishes and the resulting collision operator is just a nonlinear friction operator Vv . [fVv(f , W)]. The resulting evolution equation is Of 0---[+ v . Vx f = Vv . [fVv(f * W)], (271) with U (z) = Izl3. The very same equation can also be obtained as the mean-field limit of a one-dimensional system of particles colliding inelastically, as first suggested (so it seems) by McNamara and Young [343], and rigorously proven by Benedetto, Caglioti and Pulvirenti [70]. Note that (271) is a Boltzmann-like, not an Enskog-like equation! Of course, we have only presented some of the most mathematically oriented models. In the physical literature one encounters dozens of other equations, derived from physical or phenomenological principles, which we do not try to review. Hydrodynamics. Another topic where the separation of scales is problematic is the hydrodynamic limit. This subject is important for practical applications, but nobody really knows, even at a formal level, what hydrodynamic equations should be used. Due to the possibility of long-range correlations, there generally seems to be no clear separation of scales between the kinetic and hydrodynamic regimes [236]. The Chapman- Enskog expansion works terribly bad: each term of the series should be of order 1! Resummation methods have been tried to get some meaningful fluid equations [236]. From the mathematical viewpoint this procedure is rather esoteric, which suggests to look for alternative methods .... To add to the confusion, due to the tendency of granular media to cluster, it is not clear what should be considered as local equilibrium! See the discussion about Homogeneous Cooling States in Section 2.4 below. Also, to be honest we should add that even in the most favorable situation, i.e., a simplified model like (271), with some additional diffusion term to prevent clustering as much as possible, and under the assumption of separation of scales, then limit hydrodynamic equations can be written formally [69], but rigorous justification is also an open problem, mainly due to the absence of Lyapunov functional .... 2.2. Spatial inhomogeneities In the classical, elastic theory of the Boltzmann equation, spatial homogeneity is a mathematical ad hoc assumption. However, it is not so unrealistic from the physical viewpoint in the sense that it is supposed to be a stable property: weakly inhomogeneous initial data should lead to weakly inhomogeneous solutions (recall [32]). On the other hand, in the case of granular media, some physicists think that severe inhomogeneities may develop from weakly inhomogeneous states, particularly because
- 277. A review of mathematical topics in collisional kinetic theory 277 of the possibility of collapse by loss of energy [198]. Also, numerical experiments seem to indicate the unstability of a homogeneous description (see the references and comments in [70, Section 4]). Besides shedding more doubts on mathematical studies based on the assumption of spatial homogeneity, these remarks raise a very interesting mathematical challenge, namely prove that weak inhomogeneity may break down in finite time for some realistic initial configurations. It is not very clear whether this study should be performed with a Boltzmann-like, or an Enskog-like equation .... Some related considerations about blow-up: first of all, there are initial configurations of n inelastic particles which lead to collapse in finite time, and they are not exceptional (some nice examples are due to Benedetto and Caglioti [67]). Next, for the inelastic Boltzmann equation, there is no entropy functional to prevent dramatic collapse, as in the elastic case. Even the DiPema-Lions theory cannot be adapted to inelastic Boltzmann equations, so the Cauchy problem in the large is completely open in this case! We further note that Benedetto, Caglioti and Pulvirenti prove that there is no blow-up for Equation (271) when W(z) = )~lzl3 with )~ very small; in this case Equation (271) can be treated as a perturbation of vacuum. Of course it is precisely when )~ is of order 1 that one could expect blow-up effects. Also, Benedetto and Pulvirenti [73] study the one- dimensional Boltzmann equation for a gas of inelastic particles with a velocity-dependent inelasticity parameter e = e(lv - v.I) behaving like (1 + air - v.l• -1 for some a, y > 0, and they show, by an adaptation of the one-dimensional techniques of Bony [95], that blow-up does not occur in that situation. 2.3. Trend to equilibrium For the moment, and in spite of our remarks in the previous paragraph, we restrict to the spatially homogeneous setting (or say that we are only interested in trend to local equilibrium, in some loose sense). We consider two cases: (1) the inelastic Boltzmann collision operator alone; then, because of energy loss, the equilibrium is a Dirac mass at some mean velocity; (2) the inelastic Boltzmann collision operator together with some diffusion. Then there is a nontrivial, smooth stationary state (see Gamba, Panferov and Villani [228]). It is not explicit but some qualitative features can be studied about smoothness, tails, etc. The same topic has also been recently studied by Carrillo and Illner in the case of the pseudo- Maxwellian collision kernel. In both cases entropy methods do not seem to apply because no relevant Lyapunov functional has been identified (apart from the energy). What is worse, in the second case, even uniqueness of the stationary state is an open problem. The study of trend to equilibrium is therefore open. However, such a study was successfully performed for two simplified models: - in the pseudo-Maxwellian variant of the inelastic Boltzmann operator, with or without diffusion [86,80]. Then the behavior of all moments can be computed, and trend to equilibrium can be studied from the relaxation of moments;
- 278. 278 C. Villani - in the simplified model considered by Benedetto, Caglioti and Pulvirenti [70], or its diffusive variant. In this case, the appearance of a new Lyapunov functional, with an interaction energy 1 [ Iv - wl 3 - JR f (v)f (w) 2 • 3 dvdw enables a study of rate of convergence by entropy dissipation methods. 8 In the spatially inhomogeneous case, the situation is still worse. In the non-diffusive case, all states of the form p~(x)6o(v), for instance, are global equilibria. But are there some preferred profiles p~ ? Even for the simplified Equation (271) or its diffusive variants, the Lyapunov functional which worked fine in the spatially homogeneous context, now fails to have any particular behavior. At a more technical level, the method of Desvillettes and Villani [184] cannot be applied because of the non-smoothness of the equilibrium distribution. On the whole, trend to equilibrium for granular media is a really challenging problem. 2.4. Homogeneous Cooling States For most physicists, a Dirac mass is not a relevant steady state, and the role played in the classical theory by Maxwellian distributions should rather be held by particular solutions which "attract" all other solutions. 9 They often agree to look for these particular solutions in the self-similar form 1 (v-vo) Ru fs(t, v) = ~ N (t----~ F or(t) ' t )O, v 9 . (272) In the sequel, we set v0 = 0, i.e., restrict the discussion to centered probability distributions. A solution of the spatially homogeneous inelastic Boltzmann equation which takes the form (272) is called a Homogeneous Cooling State (HCS). Though the existence of HCS is often taken for granted, it is in general a considerable act of faith. Sometimes HCS are considered under some scaling where also the elasticity coefficient e goes to 1 as t --+ +cx~ .... For physical studies of these questions one can consult [104,238]. For the pseudo-Maxwellian variant of the inelastic Boltzmann operator, Bobylev, Carrillo and Gamba [86] have shown that HCS do not exist: one can construct a self-similar distribution function which captures the behavior of all moments of all solutions, but this distribution function is not nonnegativeI HCS exist only in the following weakened sense: for any integer no, there exists a self-similar solution of the inelastic Boltzmann equation which gives the fight behavior for all moments of order ~<no of all solutions. And yet, this 8See the discussionin Section6.3 of Chapter 2C. 9By the way, in the classical setting it was once conjectured that the Bobylev-Krook-Wu explicit solutions would attract all solutions of the spatially homogeneousBoltzmann equation with Maxwellian molecules. But this has been shownto be false ... except,in somesense,in the unphysicalregimeof negativetimes [81]!
- 279. A review of mathematical topics in collisional kinetic theory 279 weaker statement is still false for some particular values of the inelasticity parameter e. See [86] for more details, in particular a discussion of the stability of the HCS description. For model (271), in a spatially homogeneous setting, i.e., ~f = Vv. [fVv(f 9 U)], (273) Ot with U(z) -Izl3/3, then HCS do exist and are just made of a combination of two Dirac masses: up to some change of scales, 113 -+-3~]. f s(t, v)- -~ ~, z, This solution is obtained via the search for steady states to the rescaled equation ~f = Vv. (fVv(f * W)) - Vv. (fv). (274) Ot One finds that a distinguished steady state is Fs = (3 1 + 31)/2. 2 2 It is also true [70] that this HCS is a better approximation to solutions of (273), than just the Dirac mass (at least if the initial datum has no singular part). This means that solutions of (274) do converge towards the steady state Fs. But this approximation is in general quite bad! It was shown by Caglioti and Villani [116] that the improvement in the rate of convergence is essentially no better than logarithmic in time. For instance, if W stands for the Wasserstein distance (244), then solutions of (273) satisfy f+~ W(f(t), fs(t))dt = +cx~. Since also W(f(t),3o)= O(1/t), this shows that the improvement in the rate of convergence cannot be O(log l+e t), for any e > 0 - which means negligible by usual standards, l~ The preceding considerations cast further doubts about the mathematical relevance of HCS, which however are at the basis of several hydrodynamical equations for granular media. Further clarification is still badly needed; an attempt is done in [86] for the simplified pseudo-Maxwellian model. This concludes our brief discussion of the kinetic theory of granular media. In the sequel, we shall enter a completely different physical world. 3. Quantum kinetic theory Very recently, Lu [328], Escobedo, and Mischler [210,211] have begun to apply the techniques of the modem theory of the spatially homogeneous Boltzmann equation, to l~ maybeshouldoneuse anevenweakernotionofdistanceto measurethe rateofconvergence??
- 280. 280 C. Villani quantum kinetic models, thus opening up the path to a promising new direction of research. This is part of a general trend which has become increasingly active over last years: the mathematical derivation and study of quantum statistical models. This also coincides with a time when the interest of physicists in Bose condensation is enhanced by the possibility of experiments with very cold atoms. Most of the explanations which follow come from discussions with Escobedo and Mischler, and also from Lu's paper [328]. First of all, we should clarify the meaning of quantum kinetic theory: it does not rest on the traditional quantum formalism (wave function, Wigner transforms, etc.). Instead, it is rather a classical description of interacting particles with quantum features. This approach was initiated by the physicists Nordheim, and Uehling and Uhlenbeck, in the thirties. Thus, the basic equation still looks just like a Boltzmann equation: ~f Ot + v(p). Vxf- Q(f, f), t ~>0, x elR 3 pER 3 (275) Here p stands for the impulsion of the particle and v is the corresponding velocity: v(p) = Vp E (p), where E (p) is the energy corresponding to the impulsion p. The unknown is a time-dependent probability density on the phase space of positions and impulsions. When dealing with massive particles, we shall consider a non-relativistic setting (to simplify) and identify v with p. On the other hand, when dealing with photons, which have no mass, we assume that the energy is proportional to IPl, and the velocity to p/lpl. Now, all the quantum features are encoded at the level of the collision operator Q in (275). One traditionally considers three types of particles: , fermions, which satisfy Pauli's exclusion principle. In this case the transition from state p~ to state p is easier if f (p) is low. Accordingly, the "Boltzmann-Fermi" collision operator reads QF(f, f)- fR3 dp, fs2 do- B(v-v,, o-)[f'fJ(1 + ef)(1 +ef,) - ff,(1 + sf')(1 + el,)], (276) where e is a negative constant, ll Up to change of units, we shall assume that e = -1. Moreover, as in the classical case, ft= f(pl) and so on, and pl, p~ are given by the formulas , P+P. IP- P,] p = + ~ o . , 2 2 I P+P. IP- P.I P*= 2 2 o." (277) llln physicalunits,e shouldbe -(h/m)3/g, whereh is Planck'sconstant,m themassofaparticleand g the so- called"statisticalweight"ofthisspeciesofparticles.Forthederivationof (276)see Chapmanand Cowling[154, Chapter 17].
- 281. A review of mathematical topics in collisional kinetic theory 281 Equation (275) with Q = Q F will be called the Boltzmann-Fermi equation. It is supplemented with the a priori bound O~f~l [=-l/el; , bosons, which, on the contrary to fermions, do like to cluster. The collision operator, Q8, is just the same as (276), but now e = + 1. The corresponding equation will be called the Boltzmann-Bose equation; , photons, which are mass-free particles exchanging energy. Usually they are considered only in interaction with bosons or fermions. For instance, here is the Boltzmann-Compton model: Qc(f, f) = fo~176 k2 + f) e-~ - f( k'2 + g')e-~:'] dk', t~O,k~O. (278) Here the phase space is just R+, the space of energies, because the distribution of photons is assumed to be spatially homogeneous and isotropic. Thus the corresponding evolution equation is just af = Qc(f, f), t ~ O, k ~ O. (279) 0t We quote from Escobedo and Mischler [211]: Equation (279) describes the behavior of a low-energy, spatially homogeneous, isotropic photon gas interacting with a low- temperature electron gas with Maxwellian distribution of velocities, via Compton scat- tering. This model will be called Boltzmann-Compton. We now survey some of the main problems in the field. 3.1. Derivation issues The derivation of equations like Boltzmann-Fermi or Boltzmann-Bose is not a tidy business (see Chapman and Cowling [154] ...). Therefore, the expected range of applicability and the precise form of the equations are not so clear. A better understood situation is the linear setting: description of the effect of a lattice of quantum scatterers on a density of quantum particles. Not only is the exact equation well understood, but also a theoretical basis, in the spirit of Lanford's theorem, can be given. Starting from the many-body Schrtidinger equation as microscopic equation, Erd6s and Yau [205,206] have been able to retrieve the expected linear Boltzmann equation as a macroscopic description. Related works are performed in [132-134] .... In the sequel we do not consider these issues and restrict the discussion to the nonlinear equations written above. Here are a few problematic issues about them.
- 282. 282 c. Villani Cross-sections. It seems, nobody really knows what precise form of the cross-section, or equivalently of the collision kernel B(v - v,, or) in (275) (or b(k, U) in (278)) should be used - except in some particular cases with photon interaction .... Some formulas can be found in [154] but they are not very explicit. This makes it difficult to give an interpretation of some of the mathematical results, as we shall see. It would be desirable to identify some model collision kernels playing the same role as the ones associated with inverse-power interactions in the classical theory. According to certain physicists, it would be not so bad to understand the case of a simple hard-sphere collision kernel. Grazing collisions. Some variants of these equations are obtained by a grazing collision asymptotics. To this class belong the quantum Landau equation (see Lemou [298] and references therein), or the well-known Kompaneets equationl2 [290], O fOlk20 f ]OF Ot = 0--k ~ -4-(k2 - 2k)f + f2 =- Ok' t >~O, k >>. O. (280) A flux condition must be added at the boundary: lim F(k) = 0. (281) k--+0 Equation (280) describes the same kind of phenomena as the Boltzmann-Compton equation, and can in fact be obtained from it by an asymptotic procedure similar to the one leading from the Boltzmann to the Landau equation (see Escobedo and Mischler [211]) under some assumptions on the initial datum. However, the validity of this approximation cannot be universally true, because the Kompaneets equation has some strange "blow-up" properties: Escobedo, Herrero and Velazquez [208] have shown that the flux condition (281) may break down in finite time for arbitrarily small initial data. Also the long-time behavior of the Kompaneets equation can be non-conventional; this is consistent with the remark by Caflisch and Levermore [114] that for large enough mass there are no stationary states .... Besides physical interest, all these considerations illustrate the fact that the asymptotics of grazing collisions may destroy (or create?) some important features of the models. Consistency with classical mechanics. All these quantum models involve the Planck constant as a parameter; of course when one lets the Planck constant go to 0 (which would in fact be the formal consequence of a change in physical scales, from microscopic to macroscopic), one expects to recover the Boltzmann-like equations of classical mechanics. This can be justified in some cases, see, for instance, [196]. Hydrodynamics. Physicists expect that some hydrodynamic limit of the Boltzmann-Bose equation leads to the Gross-Pitaevski, or Ginzburg-Landau, model (based on a cubic nonlinear Schr6dinger equation) for the evolution of the Bose condensate 13 part. Again, 12Thisequationis oftenwrittenwithk2f as unknown. 13Seethe nextsection.
- 283. A reviewof mathematical topics in collisionalkinetic theory 283 this would need clarification .... We note however that the justification of this limit would look more interesting if the derivation of the Boltzmann-Bose equation was first put on a more rigorous basis. 3.2. Trend to equilibrium Equations such as Boltzmann-Fermi, Boltzmann-Bose or Boltzmann-Compton all satisfy entropy principles, and equilibrium states are entropy minimizers 14: (1) For the Boltzmann-Fermi equation, the entropy is HBF(f)- f[flogf- (1 - f) log(1 - f)]. Equilibrium states are of the form 1 .T'(p) = (or > 0, fl E R) (282) eulP-po I2+fi -n t- 1 or ~'(p) = llp_pol~ R (R > 0). A state like (282) is called a Fermi-Dirac distribution. Here P0 is the mean impulsion. (2) For the Boltzmann-Bose equation, the entropy is HBB(f) = f[flogf- (1 4- f)log(1 4- f)] (here s = + 1) and the shape of equilibrium states depends on the temperature. There is a critical condensation temperature Tc such that the equilibrium state/3 takes the form 1 -- (or > 0, /3 >~0) when T >~Tc, (283) 13(p) e~IP-Po12+fl -- 1 1 - + #6po (or > 0, # > 0) when T < Tc. (284) B(p) e~ -po12 -- 1 These distributions are called Bose-Einstein distributions. The singular part of (284) is called a Bose condensate. (3) Finally, for the Boltzmann-Compton equation, the entropy is given by f0 ~ Hsc(f)- [(k e + f)log(k e + f)- f log f- kf- k e log(k2)] dk, 14Underthe constraint0 ~<f ~<1for the Boltzmann-Fermimodel.
- 284. 284 C. Villani and, according to Caflisch and Levermore [114], the minimizers are of the form 1 B(k) = ek+Z _ 1 + c~60, where )~ and c~ are nonnegative numbers, at least one of them being 0. For )~ > 0 this is a Bose distribution, for ~. = 0 it is called a Planck distribution. As in the Boltzmann case, these distributions, obtained by a minimization principle, also coincide with the probability distributions which make the collision operator vanish. There are some technicalities associated with the fact that singular measures should be allowed: they have recently been clarified independently by Escobedo and Mischler, and by Lu. Now, let us consider the problem of convergence to equilibrium, in a spatially homogeneous setting for simplicity. As far as soft methods (compactness and so on) are concerned, Pauli's exclusion principle facilitates things a great deal because of the additional L ~ bound. Therefore, convergence to equilibrium in a (very) weak sense is not very difficult [211]. However, no constructive result in this direction has ever been obtained, neither has any entropy-entropy dissipation inequality been established. In the Bose case, this is an even more challenging problem since also soft methods fail, due to the lack of a priori bounds. The entropy is now sublinear and fails to prevent concentration, which is consistent with the fact that condensation may occur in the long- time limit. Actually, as soon as T < Tc, a given solution cannot stay within a weakly compact set of L 1 as t --+ +e~z .... Lu [328] has attacked this problem with the well- developed tools of modern spatially homogeneous theory, and proven that - when the temperature is very large (T >> Tc), solutions of the spatially homogeneous Boltzmann-Bose equation are weakly compact in L 1 as t --+ +cxz, and converge weakly towards a Bose distribution of the form (283); - when the temperature is very low (T < Tc), solutions are not weakly compact in L 1, but converge to equilibrium in the following extremely weak sense [328]: if (tn) is a sequence of times going to infinity, then from f (tn, ") one can extract a subsequence converging in biting-weak L 1 sense towards a Bose distribution of the form (284). In this theorem, not only is biting-weak L 1 convergence a very weak notion (weaker than distributional convergence), but also the limit may depend on the sequence (t,). Furthermore, it is not known whether weak L 1 compactness as t ~ oo holds true when T is greater than Tc, but not so large. Lu's theorem is proven for isotropic homogeneous solutions. Isotropy should not be a serious restriction, but seems compulsory to the present proof. What is more, Lu's work relies on a strong cut-offassumption for the kernel B" essentially, 8(iv - v,i, coso) co(lo - o,i ,, Io - o,I), c > o, (285) where 0 is as usual the deviation angle. This assumption enables a very good control of the Q+ part, but may do some harm for other, yet to be found, a priori estimates.
- 285. A review of mathematical topics in collisional kinetic theory 285 3.3. Condensation infinite time Physical experiments with very cold atoms have recently become possible, and have aroused a lot of interest. For instance, a few years ago it became possible to experimentally create and study Bose condensates. Among other phenomena, physicists report the formation of a condensate in finite time. However, Lu has proven that there is no finite time clustering for the Boltzmann-Bose equation studied in [328]. This seems to leave room for two possibilities, both of which may lead to exciting new research directions: 9 either the Boltzmann-Bose equation should be discarded for a more precise model when trying to model Bose condensation; 9 or the Bose condensation is excluded by the strong cut-off assumption (285), which penalizes interactions with v _~ v. (supposedly very important in condensation effects). On this occasion we strongly feel the need to have a better idea of what collision kernels would be physically realistic. Proving the possibility (or genericity) of finite-time condensation for "bigger" collision kernels (say B = 1?) would be a mathematical and physical breakthrough for the theory of the Boltzmann-Bose model. 3.4. Spatial inhomogeneities So far we have only considered spatially homogeneous quantum Boltzmann equations, now what happens for spatially inhomogeneous data? Due to the additional L ~ bound, the Boltzmann-Fermi model seems easier to study than the classical Boltzmann equation; in particular existence results can be obtained without too much difficulty [196,309]. The situation is completely different for the Boltzmann-Bose model, since one would like to consider singular measures as possible data. A completely new mathematical theory would have to be built! A particularly exciting problem would be the understanding of the space- time evolution for the Bose condensate. It was communicated to us by Lu that for small initial data in the whole of R 3, one can prove that Bose-Einstein condensation never occurs .... This should be taken as a clue that the underlying mathematical phenomena are very subtle.
- 286. 286 C. Villani Bibliographical notes General references. Standard references about the kinetic theory of rarefied gases and the Boltzmann equation are the books by Boltzmann [93], Carleman [119], Chapman and Cowling [154], Uhlenbeck and Ford [433], Truesdell and Muncaster [430], Cercig- nani [141,148], Cercignani, Illner and Pulvirenti [149], as well as the survey paper by Grad [250]. The book by Cercignani et al., with a very much mathematically oriented spirit, may be the best mathematical reference for nonspecialists. The book by Uhlenbeck and Ford is a bit outdated, but a pleasure to read. There is no up-to-date treatise which would cover the huge progress accomplished in the theory of the Boltzmann equation over the last ten years. For people interested in more applied topics, and practical aspects of modelling by the Boltzmann equation, Cercignani [148] is highly recommended. We may also suggest the very recent book by Sone [407], which is closer to numerical simulations. The book by Glassey [233] is a good reference for the general subject of the Cauchy problem in kinetic theory (in particular for the Vlasov-Poisson and Vlasov- Maxwell equations, and for the Boltzmann equation near equilibrium). Also the notes by Bouchut [96] provide a compact introduction to the basic tools of modem kinetic theory, like characteristics and velocity-averaging lemmas, with applications. To the best of our knowledge, there is no mathematically-oriented exposition of the kinetic theory of plasma physics. Among physicists' textbooks, Balescu [46] certainly has the most rigorous presentation. The very clear survey by Decoster [160] gives an accurate view of theoretical problems arising nowadays in applied plasma physics. There are many, many general references about equilibrium and non-equilibrium statistical physics; for instance, [49,227]. People who would like to know more about information theory are advised to read the marvelous book by Cover and Thomas [156]. A well-written and rather complete survey about logarithmic Sobolev inequalities and their links with information theory is [16] (in french). Historical references. The founding papers of modem kinetic theory were those of James Clerk Maxwell [335,336] and Ludwig Boltzmann [92]. It is very impressive to read Maxwell's paper [335] and see how he made up all computations from scratch! The book [93] by Boltzmann has been a milestone in kinetic theory. References about the controversy between Boltzmann and his peers can be found in [149, p. 61], or Lebowitz [293]. Some very nice historical anecdotes can also be found in Balian [49]. Certainly the two mathematicians who have most contributed to transform the study of the Boltzmann equation into a mathematical field are Torsten Carleman in the thirties, and Harold Grad after the Second World War. Derivation of the Boltzmann equation. For this subject the best reference is certainly Cercignani, Illner and Pulvirenti [149, Chapters 2 and 4]. A pedagogical discussion of slightly simplified problems is performed in Pulvirenti [394]. Another excellent source is the classical treatise by Spohn [410] about large particle systems. These authors explain in detail why reversible microdynamics and irreversible macrodynamics are not
- 287. A review of mathematical topics in collisional kinetic theory 287 contradictory- a topic which was first developed in the famous work by Ehrenfest and Ehrenfest [202], and later in the delightful book by Kac [284]. Further information on the derivation of macroscopic dynamics from microscopic equations can be found in Kipnis and Landim [287]. Hydrodynamic limits. A very nice review of rigorous results about the transition from kinetic to hydrodynamic models is Golse [239]. No prerequisite in either kinetic theory, or fluid mechanics is assumed from the reader. Note the discussion about ghost effects, which is also performed in Sone's book [407]. The important advances which were accomplished very recently by several teams, in particular, Golse and co-workers, were reviewed by the author in [441 ]. There is a huge probabilistic literature devoted to the subject of hydrodynamic limits for particle systems, starting from a vast program suggested by Morrey [352]. Entropy methods were introduced into this field at the end of the eighties, see in particular the founding works by Guo, Papanicolaou and Varadhan (the GPV method, [263]), and Yau [466]. For a review on the results and methods, see the notes by Varadhan [439], the recent survey by Yau [467] or again, the book by Kipnis and Landim [287]. Mathematical landmarks. Here are some of the most influential works in the mathemati- cal theory of the Boltzmann equation. The very first mathematical steps are due to Carleman [118,119] in the thirties. Not only was Carleman the very first one to state and solve mathematical problems about the Boltzmann equation (Cauchy problem, H theorem, trend to equilibrium), but he was also very daring in his use of tools from pure mathematics of the time. In the seventies, the remarkable work by Lanford [292] showed that the Boltzmann equation could be rigorously derived from the laws of reversible mechanics, along the lines first suggested by Grad [249]. This ended up a very old controversy and opened new areas in the study of large particle systems. Yet much remains to be understood in the Boltzmann-Grad limit. At the end of the eighties, the classical paper by DiPerna and Lions [192] set up new standards of mathematical level and dared to attack the problem of solutions in the large for the full Boltzmann equation, which to this date has still received no satisfactory answer. A synthetical review of this work can be found in G6rard [231 ]. Finally, we also mention the papers by McKean [341] in the mid-sixties, and Carlen and Carvalho [121] in the early nineties, for their introduction of information theory in the field, and the enormous influence that they had on research about the trend to equilibrium for the Boltzmann equation. Acknowledgements It is a pleasure to thank D. Serre for his suggestion to write up this review, and E. Ghys for his thorough reading of the first version of the manuscript. I also warmly thank L. Arkeryd, E. Caglioti, C. Cercignani, L. Desvillettes, X. Lu, F. Malrieu, N. Masmoudi, M. Pulvirenti, G. Toscani for providing remarks and corrections, and J.-E Coulombel for his job of
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- 306. CHAPTER 3 Viscous and/or Heat Conducting Compressible Fluids Eduard Feireisl* Institute of Mathematics AV CR, Zitnd 25, 115 67 Praha 1, Czech Republic Contents 1. Basic equations of mathematical fluid dynamics .............................. 309 1.1. Balance laws .............................................. 309 1.2. Constitutive relations ......................................... 309 1.3. Barotropic models ........................................... 311 1.4. Boundary conditions ......................................... 311 1.5. Bibliographical comments ...................................... 313 2. Mathematical aspects of the problem .................................... 313 2.1. Global existence for small and smooth data ............................. 315 2.2. Global existence of discontinuous solutions ............................. 317 2.3. Global existence in critical spaces .................................. 318 2.4. Regularity vs. blow-up ........................................ 319 2.5. Large data existence results ...................................... 319 2.6. Bibliographical comments ...................................... 321 3. The continuity equation and renormalized solutions ............................ 321 3.1. On continuity of the renormalized solutions ............................. 322 3.2. Renormalized and weak solutions .................................. 323 3.3. Renormalized solutions on domains with boundary ........................ 324 4. Weak convergence results .......................................... 325 4.1. Weak compactness of bounded solutions to the continuity equation ................ 326 4.2. On compactness of solutions to the equations of motion ...................... 328 4.3. On the effective viscous flux and its properties ........................... 330 4.4. Bibliographical remarks ....................................... 332 5. Mathematical theory of barotropic flows .................................. 333 5.1. Energy estimates ........................................... 336 5.2. Pressure estimates for isentropic flows ................................ 336 5.3. Density oscillations for barotropic flows ............................... 338 5.4. Propagation of oscillations ...................................... 340 5.5. Approximate solutions ........................................ 343 6. Barotropic flows: large data existence results ............................... 344 *Work supported by Grant 201/98/1450 of GA (~R. HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME I Edited by S.J. Friedlander and D. Serre 9 2002 Elsevier Science B.V. All rights reserved 307
- 307. 308 E. Feireisl 6.1. Global existence of classical solutions ................................ 344 6.2. Global existence of weak solutions ................................. 345 6.3. Time-periodic solutions ........................................ 347 6.4. Counter-examples to global existence ................................ 348 6.5. Possible generalization ........................................ 348 7. Barotropic flows: asymptotic properties .................................. 349 7.1. Bounded absorbing balls and stationary solutions ......................... 349 7.2. Complete bounded trajectories .................................... 350 7.3. Potential flows ............................................. 353 7.4. Highly oscillating external forces .................................. 355 7.5. Attractors ............................................... 356 7.6. Bibliographical remarks ....................................... 357 8. Compressible-incompressible limits .................................... 357 8.1. The spatially periodic case ...................................... 358 8.2. Dirichlet boundary conditions .................................... 359 8.3. The case ~'n ~ c~ ........................................... 359 9. Other topics, directions, alternative models ................................ 361 9.1. Models in one space dimension ................................... 361 9.2. Multi-dimensional diffusion waves ................................. 361 9.3. Energy decay of solutions on unbounded domains ......................... 363 9.4. Alternative models .......................................... 364 10. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 10.1. Local existence and uniqueness, small data results ......................... 365 10.2. Density estimates ........................................... 365 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
- 308. Viscous and~orheat conducting compressiblefluids 309 1. Basic equations of mathematical fluid dynamics 1.1. Balance laws Let S2 C R N be a domain in two- or three-dimensional space (N = 2, 3) filled with a fluid. We shall assume the fluid is a continuous medium the state of which at a time t 6 I C R and a spatial point x E I2 is characterized by the three fundamental macroscopic quantities; the density 0 = O(t,x), the velocity u = u(t, x), and the temperature 0 = O(t,x). The fluid motion is governed by a system of partial differential equations expressing the basic principles of classical continuum mechanics. Conservation of mass: O0 Ot + div(0u) = 0 (1.1) Balance of momentum (Newton's second law of motion): O(ou) Ot + div(ou | u) + Vp = div Z + of (1.2) Conservation of energy (the first law of thermodynamics): OE Ot + div((E + p)u) - div(2Su) - div q + of. u (1.3) Here p is the pressure, 27 denotes the viscous stress tensor, E stands for the specific energy, q is the heatflux, and f denotes a given external force density. We have chosen the spatial description where attention is focused on the present config- uration of the fluid and the region of physical space currently occupied. This description was introduced by d'Alembert and is usually called Eulerian in hydrodynamics. There is an alternative way - the referential description - introduced in the eighteenth century by Euler that is called Lagrangean. In this description the Cartesian coordinate X of the posi- tion of the particle at the time t -- to is used as label for the particle X (see, e.g., Truesdell and Rajagopal [105]). 1.2. Constitutive relations The general system (1.1)-(1.3)of N + 2 equations must be complemented by constitutive relations reflecting the diversity of materials in nature. An important class of fluids that occupies a central place in fluid mechanics is the linearly viscous or Newtonian fluid, whose viscous stress tensor I7 takes the form 57 -- Z(Vu) -- #(Vu + (Vu) t) + Adivuld,
- 309. 310 E. Feireisl where # and ~ are the viscosity coefficients assumed to be constant unless otherwise specified. The full stress characterized by the Cauchy stress tensor T is related to 27 by the Stokes law T=Z-pId, where the pressure p = P(O, O) is a general function of the independent state variables O and 0. The specific energy E can be written in the form E = Ekinetic + Einternal, 1 Ekinetic = =Olu] 2, ginternal = oe, z where e is the specific internal energy related to the density and the temperature by a general constitutive law e = e(Q, 0). In accordance with the basic principles of thermodynamics, we postulate the existence of a new state variable- the specific entropy S = S(O, O) - satisfying OS_ lOe OS=_l(Oe p) 00 -- 0 00' 0Q 0 00 O2 9 (1.4) Consequently, one can replace (1.3) by the entropy equation q) 27(Vu) 9Vu Ot(QS) + div QSu + -~ - 0 q. V0 02 (1.5) where, by virtue of the second law of thermodynamics, the right-hand side should be non- negative which yields the restriction 2 ~z~>0, z+~tt~>o together with q. VO ~ O. We focus on viscous fluids assuming always 2 /z > 0, )~+ -:-:# ~>0. (1.6) PC Finally, the heat flux q is related to the temperature by the Fourier law q = -x VO , tc >~O, where the heat conduction coefficient K may depend on 0, O and even on V0 though it is assumed constant in most of the cases we shall deal with.
- 310. Viscous and~orheat conducting compressible fluids 311 1.3. Barotropic models The flow is said to be barotropic if the pressure p depends solely on the density 0. There are several situations when such a hypothesis seems appropriate. For instance, the ideal gas consitutive relation for the pressure reads p= (y -1)oe, e =cvO, cv > O, (1.7) where ~, > 1 is the adiabatic constant. Accordingly, the entropy S takes the form S = log(e) + (1 - y) log(o). Substituting # = )~= x = 0 in (1.5) and assuming a spatially homogeneous distribution So of the entropy at a time to E I we easily deduce S(t) = So for any t E I and, consequently, P(o) =ao • a=(g-1)exp(So)>O. Under such circumstances, Equations (1.1), (1.2) represent a closed system describing the motion of an isentropic compressible viscous fluid. A similar situation occurs in the isothermal case when we suppose O(t) -- 00 and (1.7) reduces to P(O) = rOoo, r > O. For a general barotropic flow, the specific energy E can be taken in the form 1 E[ O, u]- ~O]u] 2 + P(O) (1.8) z with e' (z)z - e(z) = p(z). The energy of a barotropic flow satisfies the equality OtE + div((E + p)u) -- div(Z:u) - r" Vu + of. u (1.9) which is now a direct consequence of (1.1), (1.2). From the mathematical point of view, the barotropic flows represent an interesting class of problems for which an existence theory with basically no restriction on the size of data is available (see Section 6 below). 1.4. Boundary conditions To obtain mathematically well-posed problems, the equations introduced above must be supplemented by initial and/or boundary conditions. The boundary 012 is assumed to
- 311. 312 E. Feireisl be an impermeable rigid wall, i.e., the fluid does not cross the boundary but may move tangentially to the boundary. Accordingly, we require u.n=O onlxO~, (1.10) where n stands for the outer normal vector. For both experimental and mathematical reasons, (1.10) should be accompanied by a condition for the tangential component of the velocity. As observed in experiments with viscous fluids, the tangential component approaches zero at the boundary to a high degree of precision. This can be expressed by the no-slip boundary conditions: u=0 on I x OS2. (1.11) On the other hand, in vessels with frictionless boundary (cf. Ebin [25]), condition (1.10) is usually complemented by the requirement that the tangetial component of the normal stress is zero, which can be written in the form of the no-stick boundary conditions: u.n=0, (rn) xn=0 onlx0~2. (1.12) Similarly, one prescribes either the heat flux or the temperature. For a thermally insulated boundary, the condition reads q.n=O on/xOl2 while O= Ob on/xOS2 when the boundary distribution of the temperature is known. If S-2is unbounded, it is customary to prescribe also the limit values of the state variables for large x 9 S2, e.g., ~ 0~, u--+ u~, 0 ~ 0~ as Ix l ~ ~. In the in-flow and/or out-flow problems, the homogeneous Dirichlet boundary condi- tions (1.11) are replaced by a more general stipulation U=Ub onlzOI2. Moreover, the density distribution must be given on the in-flow part of the boundary, i.e., Q(t, x) -- Qb(X) for (t, x) 9 I x 012, Ub(X) 9n(x) < 0. Other types of boundary conditions including unilateral constraints and free boundary problems are treated in the monograph by Antontsev et al. [4, Chapters 1, 3].
- 312. Viscousand~orheatconductingcompressiblefluids 313 1.5. Bibliographical comments An elementary introduction to the mathematical theory of fluid mechanics can be found in the book by Chorin and Marsden [12]. More extensive material is available in the monographs by Batchelor [7], Meyer [77], Serrin [93], or Shapiro [94]. A more recent treatment including the so-called alternative models is presented by Truesdell and Rajagopal [105]. A rigorous mathematical justification of various models of viscous heat conducting fluids is given by Silhav~ [96]. Mainly mathematical aspects of the problem are discussed by Antontsev et al. [4], M~ilek et al. [68], and more recently by Lions [61,62]. 2. Mathematical aspects of the problem The first and most important criterion of applicability of any mathematical model is its well-posedness. According to Hadamard, this issue comprises a thorough discussion of the following topics. 9 Existence of solutions for given data. The data for the problem in question are usually the values of the state variables Q, u, and 0 specified at a given time t = to and/or the driving force f together with the boundary values of certain quantities as the case may be. The problem is whether or not there exist solutions for any choice of the data on a given time interval I. 9 Uniqueness. The model is to be deterministic, specifically, the time evolution of the system for t > to must be uniquely determined by its state at the time to. 9 Stability. Small perturbations of the data should result in small variation of the corresponding solution at least on a given compact time interval. On the other hand, experience with much simpler systems of ordinary differential equations suggests that chaotic behaviour may develop with growing time. Roughly speaking, the solutions may behave in a drastically different way in the long run no matter how close they might have been initially. To begin, let us say honestly that a rigorous answer to most of the issues mentioned above is very far from being complete. Global existence and uniqueness of solutions to the system (1.1)-( 1.3) is still a major open problem and only partial results shed some light on the amazing complexity of the problem. In this introductory section, we review the presently available results on the existence of classical as well as weak or distributional solutions to the full system of equations of a compressible Newtonian and heat-conducting fluid. Equations (1.1)-(1.3) written in Cartesian coordinates take the form OkO O(~OU j ) t =0; at OXj O(OUi) O(ouJu i) Ot OXj Op Oxi -[--~) aXj /~] -[- Ofi' i--1 ..... N; (2.1) (2.2)
- 313. 314 E. Feireisl O(oO) + Cv Ot -~-~ OXj " (2.3) Here and always in what follows, the summation convention is used. The term classical solution means that the state variables have as many derivatives as necessary to give meaning to (2.1)-(2.3) on S2 • I and are continuous up to the boundary 0S2 to satisfy the boundary conditions as the case may be. Usually, we make no distinction between classical and strong solutions whose generalized derivatives are locally integrable functions and satisfy the equations almost everywhere in the sense of the Lebesgue measure. Typically, any strong solution is a classical one provided some additional smoothness of the data is assumed. Multiplying (2.1)-(2.3) by a compactly supported and smooth test function 9 and integrating the resulting expressions by parts, we get the integral identities: ffs~ 09 09 dxdt - 0; (2.4) 0-57 + Ouj Oxj fl fl2 " 09 Oui 09 OUi 09 uj 09 + P -- lZ~~ --~ + OUt OXj ~ OXj OXj OU j O~ -- ()~ + lZ)OXj OXi ~-Ofi9 dx dt - 0, i = 1 ..... N; (2.5) y,fo(ooo r Cv -~ -l-oOuJ ) -- p )9 -- K~ ~ OXj OXj O0 09 OXj OXj + -~ -~xk + ~xj 9 + )~ Oxj ,] 9dxdt-0. (2.6) We shall say that Equations (2.1)-(2.3) hold in D I(I • 12) (in the sense of distributions) or, equivalently, that 0, u, and 0 is a weak solution of the problem if the integral identities (2.4)-(2.6) hold for any test function 9 6 D(I • S2). The symbol D(Q) denotes the space of infinitely differentiable functions with compact support in an open set Q. It is not difficult to observe that the local formulation (2.1)-(2.3) and the integral formulation (2.4)-(2.6) are in fact equivalent provided the solution is smooth enough. On the other hand, (2.4)-(2.6) make sense under much weaker assumptions, namely, when the quantities 0, 0ui, 0uiuj, P, OUi/OXj,of i, 00, oOuj, pOuJ/Oxj, O0/OXi, IOui/Oxj[ 2, i, j = 1..... N are locally integrable on I • S2. We shall always tacitly suppose that this is the case whenever speaking about weak solutions.
- 314. Viscous and/or heat conducting compressible fluids 315 2.1. Global existence for small and smooth data Following the pioneering work of Matsumura and Nishida [72] we consider the system (2.1)-(2.3) where ~z = ~z(o, 0), z = z(o,0) are smooth functions satisfying # > O, )~+ 2/3/z ~>O; (2.7) the pressure p is given by a general constitutive relation Op Op -- > O; (2.8) p--p(o,O) and p, 00' O0 x=x(o,O), x>O. (2.9) In addition and in accordance with (1.4), we suppose Op(o,o) p(o,O) = ~ 0 . (2.10) O0 The problem (2.1)-(2.3) is complemented by the Dirichlet boundary conditions uilos2=O, i=1,2,3, 010s~=0b, (2.11) where 0b > 0; and the initial conditions 0(0, x) = Oo(x)> O, Ui (0, X) -- Uio(X), i -- 1, 2, 3, O(O, x) = Oo(x), x ~ S2. (2.12) The following global existence theorem holds. THEOREM 2.1. Let I2 C R 3 be a domain with compact and smooth boundary. Let the quantities #, )~, p, and tc comply with the hypotheses (2.7)-(2.10). Moreover, let the i O0 belong to the Sobolev space W3'2(I-2) and satisfy the compatibility initial data 00, uo, conditions i _ O, (90 -- Ob, Uo O p(oo, 0o) Oxi 0 OUio~ 0 ()~(00 00) -[-"lZ(OO, 00)) OXj ~]~] -- OOfi -- OXj l's 00) OXj /] -[- ~Xi '
- 315. 316 E. Feireisl OuJo p(oo, Oo) 00) x(oo, Oo) + OXj ~Xj i =1,2,3, 2 ~-Txk+ Ox-Txj +)~(0~ 0~ ~xj ' on the boundary 0I-2. Finally, let fi __ OF/Oxi, i = 1, 2, 3 where F belongs to the Sobolev space W5'2 (S2). Then there exists e > 0 such that the initial-boundary value problem (2.1)-(2.3), (2.11), (2.12) posseses a unique solution O, u, 0 on the time interval t ~ (0, oo) provided the initial data satisfy I100- 0 IIw3,2(s-2)+ Iluollw3,2(s2)+ I100-0bllw3,2(s2) + IIFIIws,2(x2)< s, where 0 -- -~1 oodx. Theorem 2.1 in its present form is taken over from Matsumura and Nishida [73] (cf. also [72]). The proof, which is rather lengthy and technical, is based on a priori estimates resulting from energy relations. Their method has been subsequently adapted by many authors to attack various problems with non-homogeneous boundary conditions (cf., e.g., Valli and Zajaczkowski [109]) as well as the barotropic models (see Valli [108]). The common feature of all these results is that they apply only to problems where the data are small and regular. The solutions the existence of which is claimed in Theorem 2.1 belong to the space o, u, o e c([o, r]; W3'2(s?)), where W k'p (~2) denotes the Sobolev space of functions whose derivatives up to order k lie in the Lebesgue space L p (S-2) (for basic properties of Sobolev spaces see, e.g., the monograph of Adams [1]). The scale Wk'2 forms a suitable function spaces framework because of the variational structure of the problem. The a priori estimates are obtained in the Hilbertian scale W k'2 in a very natural way via the "energy method" used in the pioneering paper by Matsumura [71]. Assuming more regularity of the initial data and F one could prove the same result with W3,2 replaced by W k'2 with k sufficiently large. It follows then from the standard embedding theorems that the solution would be classical. Alternatively, one can use the smoothing effect of the diffusion semigroup to deduce that the solutions constructed in Theorem 2.1 are, in fact, classical for t > 0 (cf. Matsumura and Nishida [72]).
- 316. Viscous and~or heat conducting compressible fluids 317 2.2. Global existence of discontinuous solutions Discontinuous solutions are fundamental both in the physical theory of nonequilibrium thermodynamics and in the mathematical theory of models of inviscid fluids. It seems natural, therefore, to have a rigorious mathematical theory for the system (2.1)-(2.3) which would accommodate discontinuities in solutions. Of course, one has to abandon the classical concept of solution as a differentiable function and turn to the weak solutions which satisfy the integral identities (2.4)-(2.6). It follows from (2.4)-(2.6) that the density 0, the momenta Qui, i -- 1, 2, 3, and the specific internal energy Q0 considered as vector functions of time are weakly continuous, i.e., the quantities f~ Odpdx ' Is2 o u idpdx ' i -- 1 N, [. belong to C(I) for any fixed 4~e 79(S2). f oOCdx Consequently, it makes sense to prescribe the initial conditions even in the class of weak solutions. Pursuing this path Hoff [47] examined the system (2.1)-(2.3) on the whole space R3 where the pressure p and the internal energy e obey the ideal gas constitutive relations (1.7) and the initial data 00, u0, 00 satisfy 00 - a ~ L ~ n L ~(R~), 00 - ~ ~ L:(R~), uo ~ [ws'=(R~)] ~ se(1/3,1/2), (2.13) for certain positive constants ~, 0. The Sobolev spaces Ws'2(R 3) for a general real parameter s may be defined in terms of the Fourier transform (see Adams [1]). We report the following result (see [47, Theorem 1.1]). THEOREM 2.2. Let 1-2 -- R 3. Assume that )~, lz and tc are constants satisfying > 0, ~/3 < ~ + ~ < (~ + 1/3,/i3)u, ~ > 0. Let the pressure p obey the ideal gas constitutive relation P--(V- 1)CvQ0, Y > 1, Cv > 0, (2.14) and the initial data Qo, uo, Oo satisfy (2.13) for certain positive constants ~, O. Let positive constants 0 < Ql < Q < Q2, 0 < 02 < 01 < 0 be given. Finally, set f- 0 in (2.2). Then there exists ~ > 0 depending on Qi, Oi, i - 1, 2, and s such that the initial-value problem (2.1)-(2.3), (2.12) possesses a weak solution Q, u, 0 on the set (0, cx~) x R 3
- 317. 318 E. Feireisl provided I1~0 - ~llt2nt~(R 3) + II00 -0llt2(e3) -+-Iluollws,2nt4<R3) < e, ess inf00 >/01. Moreover, the solution satisfies Q1 ~<Q(t, x) ~<~02, O(t, x) >~02 for a.a. (t, x) E (0, c~) • R 3, and Q(t) --+ b, u(t) --+ 0, O(t) --+ 0 in L p (R 3) as t --~ cx~ for any 2 < p <<.cx~. A similar result under slightly more restrictive hypotheses on the data can be proved for 12 = R2 (see Hoff [47, Theorem 1.1]). The proof of Theorem 2.2 leans, among other things, on the regularity properties of the quantity p - ()~+ 2#) divu termed the effective viscous flux. More specifically, this quantity is shown to be free of jump discontinuities. This is the first indication of the important role played by the effective viscous flux in the mathematical theory of compressible fluids. We will address this issue in detail in Section 4.3. 2.3. Global existence in critical spaces The solutions obtained in Theorem 2.2 solve the problem for a very general class of initial data but are not known to be unique. On the other hand, Theorem 2.1 yields a unique solution at the expense of higher regularity imposed on the data. A natural question to ask is how far one can get from the hypotheses of Theorem 2.1 to those of Theorem 2.2 to save uniqueness or, more precisely, what is a critical space of data for which the weak solutions are unique. Such a question was already studied for the incompressible Navier-Stokes equations by Fujita and Kato [41]. The same problem for the full system (2.1)-(2.3) under rather general constitutive relations has been addressed only recently by Danchin [16]. He obtains existence and uniqueness of global solutions in a functional space setting invariant by the natural scaling of the associated equations: - ,(v2t, vx), - where the pressure law p is changed to I)2p. A functional space for the triple [0, u, 0] is termed critical if the associated norm is invariant under the transformation Lo, u, O] ~ [~o~,u~, 0~]
- 318. Viscousand~orheat conducting compressiblefluids 319 up to a constant independent of v. Accordingly, the well-posedness for the problem (2.1)- (2.3) can be stated in terms of the Besov spaces B s (R3) whose exact definition goes 2,1 beyond the framework of the present paper (see [16]). Let us only remark that the final result is of the same character as Theorem 2.1, namely, global existence and uniqueness of (weak) solutions of the problem (2.1)-(2.3) for data which are a small perturbation of a given equilibrium state. 2.4. Regularity vs. blow-up Since the celebrated work of Leray, it has been a major open problem of mathematical fluid mechanics to prove or disprove that regular solutions of the incompressible Navier-Stokes equations in three space dimensions exist for all time. Clearly, the same problem for the general system (2.1)-(2.3) seems even more delicate. As a matter of fact, there is a negative result of XIN [110, Theorem 1.3]. He considers the system (2.1)-(2.3) posed on the whole space R3 with zero thermal conductivity tc -- 0 and the initial density 00 compactly supported: THEOREM 2.3. Let S-2 -- R 3 and m > 3 be a given number. Consider the system (2.1)-(2.3) complemented by the initial conditions (2.12) where the viscosity coefficients )~, lZ are constant and satisfy (1.6), and p obeys the constitutive law (2.14). Moreover, let tc = O, f = O, and ~O0,U0, 00 E W m'2 (e3), supp 00 compact in R 3 00/>0>0. Then there is no solution of the initial value problem (2.1)-(2.3), (2.12) such that ~0,U, 0 E C 1([0, 0<~); wm'2(R3)). It seems interesting to compare the conclusion of Theorem 2.3 with the existence result of Theorem 2.1. Obviously, the above theorem does not seem to solve (in a negative way) the question of regularity for the compressible Navier-Stokes equations because of the hypothesis of compactness of the support of 00. We remark in this regard that the Navier- Stokes system is a model of nondilute fluids in which the density is bounded below away from zero. It is natural, therefore, to expect the problem to be ill-posed when vacuum regions are present at the initial time. 2.5. Large data existence results To begin with, one should say there are practically no global existence results for the full system (2.1)-(2.3) when the data are allowed to be large. The question of local existence of classical solutions for regular initial data was addressed by Nash [79]. There is no indication, however, whether or not these solutions exist for all times.
- 319. 320 E. Feireisl Note that the problem here is of different nature than for systems of nonlinear conservation laws without diffusion terms. Indeed the equations (2.2), (2.3) are parabolic in u, 0 respectively provided the density 0 is kept away from zero. Accordingly, one can anticipate these state variables to be regular provided uniform estimates of 0 were available. On the other hand, the density solves the hyperbolic equation (2.1) which is, however, only linear with respect to 0. Consequently, no shock waves should develop in 0 provided they were not present initially and the velocity field u was sufficiently regular. Formally, one can use the standard method of characteristics to deduce: d dt --Q(t, X (t)) + Q(t, X (t))divu(t, X (t)) - 0, where X'(t) -u(t,X(t)), X (0) = X0 9 S2. We end up in a "vicious circle" as we need uniform bounds on divu to estimates the amplitude (and positivness) of Q but those are not available from the standard energy estimates. As indicated by Choe and Jin [11, Theorems 1.3, 1.4], the following three questions are intimately interrelated: 9 uniform (on compact time intervals) upper bounds on the density Q; 9 uniform boundedness below away from zero of Q; 9 uniform bounds on u. Answering one of these questions would certainly lead to a rigorous large data existence theory in the framework of distributional (weak) solutions for the problem (2.1)-(2.3) (cf. also Lions [62]). The above mentioned difficulties made several authors to search for a completely different approach to the problem. Motivated by the pioneering work of DiPerna [22], the theory of measure-valued solutions was developed by M~ilek et al. [68]. Roughly speaking, the "value" of each state variable at a fixed point (t, x) is no longer a number (or a finite component vector) but a probability measure (the Young measure) characterizing possible oscillations in a sequence of approximate solutions used to construct this particular variable. The numerical values of Q, u, 0 are centers of gravity of the corresponding Young measures and the nonlinear constitutive relations are expressed in a very simple way. These solutions are of course more general quantities than the distributional solutions and coincide with them provided one can show that the Young measures are concentrated at one point, i.e., they are Dirac masses for each value of the independent variables (t, x). One can expect positive existence results in the class of measure valued solutions whenever suitable a priori estimates are available so that the nonlinear compositions are equi-integrable and consequently weakly compact in the space of Lebesgue integrable functions. This is of course considerably less than compactness of the state variables in the strong L l-topology- an indispensable ingredient of any existence proof of distributional solutions. The major shortcoming of measure-valued solutions is certainly the almost insurmount- able problem of uniqueness solved only in the case of a scalar conservation law in [22].
- 320. Viscous and~or heat conducting compressible fluids 321 This is, of course, the price to be paid for the relatively simple existence theory and one might feel tempted to say it is the same situation as when the weak solutions were in- troduced. However, this gap between existence and uniqueness, accepted for the weak solutions, seems to be simply too large in the class of measure-valued solutions and the approach is slowly being abandoned. 2.6. Bibliographical comments Besides the results mentioned above, the small data existence problems were treated by Solonnikov [97], Tani [101], Valli [108] and others. The existence theory in critical spaces for barotropic flows was developed by Danchin [15]. As pointed out several times, the main obstacle to obtain large data existence results is the lack of suitable a priori estimates. Formal compactness results for the full system (1.1)-(1.3) were obtained by Lions [62] on condition of uniform boundedness of all state variables. 3. The continuity equation and renormalized solutions Motivated by the work of Kruzkhov on scalar conservation laws, DiPerna and Lions [23] introduced the concept of renormalized solutions as a new class of solutions to general linear transport equations. They play a similar role as the entropy solutions in the theory of nonlinear conservation laws - they represent a class of physically relevant solutions in which the corresponding initial value problems admit a unique solution. Multiplying (1.1) by b'(Q), where b is a continuously differentiable function, we obtain the identity ab(Q) Ot 4- div(b(Q)u) + (b' (Q)Q - b(o))div u -- O. (3.1) Obviously, any strong (classical) solution of (1.1) satisfies automatically (3.1). For the weak solutions, however, (3.1) represents an additional constraint which may not be always satisfied. Following [23] we shall say that Q is a renormalized solution of (1.1) on the set I x $2 if 0, u, Vu are locally integrable and (3.1) is satified in the sense of distributions (in 79'(I x s i.e., the integral identity b(o)--~ 4- b(p)u 9V9 4- (b(o) - b'(o)Q) divu~o dx dt - 0 (3.2) holds for any test function q99 79(1 x s and any b 9 C 1(R) such that b'(z) = 0 for all z large enough, say, Izl/> M. (3.3) Let us emphasize here that unlike the entropy solutions that can be characterized as satisfying a certain type of admissible jump conditions on discontinuity curves, the renormalized solutions characterize the so-called concentration phenomena (cf. Section 3.2 below).
- 321. 322 E. Feireisl 3.1. On continuity of the renormalized solutions The renormalized solutions enjoy many remarkable properties most of which can be proved by means of the regularization technique developed by DiPerna and Lions [23]. The following auxilliary assertion is classical (cf. Lions [61, Lemma 3.2]). LEMMA 3.1. Let 12 C R N be a domain and u, Vu~L p loc(ff2), Lqoc(n), where 1 <~p,q <. oo, l/r-- lip 4- 1/q <~1. Let Oe be a regularizing sequence, i.e., Oe ~ 7)(RN), Oe radially symmetric and radially decreasing, f/~ Oedx = 1, N Oe(x) --+ 0 as e --+ Ofor anyfixed x E R N {0}. Then [lOe 9 [div(cru)] - div([Oe 9 or]u)[[tr(g ) ~ c(g)llUllw,,p(g)ll~rlltq(g ) for any compact K C ~2 and re = Oe * [div(o'u)] - div([Oe 9 o-]u) --+ 0 in L,roc(s as e --+ O, where 9 stands for convolution on R N. Now one can regularize (3.1), more precisely, take ~(x) = Oe(x - y) in (3.2) to deduce OOe * b(o) Ot 4-div([Oe 9 b(o)]u ) 4- Oe * [(b'(o)O - b(o))divu] -- re (3.4) for t ~ I and x ~ s such that dist[x,012] > e. Here b is an arbitrary function satisfying (3.3) and re(t) as in Lemma 3.1, i.e., rE --~ 0 in L~oc(I • 12) as e -+ 0 provided u is locally integrable. The first consequence of (3.4) is continuity in time of the renormalized solutions. PROPOSITION 3.1. Let u, Vu be locally integrable on I • S-2 where I C R is an open time interval and s C R N a domain. Let O - a locally integrable function - be a renormalized solution of the continuity equation (2.1) on I • S2.
- 322. Viscous and~or heat conducting compressible fluids 323 Then for any compact B C F2 and any function b as in (3.3), the composition b(Q): t 9 I w->b(Q)(t) is a continuous function oft with values in the Lebesgue space LI(K), i.e., b(o) 9 C(J; Ll(B)) forany compact J C I. Moreover, we have the following corollary. COROLLARY 3.1. In addition to the hypotheses of Proposition 3.1, assume that I C R, C R N are bounded; and Q 9 Lc~(O, T; LP(~2)) fora certain p > 1, u, Vu 9 L I(I • ,(-2). Then 0 as a function of t 9 I is continuous with values in L 1(~2 )" o c c(i; Ll(n)). The proof of both Proposition 3.1 and Corollary 3.1 can be done via the regularization technique as in [23]. 3.2. Renormalized and weak solutions Another conclusion which can be deduced from Lemma 3.1 and (3.4) is that the class of weak and renormalized solutions coincide provided Q or Vu or both are sufficiently integrable. PROPOSITION 3.2. Assume 0 9 LlPoc(I x ~), u, Vu 9 Lqoc(I x ~), where 1 ~ p,q <~ oo, 1/p+ 1/q <~ 1. Then Q is a renormalized solution of (2.1) if and only if Q satisfies (2.1) in 73'(1 x F2), i.e., the integral identity (2.4) holds for any test function cp 9 73(1 x F2). Integrating (1.9) we can see that the typical regularity class for the velocity gradient is Vu 9 L2oc(I x ~2). Accordingly, to apply Proposition 3.2, one needs 0 9 L2oc(I x ~2). There is another reason why the density "should be" square integrable. The continuity equation can be (formally) written in the form Dto + odivu =0, 0 where Dt -- 0-t + u. VQ
- 323. 324 E. Feireisl is the so-called material derivative. The quantity 0 div u plays the role of a forcing term in the above equation. Thus if we want to keep, at least in a certain weak sense, the structure given by characteristics (cf. Section 2.5), we should have Qdiv u locally integrable. Taking the square integrability of Vu for granted we are led to require 0 ~ L2oc (cf. also DiPerna and Lions [23]). However, as we will see in Proposition 4.1 below, the square integrabilty of the density is not necessary for a weak solution of (1.1) to be a renormalized one. 3.3. Renormalized solutions on domains with boundary Assume that I2 C R N is a domain with Lipschitz boundary. As for the velocity field u, we suppose u 6 L~oc(l • S-2), Vu 6 L~oc(l • S2 ) for a certain q > 1. Although u need not be continuous, one can still consider the no-slip boundary conditions (1.11) in the sense of traces. Accordingly, assuming (1.11) and extending u to be zero outside S2, one has u E Wllo'q (RN). Equivalently, by virtue of the Hardy inequality (see, e.g., Opic and Kufner [85]), one can replace (1.11) by the following stipulation: lul dist[x, 8S2] Lqoc( r Using (3.5) we can show a continuation theorem for renormalized solutions. PROPOSITION 3.3. Let I-2 C R N be a Lipschitz domain and let u, Vu belong to L~oc(I • I-2 )for a certain q > 1, and let (1.11) be satisfied in the sense of traces. Let Q be a renormalized solution of (2.1) on I • I2. Then ~ is a renormalized solution of (2.1) on I • R N provided 4, u are extended to be zero outside I-2. Proposition 3.3 together with Propositions 3.1, 3.2 yield an interesting corollary, namely, the principle of total mass conservation for the weak solutions of (2.1). Consider a bounded domain I2 C R N with Lipschitz boundary on which u satisfies the no-slip boundary condition (1.11). Formally, one can integrate (2.1) over S-2to deduce -~ o dx - O, i.e., the total mass m - fs2 odx is a constant of motion. By virtue of Propositions 3.1-3.3, we have the same result for distributional solutions:
- 324. Viscousand~orheat conducting compressiblefluids 325 PROPOSITION 3.4. Let ~ C R N be a bounded Lipschitz domain. Let 0 E L~ LP(S-2)), u, VU E L q (I x I2), l < p,q <. oc, 1/p+ l/q <, l, solve (2.1) in 79'(1 x $-2)and ulo~ =0. Then the total mass m -- fs? O(t)dx is constant for t E I. 4. Weak convergence results Many of the most important techniques set forth in recent years for studying the problem (2.1)-(2.3) are based on weak convergence methods. To establish the existence of a solution, an obvious idea is first to invent an appropriate collection of approximating problems, which can be solved; and then to pass to the limit in the sequence of approximate solutions to obtain a solution of the original problem. The overall impediment of this approach is of course the nonlinearity. Whereas it is very often true that one can find certain uniform estimates on the family of approximate solutions, the bounds on oscillations of these quantites are usually in short supply. This is, for instance, the case of the density 0 solving the hyperbolic equation (2.1). In this section, we shall investigate the compactness properties of weakly convergent sequences of solutions of the continuity equation (2.1) and the momentum equations (2.2). More precisely, we consider a family of weak solutions {0n}, {un} of the system (2.1), (2.2), i.e., the integral identities fl fs? Oq) 9Oq9 dx dt= 0, (4.1) ff s? 9Oq) Oq9 , -5-[ + O.u, + P, - # ~ ~ OuJ Oq) -- (1~ -nt- ls OXj OXi ~-ofi cpdx dt =0, i=1 ..... N, n=l,2 ..... i Oq) OUn OXj OXj (4.2) hold for any test function r E 79(1 • I-2). Since all results we shall discuss are of local nature, we assume that both the time interval I and the spatial domain S2 are bounded.
- 325. 326 E. Feireisl Moreover, we suppose that Qn, Un, VUn, Pn, and fn are locally integrable and weakly convergent, specifically, ~On -----~~O an -----~I1 VUn ---> Vu pn --+ p fn --~ f weakly (in D'(I x s as n --+ oo. Here Vn ~ V weakly means that v is locally inegrable on I x s and flffevn~~ dxdt for any ~oE D(I x I2). Our goal in this section is to identify the limit problem solved by the quantities Q, u, p, and f. The best possible result is, of course, they satisfy the same system of equations. If this is the case, the problem enjoys the property of compactness with respect to the weak topology. Given relatively feeble a priori estimates (cf. Section 5), the weak compactness of the problem plays a decisive role in the larga data existence theory for barotropic flows presented in Section 6. 4.1. Weak compactness of bounded solutions to the continuity equation Although hyperbolic, the continuity equation exhibits the best properties as far as the weak compactness of solutions is concerned. Consider a sequence On, Un of renormalized solutions of (1.1) on I x s i.e., in addition to (4.1), we assume (3.2) holds for any b as in (3.3). Moreover, we shall assume that IlunllLl(Ixs2), IlVunlltq(1• ~ c for a certain q > 1; (4.3) and that the family ~Onis equi-bounded and equi-integrable, i.e., ff~ dx dt <<. Qn C, lim fQndxdt-O IQI~OJQ uniformly with respect to n = 1, 2..... (4.4) The failure of weak convergence to imply strong convergence is usually recorded by certain measures called defect measures. To this end, we introduce the cut-off operators Tk = Tk(Z),
- 326. Viscous and~orheat conducting compressiblefluids 327 where T e C I(R) is an odd function such that T(z)=z for0~<z~<l, T(z)=2 forz~>3, T concave on [0, oo). The amplitude of possible oscillations in the density sequence will be measured by the quantity OSCp[0n - 0](Q) - sup(lim sup{]Tk(on) - Tk(o)[[ Lp(Q))" k>/1" n--+oc Unlike the defect measures introduced by DiPerna and Majda [24], osc vanishes on any set on which 0n tends to 0 strongly in the L l-topology regardless possible concentration effects. Now we shall address the following question: Under which conditions do the limit functions 0, u solve (2.1)? Taking a function b as in (3.3) one deduces easily from (3.2), (4.3), and (4.4) that b(On)-+ b(o) inC(I P " Lweak(~Q)), 1 ~<p < oc. (4.6) Here and in what follows, we shall use the standard notation g(v) for a weak (L p) limit of a sequence g(vn) where Vn tends weakly to v. The possibility to find a subsequence of Vn such that the composition b(vn) is weakly convergent for any continuous b satisfying certain growth conditions is the basic statement of the theory of Young measures (cf. Tartar [102,103], Pedregal [88]). Such a limit, however, need not be unique unless the convergence of Vn is strong. A sequence vn converges to v in C (I" P , Lweak(~'2)) if it is bounded in L~(I; LP(~-2)), the function t ~-+fs2 Vn(t, x)dp(x)dx can be identified with a continuous function on I, and fs? v~(t' x)dp(x) dx --+ fs? v(t, x)dp(x) dx uniformly with respect to t e I for any test function 4~e D(s By virtue of the Aubin-Lions lemma (see, e.g., Lions [59, Theorem 5.1], the relations (4.3), (4.6) imply b(On)Un --+ b(o)u weakly in Lqoc(I • s (4.7) Indeed taking p large enough in (4.6) we get L p(~(2) compactly imbedded in W -l'q (~(-2) ; whence b(on) --+ b(o) in C(I; W -l'q (s which together with (4.3) yields the desired conclusion. The distributions lying in the "negative" Sobolev space W-1,q can be identified with generalized derivatives of vector functions in Lq (see, e.g., Adams [1]). In particular, if we knew that 0n is bounded in L~oc(I x s we could conclude that O, u solve (2.1) in the sense of distributions. In a general case, we report the following result (cf. [29, Proposition 7.1]):
- 327. 328 E. Feireisl PROPOSITION 4.1. In addition to (4.1), assume Qn, Un satisfy (2.1) in the sense of renormalized solutions on I x S-2.Moreover let (4.3), (4.4) hold, and OSCp[Qn -- Q](Q) ~<c(Q) for any compact Q c I x I-2, where 1 1 t <1. P q Then Q, u is a renormalized solution of (2.1) on I x 12. The main advantage of Proposition 4.1 is that the sequence On itself need not be bounded in LP. As we will see later, this is particularly convenient when barotropic fluids are studied (cf. Proposition 5.3 below). 4.2. On compactness of solutions to the equations of motion In this part, we shall assume that IlUnlIL2(I• IIVUnlIL2(I• ~ C foralln-- 1,2 ..... (4.8) In particular, the products uiuj, i, j = 1..... N, are bounded in L I(I, L2"/2(~)) where 2* is the Sobolev exponent for the embedding W 1,2 C L2* to hold, i.e., 2* is arbitrary finite for N -- 2 2N and 2"= if N=3,4 ..... N-2 i J to be at least integrable, it is neccessary that Consequently, for the cubic quantity QnUnU n ess sup II (t ll c for a certain p ~>N / 2 tel (4.9) provided N > 2. Here again, we face one of the major obstacles to build up a rigorous mathematical theory for the full system (2.1)-(2.3), namely, the lack of suitable a priori estimates. The only available bounds on the density are those deduced from boundedness of the total energy. In general, these "energy" estimates are not sufficient to get (4.9). Of course, the barotropic case offers a considerable improvement as the energy is given by formula (1.8) and, consequently, the desired estimates follow provided y >~N/2. The situation is more delicate in the physically relevant two-dimensional case. Here, the Sobolev space W 1'2 is embedded in the Orlicz space L ~ generated be the function q0(z) - exp(z e) - 1
- 328. Viscous and~or heat conducting compressible fluids 329 (see Adams [1]). Consequently, condition (4.9) should be replaced by ess sup fs2 ~p(On)dx <<,c tel with qJ(z) ~>z log(z). (4.10) Since On, Un satisfy also the continuity equation (4.1) we deduce from (4.9), (4.10) respectively that Onto inC(I; P Lweak(I2)), p ~ N/2 if N -- 3..... (4.11) or On --+ O inC(I; q' Lweak(S2)) for N -- 2. (4.12) In both cases this implies compactness of Qn in L 2(I, W-1,2 (S-2)), and we get OnUn --+ 0u weakly in, say, L j (I x s i Yrepresents a more difficult problem. In The weak compactness of the cubic term OnUnU n addition to the above hypotheses, we assume the kinetic energy to be bounded uniformly in n, i.e., o~lu~l 2 is bounded in L 1(I • s uniformly with respect to n. Supposing (4.9) holds for p > N/2 we have, similarly as above, 2N OnUn ~ 0u in C(I; Lreak(a"2)) for a certain r > N ~>2; (4.13) N+2' whence i UJn~ ~OUtUj weaklyin, say, Ll(I x I2) i j = 1 ... N, N >~2. D..nU n , , , , As a matter of fact, the result is not optimal for N -- 2; in that case one could use directly (4.12) provided q~ was a function dominating z log(z). Summing up the previous considerations we get the following conclusion: PROPOSITION 4.2. Let the quantities Qn, Un satisfy the estimates (4.8), (4.9) with p > N/2. Moreover, let the kinetic energy be bounded, specifically, ess sup fs? On(t)lUn (t)12 tel dx <, c foralln= l,2 ..... (4.14) Finally, assume fn are bounded and fn ~ f uniformly on I x I2. (4.15)
- 329. 330 E.Feireisl Then the limitfunctions O, u, p, and f satisfy (2.1), (2.2) in 79t(I • $2), i.e., the integral identities (2.4), (2.5) hold for any testfunction q9~ 79(1 • I-2). Let us repeat once more that it is an open problem whether or not the bounds required for the density component are really available. As we have seen in Section 2.5, uniform bounds on the density are equivalent to uniform boundedness of u - a situation reminiscent of the classical regularity problem for the incompressible Navier-Stokes equations. 4.3. On the effective viscous flux and its properties Consider the quantity p - ()~4- 2#) div u called usually the effective viscous flux. Formally, assuming all the functions in (2.2) smooth and vanishing for Ixl --+ oc we can compute Pn - (~, + 2#)divun = A -1 div(Onfn) - A -1 div(OnUn)t - "~i,j[OnUinuj ] (4.16) (summation convention). The symbol T~i,j denotes the pseudodifferential operator T~i,j - OxiA-10xj or, in terms of symbols, TP~,ij[V] -- ~--1 [~i~j .)L-.[V](~) ] where .T denotes the Fourier transform in the x-variable. The effective viscous flux enjoys certain weak compactness properties discovered by Lions [62] which represent the key point in the global existence proof for barotropic flows. Following [62] we can use (formally) (4.16) to obtain (Pn - (~ + 2#)divun)b(On) --b(On)A -1 div(Onfn) - Ot(b(On)A -1 div(OnUn)) 4- b(On)t A-1 div(OnUn) - b(On)~i,j[On UnUn],ij (4.17) where b is as in (3.3). One should keep in mind that 0n, Un here are defined on a bounded domain I2 and, consequently, a localization procedure is needed to justify this argument. Using Proposition 4.2 we can now pass to the limit for n --+ cxz in (4.16) and multiply the resulting expression by b(o) to deduce (p - (~ + 2#)divu)b(o) = b(o)A -1 div(of)- Ot(b-~A -1 div(ou)) 4- b(o)t A-1 div(ou) - b(o)T4.i,j[Ou iuJ]. (4.18)
- 330. Viscousand~orheatconductingcompressiblefluids 331 Assuming, in addition to the hypotheses already made, that On, Un are also renormalized solutions of (2.1), we can use (4.7), (4.18) together with the smoothing properties of A -1 , to pass to the limit in (4.17) for n --+ oo to obtain: lim f fs2(p. - ()~ + 2u)divu,)b(o,)~odxdt n--->(~ _ff =nli~m~flf. b(On)(UinJ~i'j[OnuJ]-J~i'j[OnuiuJ])~~ - ff. ~( uiJ~i,j [Ot/j] --J-~i,j [ouit/J])~ 0dx at (4.19) for any test function q9e 79(1 • Y2). It is a remarkable result of Lions [62] that the right-hand side of (4.19) is in fact zero. To see this we offer two rather different techniques in hope to illuminate a bit the connection of such a result with the theory of compensated compactness. Following the proof in [62] one can make use of the regularity properties of the commutator UiJ~i,j [~OUj ] -- "]-'~'i,j[Oui UJ] discovered by Coifman and Meyer [13]. Specifically, this quantity belongs to the Sobolev space W l'q provided ui e W 1'2 and Ouj e L r with r > 2 in which case 1/q- 1/p + 1/2. Of course, this hypothesis requires 0 e LP with p > 3 for N >~3 which is too strong for our purposes, but a simple interpolation argument shows one can treat the general case 0 E L p, p > N/2, by the same method. Pursuing a different path we can write f fs b(Qn)(uJJ"~i'j[QnUin]- T2~i'j[Qn i j ....])~ dx at - f f..,:(x, (0.).,(o.-.)- v, (0.))dx ,,, where the vector fields X j , Y, U, Vj are given by formulas X[ (On) - (~ob(On)6j,k - 7~k,l[~Pb(o~)&,j]), k _~2~k,i[On i V/ (On) - T~k,l[~ob(On)31,j], j - 1..... N, and ~i,j stands for the Kronecker symbol.
- 331. 332 E. Feireisl Now, it is easy to check that divXJ--divU--O and Y--V(A-l[div(OnUn)]), V j - V(A -1 [div(~ob(On)&,j)]), i.e., curl Y = curl V j -" O. Applying the L P-L q version of Div-Curl Lemma of the compensated compactness theory (cf. Murat [78] or Yi [111]) together with (4.6), (4.13), we conclude xJ (On)" Y(OnUn) ~ ((flb(o)Sj,k - ~k,l[(flb(O)Sl,j]) "JP~.k,i[Oui] in L2(I; W-I'2(~Q)), and, similarly, U(OnUn) "vJ (On) --+ (0 uk -- JP~.k,i[oui])T~k,l[(flb(O)Sl,j] in L2(I; W-1'2(;2)) provided p > N/2. This yields, similarly as above, the desired conclusion, namely, the right-hand side of (4.19) equals zero. Thus we have obtained the following important result (see [62]): PROPOSITION 4.3. Let the quantities On, Un, and fn satisfy the hypotheses (4.8), (4.9) for p > N/2, togetherwith (4.14), (4.15). Let, moreover, IlpnllL~(Z• ~ c fora certain r > 1. Then we have lim fifs2(Pn-()~+2lz)divun)b(Qn)qgdxdt 11---+oo =ffs(P-(X+2.)clivu)b(V)~~ for any b satisfying (3.3) and any testfunction ~pE 7)(I x ~). 4.4. Bibliographical remarks The theory of compensated compactness has played a crucial role in the development of the first large data existence results for systems of nonlinear conservation laws (see Dafermos [14], DiPerna [21], Tartar [102]). A good survey on weak convergence methods can be found in the monograph by Evans [26]. One of the well-known results is the so- called Div-Curl Lemma refered to above (cf. Murat [78]):
- 332. ~scous and~orheat conducting compressiblefluids 333 LEMMA 4.1. Let Un, Vn be two sequences of vector functions defined on some open set Q c R u such that Un --+ U weakly in L p (Q), Vn --+ V weakly in L q (Q)" and div Un precompact in W- l, p (Q), curl Vn precompact in W- 1,q (Q), where 1 1 1 <p,q <ec, -+-~<1. P q Then Un . Vn --+ U. V in 79'(Q). Note that the situation in Proposition 4.3 is particularly simple as div Un = curl Vn = 0 and the proof of Lemma 4.1 is elementary. There is yet another way to show Proposition 4.3 presented in [27, Lemma 5]. The defect measures similar to osc were introduced by DiPerna and Majda [24] in their study of the Euler equations. 5. Mathematical theory of barotropic flows We review the recent development of the mathematical theory of barotropic flows, specifically, we shall discuss some large data existence and related results originated by the pioneering work of Lions [62]. Accordingly, the crucial hypothesis we cannot dispense with is that the pressure p and the density 0 are functionally dependent and the relation between them is given by formula P--P(O) with p-[0, ec) --~ [0, ec) - a nondecreasing and continuous function. (5.1) As a matter of fact, most of the results will be stated for the simpler isentropic pressure- density relation P(O)--ao • a>O, y~>l, (5.2) and possible generalizations discussed afterwards.
- 333. 334 E. Feireisl The temperature 0 being eliminated from the pressure constitutive law, the system (2.1)- (2.3) reduces to O0 Ot + div(ou) = 0, (5.3) OOu Ot + div(ou | u) + Vp(o) =/zAu + (Z +/.t)V(divu) + of. (5.4) The spatial variable x will belong to a regular boundeddomain ~2 C RN, N = 2, 3, and the velocity u will satisfy the no-slip boundary conditions ulas2 =0. (5.5) Taking (formally) the scalar product of (5.4) with u and integrating by parts we obtain the energy inequality: d fs E(t) dx -+-fs2lZlVul2 dt + 0~ +/z) Idivul 2dx ~<fr2 of. u dx, (5.6) where the specific energy E satisfies (1.8). If p is given by (5.2), we have 1~ e + o log(0) E=~ 1 a for 9/= 1, E- x01ul 2 + 0 • y-1 z if y> 1. As already agreed on in Section 1, the fluids under consideration are viscous, i.e., /z>0 and )~+/z~>0. Note that the restrictions imposed on )~allow for all physically relevant situations. In what follows, we consider thefiniteenergyweaksolutionsof the problem (5.3)-(5.5) on the set I x s more specifically, 0, u will meet the following set of conditions: 9 the density 0 and the velocity u satisfy 0~>0, o ~L~176L• (~)), u ~ L2(I; [W0'2(I2)]N); 9 the specific energy E belongs to L~oc(l; L1 (~)) and the energy inequality (5.6) holds in D' (I), i.e., fl OtTr(fs~Edx) dt - f Tr(fs21zlvule+ ()~+lz)ldivulZdx)dt ~fTrfs2of'udxdt holds for any function ~ E D(I), ~ >~0;
- 334. Viscous and~or heat conducting compressible fluids 335 9 the functions 0, u extended to be zero outside C2 solve the continuity equation (5.3) in D'(I x R N) (cf. (2.4)); moreover, (5.3) is satisfied in the sense of renormalized solutions, i.e., (3.2) holds for any b as in (3.3); 9 the equations of motion (5.4) are satisfied in 79'(1 x S-2)(cf. (2.5)). As the reader will have noticed in Section 4, the value of the adiabatic constant V will play an important role in the analysis. In most cases, we shall assume V > N/2, where N = 2, 3 are the physically relevant situations. The external force density f is assumed to be a bounded and measurable function such that ess sup [f(t,x)[<~F. tEI, xEU2 In what follows, we shall give an outline of the large data existence results in the class of finite energy weak solutions. We shall also discuss the long-time behaviour and related asymptotic problems. To this end, we pursue the classical scheme for solving nonlinear problems: 9 First of all, we find a priori estimates, i.e., the bounds imposed formally on any classical solution and depending only on the data (cf. Sections 5.1, 5.2). 9 Given a family of solutions satisfying the bounds induced by a priori estimates, we examine the question of compactness, i.e., whether or not any accummulation point of this family in suitable topologies is again a solution of the original problem (see Sections 5.3, 5.4). 9 Finally, one has to find a suitable approximation scheme solvable, say, by a classical fixed-point technique, and compatible with both the estimates and compactness properties mentioned above (Section 5.5). To conclude this introduction, let us note that any finite energy weak solution satisfies 0 E C(I; • L ~ , Lweak(S2)) (-1C(I; (.C2)) 1 <~ot < V, 2V Qu E C(I" L• , weak(n)) (5.7) provided y > N/2 (cf. Proposition 3.1). In particular, the density and the momenta are well defined at any specific time t E I. Moreover, the total mass m -- fs~ 0 dx is independent of t E I; (5.8) and the total energy E defined for any t E I by formula g(t) - g[0, (0u)] (t) = [ 1 I(0u)l 2 Jo(t)>0 2 0 a - ~ ( t ) + 0 • (t) dx (5.9) y-1 is a lower semi-continuous function of t E I (see [27, Corollary 2]).
- 335. 336 E. Feireisl 5.1. Energy estimates Besides the total mass m, the total energy s is another quantity which can be shown bounded in terms of the data at least on compact time intervals. PROPOSITION 5.1. Let S-2 C R N be a bounded Lipschitz domain. Let Q, u be a finite energy weak solution of (5.3)-(5.5) where the pressure satisfies the isentropic constitutive law (5.2) with )1 > N/2. Then L f:nL IIo(t)ll + o(t)lu(t)l 2dx + IVul 2dxds LY(S-2) f{i} <, c(s m, F, t - inf{I}), (5.1o) where the quantity c is bounded for bounded values of arguments and s limsup s (Qu)](t). t--+inf{l}+ The bound (5.10), which can be easily obtained combining the energy inequality (5.6) and the Gronwall lemma, can be viewed as an a priori estimate though it holds for any finite energy weak solution of the problem. It is not difficult to see that similar results can be derived provided p is given by a general constitutive relation (5.1) and satisfies suitable growth conditions for large values of the density. On the other hand, as already mentioned in Section 2.5, uniform a priori estimates of Q seems to be out of reach of the standard techniques and represent a major open problem of the present theory. 5.2. Pressure estimates for isentropic flows By virtue of (5.10), the isentropic pressure p(Q) belongs automatically to the set L 1 (I x 12) at least for bounded time intervals I. On the other hand, the weak compactness results like Proposition 4.3 require p in a weakly complete (reflexive) space L r (I x ~2) with r > 1. Such a bound is indeed available as the following result shows: PROPOSITION 5.2. Assume S-2 C R N, N >~2, is a bounded Lipschitz domain. Let O, u be a finite energy weak solution to the problem (5.3)-(5.5) on I x I-2 where the isentropic pressure p is given by (5.2) with y > N/2. Let 0<rl<min ~,~ --~--1 be given. Denote by m = fs~ Qdx the (conserved) total mass and let F = ess sup/• if[.
- 336. Viscous and~orheat conducting compressiblefluids 337 Thenfor any bounded time interval J C I, we have V+I fJf~ OY+rJdxdt<'c(m'F'rl'lJl)(1-+-sup'Y(t)) • " t e J (5.11, A local version of the above estimates was obtained by Lions [62]. In fact, the bounds on r/in Proposition 5.2 are not optimal. Similarly as in the local case (see [62]), one could verify the best values for r/: 2 O<r/~y-1. However, for further purposes, it is convenient to have the integrals containing 07 bounded in terms of the total mass m equivalent to the Ll-norm rather than the total energy s proportional to the L • of O. The validity of (5.11) up to the boundary of I-2 was proved in [39] by means of a multiplier technique. More specifically, the main idea is to take the quantities ~i(t,x) -- 1/e(t)~i[b~e * Tk(~Or/)], i -- 1 ..... N, as test functions for (2.5). Here c D(1), 0~< ~ ~< 1, Z lOtaPldt <<.2, O~(x) is a regularizing sequence as in Lemma 3.1, Tk are the cut-off operators introduced in (4.5), and, most importantly, the symbol B stands for an inverse of the div operator, i.e., v = B[g] solves the equation div v -- g - ~ g dx, vlos2 =0. (5.12) Equation (5.12) has been studied by many authors. Here, we have adopted the approach which is essentially due to Bogovskii [8]. It can be shown that the problem (5.12) admits a solutions operator B: g ~ v enjoying the following properties: 9 B- [B1 ..... BN] is a bounded linear operator from LP(S2) into [w;'P(s2)] N, specifically, c(p)llglltp(~) for any 1 < p < ec. 9 The function v = B[g] solves the problem (5.12). 9 If, moreover, g ~ LP(S-2) can be written in the form g =divh where h 6 [Lr(S-2)]N, h- n = 0 on 0s then [ILr( c(p,r) llhllLr( ).
- 337. 338 E.Feireisl The proof of the existence of the operator/3 as well as the above properties can be found in Galdi [42] or Borchers and Sohr [9]. An alternative approach to show (5.11) based on properties of the Stokes operator was proposed by Lions [63]. The estimates (5.8), (5.10), (5.11) are the only available for the problem (5.3)- (5.5) unless some smallness assumptions are imposed on the data, i.e., on go, m, and f. Thus in accordance with Proposition 4.2, the main stumbling block to show compactness of solutions is the pressure term. Indeed the above estimates guarantee only weak compactness of the density 0 while for p = P(O) to be compact we need strong compactness of 0 in L p. As we shall see later, the way out this vicious circle is provided by Proposition 4.3, namely, by the compactness properties of the effective viscous flux. 5.3. Density oscillations for barotropic flows We show how Proposition 4.3 can be used to describe the amplitude of density oscillations for barotropic flows discussed in Section 4.1. In addition to (5.1), we suppose p(O) = O, P(O) ~ 0 for 0 i> O, p convex on [0, co). It is easy to verify that p(y)-p(z))p(y-z) for all0~<z~<y which yields immediately (p(y) - p(z))(Tk(y) -- Tk(z)) ) P(lT~(y)- Z (z l)I rk(z~] for all y, z ~>0. (5.13) Under the hypotheses (and notation) of Proposition 4.3, we have p = P(O) and we can write lim fQ P(On)Tk(On) - P(O) Tk(o) dxdt n--'+O0 -- nlirn fQ (P(On) - P(O))(Tk(On) -- Tk(o)) dx at + fQ (P(O) - P(O))(T~(o) - Tk(o) ) dx at (5.14) for any bounded Q c I x I2.
- 338. Viscous and~orheat conducting compressiblefluids 339 As p is convex and Tk concave on [0, oo), the second integral on the right-hand side of (5.14) is non-negative and, making use of (5.13) we infer nlim fQ P(On)Tk(On) -- P(O) Tk(o) dx dt > limsup fo~__,~ P(lrk(o~)- r (o>l dx dt. (5.15) On the other hand, we have lim JQ divun Tk(on) - divuTk (0) dx dt n---+OO = n--->~lim fodivun(Tk(On) - Tk(o) ) dx dt ~<sup Ildivun IIL2(Q)limsup Ilr (o ) - Tk(o)llLe(Q). n~/1 n-+oo (5.16) Thus if the pressure is superlinear at infinity, the relations (5.15), (5.16) together with Proposition 4.3 enable to estimate the amplitude of oscillations OSCp[Qn -- Q](Q) intro- duced in Section 4.1. In particular, the following result holds (see [29, Proposition 6.1]). PROPOSITION 5.3. Let S-2 C R N, N ~ 2 be a bounded Lipschitz domain and I C R a bounded interval. Let the pressure p be given by the formula (5.1), p(O) = O, p convex, and P(O) ~ ao • a > O, Y > N/2. Let On, Un be a sequence of finite energy weak solutions of the problem (5.3)-(5.5) with f = fn and such that mn--f On<~m, limsup C[On,(OnUn)](t)<~CO, t-+inf{l }+ and ess sup If~l~ F 1•163 independently of n. Then osc• - o](Q) ~<c(Q)(sup Ildivu, IIL2(Q~)1/y "n>/1
- 339. 340 E. Feireisl for any weak limit O of the sequence On and any bounded Q c I x 12. As a straightforward consequence of Propositions 4.1, 5.3 we get the following: COROLLARY 5.1. Under the hypotheses of Proposition 5.3, let On ---§ O Un --~U weakly star in L ~ (I; L • (I-2)), weakly in L2(I; W1'2($2)). Then O, u solve (5.3) in the sense of renormalized solutions, i.e., Equation (3.2) holds for any b satisfying (3.3). 5.4. Propagation of oscillations For simplicity, we suppose the pressure p is given by the isentropic constitutive relation (5.2) with y > N/2. Similarly as in Proposition 5.3, let On, Un be a sequence of finite energy weak solutions of (5.3)-(5.5) on some bounded time interval I such that limsup $[0~, (O~u~)] ~ Co, t---~inf{l}+ Ilfn []L~176 • ~ F uniformly in n. The issue we want to address now is the time propagation of oscillations in the density component. To begin with, it seems worth-observing that any reasonable solution operator we could associate with the finite energy weak solutions cannot be compact with respect to O-This is due to the hyperbolic character of the continuity equation (5.3). In accordance with the observations made by Lions [60], the oscillations should propagate in time. Serre [92] studied this phenomenon and showed the amplitude of the Young measures associated to the sequence On(t) is a non-increasing function of time. His proof is complete in the dimension N -- 1 and formal for N >~2 taking the conclusion of Proposition 4.3 for granted. Having proved Proposition 4.3 Lions [62] completed the proof for N ~>2. The fact that oscillations cannot be created in On unless they were present initially plays the crucial role in the existence theory developed in [62]. Here we go a step further by showing that the amplitude of possible oscillations decays with time at uniform rate depending solely on the value of the initial energy g0 (see [37]). In particular, the time images of bounded energy initial data are asymptotically compact with respect to the density component. This is precisely what is needed to develop a meaningful dynamical systems theory associated to the problem. In accordance with our hypotheses, we can show On "-+ O in C(I; • Lweak(~Q)) , Tk(On) ~ Tk(O) in C(I; L~weak(f2))for any a t> 1, k/> 1, Un --~ u weakly in L2(I; Wo'2(ff2)).
- 340. Viscous and~or heat conducting compressible fluids 341 To measure the amplitude of oscillations of the sequence ~On,we introduce a defect measure dft, dft[0n - 0](t) -- fs? v(t, x) dx, where v - 0 log(0) - 0 log(0). (5.17) By virtue of Corollary 5.1, both On and 0 are renormalized solutions of (5.3) and we have ; ot 5"2 , Onlog(On)-~ol~ inC(J Lweak()) l~<ct<y, 0 log(0) E C(J; Lweak~ (if2)) , 1 ~<ot < 9/. Consequently, dft[0n - 0] is a continuous function of t 6 I. Mainly for technical reasons, we are not able to deal directly with the function dft. We consider instead a family of approximate functions: z log(z) Lk(z) -- ok Tk(s)/s e Zlog(k) + z J~" ds forO ~<z ~<k, forz ) k. It is easy to observe that Lk(z) -- ~kZ + bk(z) where bk satisfy (3.3) and L'k(z)z- L~(z)= T~(z). Since both On, ~oare renormalized solutions of (5.3) on I x R 3, we deduce fs2(Lk(On) -- Lk(O))(t2) dx - Js2 (Lk(On) -- Lk(o))(tl) dx f f -- Tk (0) div u - Tk(0) div Un dx dt + (Tk(O) -- Tk(On)) divun dx dt for any t~, t2 E I. Letting n ~ oo and using Proposition 4.3 together with (5.15), we obtain fn (Lk(O) -- Lk(o))(t2) dx - fn (Lk(o) - Lk(o)) (tl) dx a lim sup ITk(On) -- Tk(o)l • dx dt t X+21z n~oc <~ (Tk(O) -- Tk(O)) divudx dt for any tl ~ t2, tl, t2 e I. (5.18)
- 341. 342 E. Feireisl Our aim now is to pass to the limit for k ~ oo in (5.18). Clearly, lye(Lk(o) - Lk(o))(t) dx dt --+ dfl[On - O](t) for k --+ oo while II o1(1 x n)) y-1 2y By virtue of Proposition 5.3, the right-hand side of the above inequality tends to zero for k --+ oo and so does the right-hand side of (5.18). Finally, we have lim sup [Tk(On)- Tk(0)1• dx dt /7-+00 ot-y+l f lt2 >~ 1$21 '~ limsup IIT (On)- r*(0)ll '+' L~(n) dx dt, /7-----~(x) and (5.18) yields: dft[o/7 -- 0](t2) --dft[On -- 0](tl) ( ~-~+') f'~ • + all2[)~ + 2# ~ limsUPn____~oo [[0n - 0liLt(n) dx dt ~<0 (5.19) for any tl ~<t2, tl, t2 6 I and 1 ~<c~ < y. To conclude, we shall need the following auxiliary result (cf. [28, Lemma 2.1]). LEMMA 5.1. Given ~ E (1, y) there exists c = c(a) such that z log(z) - y log(y) ~< (1 + log+(y))(z - y)+ c(ot)(lz - y11/2 + [z - yl ~) for any y, z ~ O. In accordance with Lemma 5.1, we can write n Onl~ - fn (1 + log+(o))(On - o) dx 2~- 1 ~<c(cO (IS21-~ lion - oIIZL ~ '~ (n) + lion - ollL,~(n)) which together with (5.19) yields L t2 dft[0/7 - O](t2) -dft[0/7 - 0](tl) + q~(dft[0/7 - 0](t))dt ~<0,
- 342. Viscous and~orheat conducting compressiblefluids 343 where the nonlinear function q~ depends only on the structural properties of the logarithm and can be chosen independently of the data to satisfy : R w-~R is continuous and strictly increasing, q~(0) -- 0. (5.20) Summing up the above considerations we have arrived at the following conclusion: PROPOSITION 5.4. Let s C R N, N ~ 2 be a bounded Lipschitz domain and I C R a bounded interval. Let Qn, Un be a sequence of finite energy weak solutions of the problem (5.3)-(5.5) on I x s where pressure p is given by the isentropic constitutive relation N p--aQ • a>0, V >-, ' 2 and f = fn. Let limsup s (QnU,)](t)<~s t ~ inf{ I }+ IlfnIIL~(Z~n) ~ F independently of n. Let O be a weak limit of the sequence On. Then dft[On - O](t2) <~X(t2 - tl) for any tl, t2 9 I, tl ~ t2, where X is the unique solution of the initial-value problem X'(t) + q~(X (t)) - 0, x(O)=dft[Qn-Q](tl) and ci9 is a fixed function satisfying (5.20). It can be shown that q~ has a polynomial growth for values close to zero and, consequently, the quantity dft[on - O](t) behaves like t -# for a certain fi > 0 when t ---~ ~. 5.5. Approximate solutions The a priori estimates derived in Sections 5.1, 5.2 together with the compactness results in Propositions 4.2, 5.4 form a suitable platform for a larga data existence theory for the problem (5.3)-(5.5). The final task, as usual, is to find a suitable approximation scheme compatible with both a priori estimates and the compactness results claimed above. Needless to say there are many ways to do it. Here we pursue the approach of [32] and consider the approximate problem: OQ Ot -F div(Qu) = e AQ, (5.21)
- 343. 344 E, Feireisl ~0u + div(ou | u) + Vp(0) + ~V0 ~ + ~Vu. V0 8t =/xAu + (~ + #)V divu + 0f (5.22) complemented by the boundary conditions ulos2 : XT09nlos2:0. (5.23) The parameters e > 0, ~ > 0 are "small" and fl > 0 "large". The system (5.21)-(5.23) can be solved by means of the standard Faedo-Galerkin method to obtain approximate solutions 0e,~, ue,~ (cf. [32, Proposition 2.1]). Then one can pass to the limit, first for e ~ 0 and then for ~ ~ 0, to obtain a finite energy weak solution of the problem (5.3)- (5.5) (see [32]). The reason for introducing two parameters e and ~ is that the energy estimates presented in Section 5.1 and the pressure estimates in Section 5.2 are compatible only if fl > N. An alternative approach is the approximation scheme introduced by Lions [62] or the method of time-discretization based on solving a family of stationary problems (see also Lions [62]). 6. Barotropic flows: large data existence results The mathematical theory presented in Section 5 can be used to obtain rigorous existence results for barotropic flows with essentially no restriction on the size of the data. We start with a very particular case posed in two space dimension where one can show even existence of strong (classical) solutions. 6.1. Global existence of classical solutions The result we are going to present is due to Vaigant and Kazhikhov [107]. Consider the system Ot + div(0u) = 0, (6.1) O(ou) ~t + div(0u | u) + aVo• =/zAu + V(0~(O) + #)divu), (6,2) where (t,x) a (0, T) x R2. The functions 0, u are for simplicity considered spatially periodic, i.e., O(t, x 4- o9) = O(t, x), u(t, x 4- o9) = u(t, x). (6.3) The problem is complemented by the initial conditions 0(0, x) = Oo(x) ~>0 > 0, u(0, x) -- u0(x). (6.4)
- 344. Viscous and~orheat conducting compressible fluids 345 Under the hypotheses /z>O, a>O, y>~O, and )~(o)--bo ~, b>O, /3>~3, (6.5) Vaigant and Kazhikhov [107] proved the following result. THEOREM 6.1. In addition to the hypotheses (6.5), let 00 E LpC~r(R2), UO E Wle2(R2). Then the initial-value problem (6.1)-(6.4) possesses a global (T = oo) weak solution. The continuity equation (6.1) holds in 79f((0, T) • R2) and the equations of motion (6.2) are satisfied a.a. on (0, T) x R 2. If, moreover, 00 E Wlerq (R2), U0 E Wp2e q (e 2) for some q > 2, then there is a unique strong solution satisfying the equations a.e. on (0, T) • R2. Finally, if (7l+ce 00 E vper (R2) , (-72+ot u0 E vper (R 2) for some ot > O, then the strong solution is classical (smooth). Theorem 6.1 is a remarkable result since it solves both the problem of existence and uniqueness as well as regularity of solutions. The obvious restrictions of applicability are due to the rather unnatural hypotheses (6.5), i.e., the viscosity coefficient # must be constant while )~ depends on 0 in a very specific way. The proof of Theorem 6.1 is based on very strong a priori estimates - much better than presented in Sections 5.1, 5.2. These estimates are available thanks to the particular form of the constitutive relations and the fact the problem is posed in two space dimensions. 6.2. Global existence of weak solutions We consider the problem (5.3)-(5.5) posed on a bounded regular domain 12 C R N, N = 2, 3. We prescribe the initial conditions p(O) = Oo >/O, (pu)(O) = qo (6.6) satisfying a compatibility condition q0(x) = 0 whenever Q0(x) = 0 (6.7)
- 345. 346 E. Feireisl and such that 00 6 L•163 Iq~ 6 Ll(~). (6.8) 00 The assumption (6.8) is nothing else but the requirement the initial data to be of finite energy. The following theorem asserts the existence of the finite energy weak solutions to the problem (5.3)-(5.5), (6.6) introduced in Section 5. THEOREM 6.2. Let s C R N, N -- 2, 3, be a bounded regular domain and T > 0 given. Consider the system (5.3), (5.4) complemented by (5.5), (6.6), where p is given by the isentropic constitutive law (5.2) with N Y>2' f is a bounded measurable function on (0, T) x X-2,and the initial data 00, q0 satisfy (6.7), (6.8). Then the problem (5.3)-(5.5) posseses a finite energy weak solution O, u on (0, T) x s satisfying the initial conditions (6.6). As already remarked in (5.7), both the density 0 and the momenta (0u) are continuous functions of t with respect to the L P-weak topology, and, consequently, the initial conditions (6.6) make sense. The existence result stated in Theorem 6.2 was first proved by Lions [62] for y ~>3/2 if N = 2 and y ~>9/5 for N = 3. The proof needs some modifications presented in [39] and [63] to accommodate the Dirichlet boundary conditions. The present version including the full range of y > N/2 was shown in [32, Theorem 1.1]. Given the weak compactness results, namely Propositions 4.1, 4.2, for solutions of (5.3), (5.4) respectively, the main ingredient of the proof of Theorem 6.2 is the strong compactness of the density stated in Proposition 5.4, the proof of which requires, among other things, convexity of the pressure. It is easy to see, however, that the same result can be obtained for a general barotropic pressure p as in (5.1) that can be written in the form P(O) =ao • + po(O), a>0, y>N/2, where P0 is a globally Lipschitz function. Note that the proof for the range ?, ~> 9/5, N = 3, t' ~>3/2, N = 2 can be modified to include a general constitutive law (5.1) where P(O) ~ ao • for all 0 large enough (cf. Lions [62]). It seems interesting to note that the physically relevant isothermal case where y = 1 seems to be completely open even if N = 2. The only large data existence result is that of Hoff [46] where the initial data (as well as the solutions) are radially symmetric. The general case g ~> 1, N = 3 for radially symmetric data was solved only recently by Jiang and Zhang [55].
- 346. Viscous and~orheat conducting compressiblefluids 347 6.3. Time-periodic solutions Similarly as above, we consider the system (5.3)-(5.5) driven by a volume force f which is periodic in time, i.e., f is a bounded measurable vector function on R • I2 satisfying f(t + co, x) = f(t, x) for a.a. t 6 R, x E s for a certain period co > 0. We are interested in the existence of a finite energy weak solution O, u enjoying the same property, i.e., O(t + co) = 0(t), (0u)(t + co) = (0u)(t) for all t E R and such that ]~0dx =m, where m is a given positive total mass. There are three main obstacles making this problem rather delicate. Given the existence results for the initial-boundary value problem presented above, only weak solutions are available, for which the question of uniqueness is highly nontrivial and far from being solved. This excludes all the so-called indirect methods based on fixed-point arguments for the corresponding period map. While the former difficulty might seem only technical, there is another feature of the problem, mentioned already in Section 5.4 namely, there is no "solution operator" or "period map" which would be compact due to possible time propagation of oscillations in the density. Last but not the least, fixing the total mass m, we have to look for solutions lying on a sphere in the space L 1 which excludes the possibility of using any fixed-point technique in a direct fashion. In the light of the above arguments, the only possibility to get positive results is to work directly in the space of periodic solutions that means to consider a genuine boundary-value problem for the evolutionary system (5.3), (5.4). This approach has been used in [31] to prove the existence of the time periodic solutions to (5.3), (5.4) on a cube in R3 complemented by the no-stick boundary conditions (1.12). Combining the method of [31] with the existence theory [32] one can prove the following result. THEOREM 6.3. Let ~ C R N, N = 2, 3, be a bounded regular domain. Consider the problem (5.3)-(5.5) where p is given by the isentropic constitutive law (5.2) with y>5/3 if N = 3, y>l for N = 2, and f is a bounded measurable function on R x I2 such that f(t + co, x) = f(t, x) for a.a. t E R, x E I-2 and a certain co > O.
- 347. 348 E. Feireisl Then, given m > 0, there exists a finite energy weak solution O, u of (5.3)-(5.5) on R x ~ such that O(t + co) = O(t), (Ou)(t + w) = (0u)(t) for all t ~ R and fs odx =m. The condition y > 5/3 in the three-dimensional case seems rather strange compared with y > 3/2 required for solving the initial-value problem. This is related to the problem of ultimate boundedness or resonance phenomena for global in time solutions. We will discuss this interesting topic in the next section. 6.4. Counter-examples to global existence It is not clear to which extent the hypothesis y > N/2 is really necessary for global existence results. Several attempts have been made to show that the barotropic model does not admit globally defined strong or even weak solutions but the results are still not very convincing in either positive or negative sense. Following the method of Vaigant [106], Desjardins [19, Proposition 1] studied the integrability properties of the density Q in the system (5.3)-(5.5). PROPOSITION 6.1. Let ~2 = B(1) C R 3 be a unit ball and let p satisfy (5.2) with 1 < y <3. Let q> lly -2 2y Then there exist f 6 L 1(0, T; L 7=-r(S2)) and a globally defined weak solution ~, u of (5.3)-(5.5) such that 1~(/, x)I q dx dt = cx~. The weakness of this result stems from the necessity to use the forcing term f which is singular at t = T. It is still an open problem whether or not the uniform upper bounds on the density can be obtained independently of the choice of y. 6.5. Possible generalization We shall comment shortly on possible improvements of Theorem 6.1 lying in the scope of the present theory.
- 348. Viscous and~orheat conducting compressiblefluids 349 To begin with, Theorem 6.1 still holds when 12 is a general (not necessarily bounded) domain with compact boundary on which the no-slip boundary conditions for the velocity are prescribed. As far as the other boundary conditions discussed in Section 1.4 are concerned, the possibility to show positive existence results seems to be closely related to the question of the boundary estimates of the pressure discussed in Section 5.2. Similarly, the hypothesis that f is bounded can be replaced by a more general condition 2V f 6 Ll(I; L• (l-2)). Other possibilities and suggestions are discussed by Lions [62]. 7. Barotropic flows: asymptotic properties Similarly as in the preceding section, we focus on the system (5.2)-(5.5) considered on a bounded regular domain S-2C R N, N = 2, 3. We shall assume that the driving force f is a bounded measurable function defined, for simplicity, for all t 6 R, x 6 12 such that [f(t,x)[~<F fora.a, t6R, x6S-2. (7.1) In accordance with Section 5, the total energy defined as fo 1 IQul 2 S[O, 0u](t) - (t)>0 2 0 a - ~(t) + 0 • (t) dx y-1 is a lower-semicontinuous function of t. 7.1. Bounded absorbing balls and stationary solutions We shall address the problem of ultimate boundedness of global in time finite energy weak solutions, the existence of which is guaranteed by Theorem 6.2. We shall see that the total energy S is the right quantity to play the role of a "norm" in these considerations. If the driving force f is uniformly bounded as in (7.1), the "dynamical system" generated by the finite energy weak solutions of the problem (5.3)-(5.5) is ultimately bounded or dissipative in the sense of Levinson with respect to the energy "norm" provided that the adiabatic constant satisfies 9/> 1 for N = 2 and y > 5/3 if N = 3. Specifically, we report the following result (see [38, Theorem 1.1]), the proof of which is based on the pressure estimates obtained in Proposition 5.2: THEOREM 7.1. Let 12 C R N, N = 2, 3, be a bounded Lipschitz domain and I C R an interval such that inf{I} > -o~. Consider the system (5.3)-(5.5) with the isentropic pressure p given by (5.2) with y > 1 ifN=2, y >5/3 forN=3, (7.2) and f satisfying (7.1).
- 349. 350 E. Feireisl Then there exists a constant Cc~, depending solely on the amplitude of the driving force F and the total mass m, with the following property: Given Eo, there exists a time T = T (Co) such that C[O, (Ou)](t) ~<Coc for all t E I, t > r + inf{I} for any O, u - a finite energy weak solution of the problem (5.3)-(5.5) - satisfying lim sup C[O,u](t) ~<Co, t~inf{1}+ fs2 Qdx - m. It seems interesting to compare Theorem 7.1 with the result of Lions [62, Theorem 6.7] on the existence of stationary solutions of (5.3)-(5.5) to shed some light on the role of the hypothesis (7.2). THEOREM 7.2. Let s C R N, N = 2, 3, be a bounded regular domain, f = f(x) a function belonging to Lc~(I-2), and m > O. Assume p = P(O) is given by (5.2) with y satisfying (7.2). Then there exists a pair of functions Q - Q(x) E LP(~), p > y, u = u(x) E wl'2(12) solving the stationary problem div(ou) = O, div(ou | u) + aVo • =/zAu + ()~+ #)V divu + of, f s2 Qdx = m in 79'(I2). As we will see later, Theorem 7.2 can be deduced from Proposition 5.4, Theorem 6.3, and Theorem 7.1. The property stated in Theorem 7.1 is evidence of the dissipative nature of the system (5.3), (5.4). In finite-dimensional setting, J.E. Billoti and J.P. LaSalle proposed it as a definition of dissipativity. Unfortunately, however, some difficulties inherent to infinite- dimensional dynamical systems make it, in that case, less appropriate. 7.2. Complete bounded trajectories ~(R;L ~ We suppose that the driving force f belongs to ~-- a bounded subset of Lloc (S'2)). To bypass the possible problem of non-uniqueness of finite energy weak solutions, we introduce a quantity U(t0, t) playing the role of the evolution operator related to the
- 350. Viscous and~or heat conducting compressible fluids 351 problem (5.3)-(5.5). [ U[s t) = [ [O(t), (Ou)(t)] [O, u is a finite energy ~](t0, weak solution of the problem (5.3)-(5.5) defined on an open interval I, (to, t]CI, with f E U and such that limsupg[o, u](t) ~<go}. t--->to+ We start with the concept of the so-called short trajectory in the spirit of M~ilek and Neeas [67]: US[Co, f'](to, t) = {[Q(t + v), (Qu)(t + v)], v e [0, 1]IQ, u is a finite energy weak solution of the problem (5.3)-(5.5) on an open interval I, (to, t + 1] C I, with f e ~-, and such that limsups ~<go[. t--+to / The following result can be viewed as a corollary of Proposition 5.4 and Theorem 7.1 (cf. [37, Theorem 1.1] or [27, Proposition 10]). PROPOSITION 7.1. Let $'2 C R N, N = 2, 3, be a bounded domain with Lipschitz boundary. Let the pressure p be given by (5.2) with y > l for N = 2, Y>5/3 if N = 3. Let .~ be bounded in L~ x $-2). Consider a sequence [On, (OnUn)] E US[Co, .T'](a, tn) for a certain tn --+ ec. Then there is a subsequence (not relabeled) such that On -+ 0 in L • ((0, 1) x S'2) and in C([O, 1]; L~(I-2)) for 1 <<.ot < y, ~o,,u~ ~ (Ou) 2y , L y+l 2V in LP((O, 1) • $2) andin C([0, 1]" weak(S2))forany 1 ~ p < -7~' and cO[On, (OnUn)] -+ s (Ou)] in L 1(0, 1), where Q, u is a finite energy weak solution of the problem (5.3)-(5.5) defined on the whole real line I = R such that s e L ~ (R) and with f e ~+ where U+ = {f If = lim hn (" + rn) weak star in L ~ (R • S-2) ~-n---->~ for a certain hn e ~ and rn --+ ec }.
- 351. 352 E. Feireisl Proposition 7.1 shows the importance of the complete bounded trajectories, i.e., the finite energy weak solutions defined on I = R whose total energy s is uniformly bounded on R. Let us define Asia] : {[~)(r), (•u)(r)], r ~ [0, 11 I~),u is a finite energy weak solution of the problem (5.3)-(5.5) on the interval I = R, with f 6 .T"+ and s (Qu)] E L ~ (R)}. The next statement is a straighforward consequence of Proposition 7.1 (see also [28, Theorem 3.1 ]). THEOREM 7.3. Let 12 C R N be a bounded Lipschitz domain. Let p be given by (5.2) with y > l for N = 2, y>5/3 if N = 3. Let ~ be a bounded subset of L e~(R x 12). Then the set As[f] is compactin L• 1) x 12) x [LP((0, 1) x 12)]3 and [ inf (11~ - sup ~ Ilt•215 [O,Ou]EU[s I_[~,~fi ]~.As [.T"] + II - 0 as t --+ oc for any 1 <<.p < 2y/(y + 1). Theorem 7.3 says that the set As[f] is a global attractor on the space of "short" trajectories. This is a result in the spirit of Mdlek and Ne~as [67] or Sell [89]. In particular, the set A s[,T'] is compact non-empty provided ~ is non-empty. Consider the special case when f = f(x) is a driving force independent of time. Accordingly, we can take y = ~+ = {f}. By virtue of Theorem 6.3, the problem (5.3)-(5.5) possesses a time-periodic solution ~)n, Un for any period ~On= 2-n such that f s2 Qn dx = m. Moreover, Theorem 7.1 implies that the restriction 0n, OnUn to the time interval [0, 1] belongs to .As . As .As is compact, the sequence On, un has an accummulation point which is a complete global solution of (5.3)-(5.5). Moreover, this solution is clearly independent of t, i.e., it is a stationary solution of a given total mass m. In other words, we have proved Theorem 7.2.
- 352. Viscousand~orheatconductingcompressiblefluids 353 7.3. Potentialflows We shall examine the flows driven by a potential force, i.e., we assume f = f(x) -- VF(x), where F is a Lipschitz continuous function. In this case, the term on the right-hand side of the energy inequality (5.6) can be rewritten as f~ d~ (Ou).VFdx= dt' where ~ (t) = fs~ 0 F dx, and, consequently, (5.6) reads as follows: d(g(t)dt - 7-[(t)) + fs~~lVul2 + (~ + 2/z)ldivu[2 dx dt ~<0. (7.3) We denote s = ess lim [s - 7-t(t)]. t---~~ By virtue of (7.3) and the Poincar6 inequality, the integral f~ Ilull2~,2(s~)dt is convergent, (7.4) in particular, fT T+I lim ~kin(t) dt= 0, T--+ oo and &in -- ~ 0[u] 2dx, f T+Ifs a Oz lim -- 0 F dx dt= gT-/oc. (7.5) r~ec y- 1 Similarly as in Proposition 7.1, one can show that any sequence tn ~ oo contains a subsequence such that ftntn+l II0(t) - 0, IIL• --+ 0, where, in view of (7.4), (7.5), Os is a solution of the stationary problem aVOZs = OsVF in X2, fs? 0s dx = m, f~ a y 1 Ozs - OsF dx = g~. (7.6)
- 353. 354 E. Feireisl Consequently, it is of interest to study the structure of the set of the static solutions, i.e., the solutions of the problem (7.6); in particular, whether or not they form a discrete set. If this is the case, any finite energy weak solution of (5.3)-(5.5) is convergent to a static state. A partial answer was obtained in the case of potentials with at most two "peaks" ([36, Theorem 1.1] and [33, Theorem 1.2]). THEOREM 7.4. Let I-2 C R N be an arbitrary domain. (i) Assume F is locally Lipschitz continuous on S2 and such that all the upper level sets [F > k] = {x 9 t"2 I F(x) > k} are connected in f2 for any k. Then given m > O, the problem (7.6) possesses at most one nonnegative solution Qs. (ii) If F is locally Lipschitz continuous and I-2 can be decomposed as ~Q -- ff21Uf22, /21N f22 -- ~, where if2 i are two subdomains (one of them possibly empty) so that [F > k] (q if2 i is connected in ~(2i for i = 1, 2for any k E R, (7.7) then, given m, s the problem (7.6) admits at most two distinct non-negative solutions. Making use of Theorem 7.4, one can show the following result on stabilization of global solutions for potential flows (cf. [34, Theorem 1.1], [27, Theorem 15]). THEOREM 7.5. Let S2 C R N, N = 2, 3, be a bounded Lipschitz domain. Let the pressure p satisfy the constitutive relation (5.2) with y > N/2. Let f = fix) = VF(x) where F is globally Lipschitz potential on S-2.Moreover, assume that S2 can be decomposed as in Theorem 7.4 so that (7.7) holds. Then for any finite energy weak solution O, u of the problem (5.3)-(5.5) defined on a time interval I = (to, oo), there exists a solution Os of the stationary problem (7.6) such that Q(t) --+ Qs strongly in L y (I2) as t --+ c~, fs2 1 fo I~u12 Ekin(t) dx-- ~ (t)>0 Q ~ (t) dx --+0 as t --+ cx~. The conclusion of Theorem 7.5 still holds if Y2 is a general (not necessarily bounded) domain with compact boundary provided F satisfies the stronger hypothesis of Theorem 7.4, namely, all upper level sets [F > k] must be connected. Similar problems on the exterior of an open ball and for radially symmetric solutions were investigated by Matu~fi-Ne~asovfi et al. [76]. Related results can be found in Novotn~ and Stra~kraba [83,84].
- 354. Viscous and~or heat conducting compressible fluids 355 It is an interesting open problem if the conclusion of Theorem 7.5 still holds when the hypothesis on the upper level sets of F is relaxed. If ~2 C R N is a bounded domain and the potential F nontrivial (nonconstant), there always exists an m - the total mass - small in comparison with F such that the solutions of the static problem (7.6) contain vacuum zones (cf. [34, Section 5]). Thus for any nonconstant F the global solutions approach rest states with vacuum regions as time goes to infinity. One should note in this context there are many formal results on convergence of isentropic flows to a stationary state under various hypotheses including uniform (in time) boundedness of the density away from zero (see, e.g., Padula [86]). As we have just observed, this can be rigorously verified only for solutions representing small perturbations of strictly positive rest states. 7.4. Highly oscillating external forces There seems to be a common belief that highly oscillating driving forces of zero integral mean do not influence the long-time dynamics of dissipative systems. Averaging a function over a short time interval should be considered analogous to making a macroscopic measurement in a physical experiment. The result of such an experiment being close to zero, the effect on the solutions to a sufficiently robust dynamical systems, if any, should be negligible at least in the long run. From the mathematical point of view, these ideas have been made precise by Chepyzhov and Vishik [10] dealing with trajectory attractors of evolution equations. They showed that the trajectory attractors of certain dissipative dynamical systems perturbed by a highly oscillating forcing term are the same as for the unperturbed system. Their results apply to a vast set of equations including the damped wave equations and the Navier-Stokes equations for incompressible fluids. Our goal now is to present comparable results for the problem of isentropic compressible flows dynamics. Highly oscillating sequences converge in the weak topology, i.e., the topology of convergence of integral means. Consider a ball Bc of radius G centered at zero in the space L ~ ((0, 1) x 12). The weak-star topology on BG is metrizable and we denote the corresponding metric de. We report the following result (see [30, Theorem 1.2]). THEOREM 7.6. Assume S-2 C R N, N = 2, 3 is a bounded Lipschitz domain. Consider the system (5.3)-(5.5) where the pressure p is given by (5.2) with Y> 1 for N = 2, y>5/3 if N = 3, and f(t, x) = VF(x) + g(t, x), where F is globally Lipschitz continuous and such that the upper level sets [F > k] are connected for any k. Then given G > O, e > 0 there exists 6 = 6(G, e) > 0 such that limsup[ll0(t)- QsIlL• [Iou(t)II,, s2 ] < l----~OO
- 355. 356 E. Feireisl for any finite energy weak solution Q, u of the problem (5.3)-(5.5) provided lim sup I[g(t)1[ L~(~, ~)• < G, /---+OO lim sup dG[g(t + s)ls~[0,1], 0] < 6. l---->OO Here Qs is the unique solution of the stationary problem (7.6). 7.5. Attractors For a general dynamical system a set ,A is called a global attractor if it is compact, attracting all trajectories, and minimal in the sense of inclusion in the class of sets having the first two properties. The theory of attractors for incompressible flows is well developed. We refer the reader to the monographs of Babin and Vishik [5], Hale [44], and Temam [104] for this interesting subject. A global or universal attractor describes all possible dynamics of a given system, and, as an aspect of dissipativity, the attractor usually has a finite fractal dimension. There seems to be at least one essential problem to develop a sensible dynamical systems theory for compressible fluids, namely, the finite energy weak solutions we deal with are not known to be uniquely determined by the initial data. On the other hand, the notion of global attractor itself does not require uniqueness or even the existence of a "solution semigroup" and plausible results in this respect can be obtained. Let r = {[Q(0), (Qu)(0)] IQ, u is a finite energy weak solution of the problem (5.3)-(5.5) on I = R with f e 9 r+ and s E L~ (R) }. Roughly speaking, the set .A contains all global and globally bounded trajectories where global means defined on the whole time axis R. The next statement shows that ,A[~] is a global attractor in the sense of Foias and Temam [40] (cf. [28, Theorem 4.1]). THEOREM 7.7. Let ~ C R N, N = 2, 3, be a bounded Lipschitz domain, and let p be given by (5.2) with y > l ifN=2, y>5/3 forN=3. Let F be a bounded subset of L ~ (R x S2). 2y L • (S2) and Then r v] is compact in Lc~(S2) x weak su, r [Q,Qu] E U [Eo,.T'] (to,t) Lm~fi],A[7] as t --+ cxz 2y for any 1 <~~ < y and any ~ e [L • (~)]3 0o) 1)]
- 356. Viscous and~orheat conducting compressible fluids 357 The apparent shortcomming of this result is that A is only a "weak" attractor with respect to the momentum component. Pursuing the idea of Ball [6], one can show a stronger result on condition that some additional smoothness of A is known (see [27, Theorem 17]). THEOREM 7.8. In addition to the hypotheses of Theorem 7.7, assume the total energy C defined by (5.9) and considered as a function the density O and the momenta Ou is (sequentially) continuous on A[F], specifically, for any sequence [On, qn] e .A[f] such that On --~ 0 in Ll(I2), qn --~ q weakly in Ll(S-2) one requires C[On, qn] ~ E[O, q]. Then [ sup inf (110 - ~llt~(~> + II0u- ~llt'(~))] 0 [O,Ou]EU[Co,F](to,t) t.[~,~fi ]E.A[.T'] as t --+ cx~. 7.6. Bibliographical remarks The existence of global attractors for the problem (5.3)-(5.5) with 9/= 1 and N = 1 was studied by Hoff and Ziane [50,51 ]. In this case, any forcing term f is of potential type so the only situation which is not covered by Section 7.3 is the case when f is time dependent. Similar results for the full system (1.1)-(1.3), still in one space dimension, were obtained recently by Zheng and Qin [112]. 8. Compressible-incompressible limits It is well-accepted in fluid mechanics that one can derive formally incompressible models as Navier-Stokes equations from compressible ones. Such a situation can be expected when letting the Mach number go to zero in the isentropic compressible Navier-Stokes equations. Following Lions and Masmoudi [64] we consider a system OOe Ot + div(oEue) = 0, (8.1) 80~ue Ot a + div(0eue | ue) + ~SV0ff = #eAue + ()~e+ #e)V divue (8.2) complemented by the initial conditions o~(o) - o ~ o, (o~u~)(o) - q~ (8.3)
- 357. 358 E. Feireisl satisfying (6.7). We shall always assume /zs --~ # > O, Xs --~ I. > -# as s --~ O. 8.1. The spatially periodic case In addition to the above hypotheses, assume the initial data are spatially periodic as in (6.3). Moreover, let qs U0 weakly in 2 N -- Lper(R ) as s ~ 0, (8.4) and fo~Ol.., f~ON Iq~[2 oo 1 O(mO)z_l (mO)• -t- ~-~[(0~ • - YOs + (y - 1) dx ~<c (8.5) where 0 m e -- (.Oi -1 f0 )1 f0 ~ 9.. o~ 1 as s--+ 0 independently of s. Let us denote, as usual, the total energy f{o 1 lOsUs 12 a nt- 62(y -- 1) (0s)• dx. The following result is due to Lions and Masmoudi [64]" THEOREM 8.1. Assume y > N/2. Let Os, us be a (spatially periodic) finite energy weak solution of the problem (8.1)-(8.3) on the time interval (0, r where the data satisfy (8.4), (8.5). Moreover, let f0 O)1 f00)N1Iqs 12 ess sup g[0s, (0sus)] ~< ... t--+0+ 2 0~ a - 1) (o~ Then 0s ~ 1 in C([0, T]; L~er(R2)) and us is bounded in L2(0, T; Wpl~2(y2)) for arbitrary T > O.
- 358. Viscous and~or heat conducting compressible fluids 359 Moreover, passing to a subsequence as the case may be we have ue ~ U weakly in L2(0, T; Wple2(R2)), where U solves the incompressible Navier-Stokes equations OtU + div(U | U) =/zAU + V P, div U = 0 (8.6) with the initial condition U(0) = 79U0 where 79 is theprojection on the space of divergence- free functions. 8.2. Dirichlet boundary conditions Now we focus on the system (8.1)-(8.3) posed on a bounded domain S-2 C R N and complemented by the no-slip boundary conditions for the velocity: u~10~ =0. (8.7) Consider the following (overdetermined) problem: -Ar = v4~ in S-2, V~ 9nlas2 = 0, 45 constant on 012. (8.8) A solution of (8.8) is trivial if v = 0 and 45 is a constant. The domain 12 will be said to satisfy condition (H) if all solutions of (8.8) are trivial. The following result was proved by Desjardins et al. [20]: THEOREM 8.2. Let 1-2 C R N, N = 2, 3, be a bounded regular domain. In addition to the hypotheses of Theorem 8.1, assume that ue satisfies the no-slip condition (8.7). Then Qe tends to 1 strongly in C([0, T]; L• and, passing to a subsequence if necessary, ue --~ U weakly in L 2((0, T) x $2) for all T > 0 and the convergence is strong if 1-2 satisfies condition (H). In addition, U satisfies the incompressible Navier-Stokes system (8.6) complemented by the no-slip boundary conditions on 012 and with U(0) -- 79U0. 8.3. The case Fn ~ cx~ Let us consider the isentropic system in the case when F -- Fn -+ ~. We follow the presentation of Lions and Masmoudi [65]. Let 12 C R 3 be a bounded regular domain. Consider the system OtOn + div(OnUn) = 0, (8.9)
- 359. 360 E. Feireisl Ot(OnUn) "+"div(OnUn | Un) q'-aVOZn" =/ZAUn -k-()~ -q-/z)V divun (8.10) with the no-slip boundary conditions for the velocity u,, I~s2 =0 (8.11) and complemented by the initial conditions o (o) = o~ >I o, (OnUn)(0) : qn, (8.12) where Iq.I 2 Io~ L• (~) <<-Ctn, On bounded in L 1(~), Oo bounded in L 1(if2), (8.13) independently of n. We are interested in the limit of the sequence On, Un of finite energy weak solutions of the problem (8.9)-(8.12) when Yn ~ c~. To this end, let us first formulate the limit problem: Otto "+-div(0u) = 0, 0 ~<0 ~< 1, (8.14) 0t(0u) + div(0u | u) + V79 =/zAu + (~. +/z)V divu, (8.15) div u = 0 a.a. on the set {0 = 1}, (8.16) 79=0 a.a. on{o<l}, 79~>0 a.a. on{o=l}. (8.17) The following result is due to Lions and Masmoudi [65]. THEOREM 8.3. Let ~ C R 3 be a bounded regular domain. Let ~On,Un be a sequence of finite energy weak solutions of the problem (8.9)-(8.12) on (0, T) x I-2 where the data satisfy (8.13). Let 0~ converge weakly to some 0 ~ and qn converge weakly to q. Then On, Un contain subsequences such that [On - 11+ --~ 0 in Lc~(O, T; L~(I2))forany 1 <~~ < c~, On ~ ~0 weakly star in L ~ (0, T; L ~(12)), 1 <<.ot < cx~, where 0~<0~<1. Moreover, OZn n is bounded in LI((0, T) • 12)and OZn " --+ 79 weakly star in M ((0, T) x ~).
- 360. Viscous and~or heat conducting compressible fluids 361 If in addition, Qo __+Qo strongly in L1 (S2), then Q, u, 79 solve theproblem (8.14)-(8.17) in 79'((0, T) x I2) where u is a weak limit of un in L 2(0, T, Wo '2(I2)). Here A//denotes the space of Radon measures. 9. Other topics, directions, alternative models 9.1. Models in one space dimension In the above analysis, we have systematicaly and deliberately avoided the case of one space dimension. Note that for compressible fluids such a situation can be physically relevant as well as interesting. From the mathematical point of view, these problems exhibit a rather different character due to the particularly simple topological structure of the underlying spatial domain. The question of global existence is largely settled in the case of one space dimension. The basic result in this direction is that of Kazhikhov [56], a more extensive material can be found in the monograph of Antontsev et al. [4]. The discontinuous (weak) solutions were studied by Hoff [45], Serre [90,91] and Shelukhin [95]. The results are quite satisfactory with respect to the criteria of well-posedness discussed in Section 2. Jiang [54] proved global existence for the full system in one space dimension when the viscosity coefficients depend on the density. Probably the most general result as well as an extensive list of relevant literature is contained in the recent paper by Amosov [2]. There is a vast amount of literature concerning the qualitative properties of solutions. Stragkraba [98,99], Stra~kraba and Valli [100] and Zlotnik [113] studied the long time- behaviour of global solutions in the barotropic case driven by a nonzero external force. Similar results for the full system were obtained in [35]. More information can be found in Amosov and Zlotnik [3], Hsiao and Luo [53], Matsumura and Yanagi [74] and many others. A complete list of references goes beyond the scope of the present paper. 9.2. Multi-dimensional diffusion waves A more detailed description of the long-time behaviour for the barotropic case in several space dimensions was obtained by Hoff and Zumbrun [52]. Following their presentation we consider the system (5.3), (5.4) with f = 0 on the whole space 12 = R3. The initial data Q(O) = po, (Qu)(O) = qo (9.1) are smooth and close to the constant state Q* -- 1, q0 = 0. Under these circumstances, the problem (5.3), (5.4) admits a global solution and the following theorem holds.
- 361. 362 E. Feireisl THEOREM 9.1. Assume that the initial data satisfy Iloo - lllt~nwl+d,2(R35+ Ilqollt~nw~+d,2(R3) < e, where e > 0 is sufficiently small and d ~ 3 is an integer Then the initial-value problem (5.3), (5.4), (9.1) possesses a global solution 0, u satisfying II~x~<~<t>- 1>11~<~ + II~x~<~u><t>ll~<~ c(d)e [ (1 + t) -r~,p for 2 <. p <. oo, <. I (1 -+-t) -r~,p+I/p-1/2 if l <. p < 2 for any multi-index loci ~ (d - 3)/2 where ra,p --1~1/2 + 3/2(1 - 1/p). Theorem 9.1 shows that perturbations of the constant state decay at the rate of a heat kernel for p >~2 but less rapidly if p < 2; in fact, the bound may even grow with time in the latter case. A more detailed picture of the long-time behaviour in the LP-norm for p ~>2 is provided by the following result. THEOREM 9.2. Under the assumptions of Theorem 9.1, we have II~x~<O- 1)(t) II/~p<~ + II~x~(<ou><t>- ~# 9 [Pqo]) IILp<~) <<.c(d)e(1 + t) -r~,p+l/p-1/2, p >/2, where P is the projection on the space of divergence free functions and 1Ca is the standard heat kernel, i.e., the fundamental solution of the problem 3tV -- lzAv = O. Thus the dynamics in LP, p > 2 is dominated by a term with constant density and a non- constant divergence-free momentum field decaying at the rate of the heat kernel. In other words, for p > 2, all smooth, small amplitude solutions are asymptotically incompressible. Finally, consider an auxiliary problem: 1 OtQ + div(ou) - ~()~ + 2/z)A0, d.1, 1 Ot(• + p'(1)VQ -/zA(Qu) H 2)~V div(ou). (9.2) Let us denote u(t) = [o(t), (ou)(t)]
- 362. Viscous and~or heat conducting compressible fluids 363 the solution of the linear problem (9.2) with the initial data 0(0) = Oo - 1, (Ou)(O) = qo. The long-time dynamics in LP, p < 2, is described as follows. THEOREM 9.3. Under the hypotheses of Theorem 9.1, we have IIa~([o(t)- 1, (Ou)(t)]- U(t))I]LP(R3 ) <, c(1, o-)e(1 + t) -r~'p+3/4(2/p-1)-l/2+~ l~<p<2, for any positive or. All results in this part are taken over from [52]. 9.3. Energy decay of solutions on unbounded domains Various authors have considered the long-time behaviour of solutions on unbounded domains. Following Kobayashi and Shibata [58] we consider the full system (2.1)-(2.3)on an exterior domain ~2 C R3 where the pressure p = P(O, O) is given by a general constitutive law conform with the basic thermodynamical principles expressed in (1.4). As for the boundary conditions, we take ulas~ -0, Olas~ --Ob, lim u(t,x)=0, lim O(t,x)=Ob. Ixl~ Ixl~ Assuming the initial data o(O) = oo, u(O) = uo, 0(0) = Oo are closed to a constant state [~, 0, 0b], Kobayashi and Shibata [58, Theorem 2] show the following decay rates: I1~(')- ~IIL~(~)+ II"(t)II L~(~)+ Ilo(t)- Obll~(~) ~ C' -~/~, II~(')-~1 L~(~ + II"(t)II L~(~> + Ilo(t)- ObllL~(~)~ C' -~/~ Related results were obtained by Deckelnick [17], Kobayashi [57], Padula [87] and many others.
- 363. 364 E. Feireisl 9.4. Alternative models Up to now, we have considered only Newtonian fluids where the viscous stress tensor 27 was a linear function of the velocity gradient Vu. However, some experimental results show that in nature there exist stronger dissipative mechanisms not captured by the classical Stokes law. Let us shortly discuss this interesting and rapidly developing area of modern mathematical physics which gives an alterantive and, given the enormous amount of open problems in the classical theory, mathematically attractive way to describe the fluid motion. In the linear theory of multipolarfluids, the constitutive laws, in particular, the viscous stress tensor 27 depend not only on the first spatial gradients of the velocity field u but also on the higher order gradients up to order 2k - 1 for the so-called k-polar fluids. In the work of Neras and Silhav~ [82], an axiomatic theory of viscous multipolar fluids was developed in the framework of the theory of elastic non-viscous multipolar materials due to Green and Rivlin [43]. Accordingly, the viscous stress tensor Z: takes a general form: k-1 -- ~-~(--1)J(/~jAJ(Vu + (Vu)t)+ )~jAJ divuId) j=O + co(Vu + (Vu) t) + f divuld, (9.3) where, in the nonlinear component, /~ - ~(IVul, divu, det(Vu)), ~o: ~o(IVul, divu, det(Vu)). The existence of the so-called measure-valued solutions of the initial value problem for isothermal flows, i.e., for the system (2.1), (2.2) with 27 given as in (9.3) and the pressure satisfying p = rOoo, was proved by Matu~fi-Nerasov~i and Novotn3~ [75]. The weak solutions for linear multipolar fluids were obtained in a series of papers by Neras et al. [81,80]. Recently, new results concerning the so-called power-law fluids, i.e., when k = 1 in (9.3), were shown by Mamontov [69,70]. 10. Conclusion Despite the enormous progress during the last two decades, we still seem to be very far from a satisfactory rigorous mathematical theory of viscous compressible and/or heat conducting fluids. There are good an bad news according to the degree of complexity of the problems considered but we still wait, for instance, for a large data existence result for, say, the isothermal flow in two and three space dimensions. On the point of conclusion, let us discuss shortly the major mathematcal difficulties presently encountered.
- 364. Viscousand~orheatconductingcompressiblefluids 365 10.1. Local existence and uniqueness, small data results As we have seen in Section 2, the initial value problem for the full system (1.1)-(1.3) complemented by physically relevant constitutive relations admits a unique global in time classical solution. From the mathematical point of view, this is nothing less or more than to say that the linearized system is well-posed. This fact seems to be the primary criterion of applicability of any mathematical model. Indeed there is only a little to say should the linearized problem be ill-posed. However, there still can remain an essential gap between "linear" and "nonlinear" provided there is no dissipative meachanism present as it is the case for nonlinear hyperbolic systems. The possibility to construct classical though only "small" solutions reveals the dissipative character of the problem, namely, the effect of the diffusion terms present in the parabolic equations (1.2), (1.3). Another aspect of dissipativity is the existence of bounded absorbing sets discussed in Section 7.1 and the existence of global attractors mentioned in Section 7.5. Although we still do not know if the attractor has a finite fractal dimension, there are strong indications (cf. Hoff and Ziane [50]) it might be the case. 10.2. Density estimates Unlike (1.2), (1.3), the continuity equation (1.1) governing the time evolution of the density is hyperbolic and linear with respect to Q. As a consequence, one cannot expect any smoothing effect as for parabolic problems or compactification phenomena as it is the case for genuinely nonlinear hyperbolic equations. We have made it clear several times in this paper that the major obstacle to develop a rigorous large data theory for our problem is the lack of a priori estimates on the density Q. The density being a non-negative function there are two aspects of the problem- boundedness from below away from zero and uniform upper bounds. Let us remark that the system (1.1)-(1.3) and, in particular, the constitutive relations for Newtonian fluids hold for nondilute fluids with no vacuum zones. Let us review the results of Desjardins [18] illuminating the role of upper bounds on Q in the well-posedness problem. Consider the isentropic model represented by the system (5.3), (5.4) where the pressure p satisfies (5.2) with y > 1. For simplicity, we consider the case of spatially periodic boundary conditions in two space dimensions. The following result is proved by Desjardins [18, Theorem 2]. THEOREM 10.1. Consider the system (5.3), (5.4) where P(0)=a0 • a>0, Y >1, in two space dimensions and with spatially periodic data Q(0) /> 0 E L~er(R2), u(O) E wpler 2(R2), f~ 0.
- 365. 366 E. Feireisl Then there exists To > 0 and a weak solution O, u of the problem such thatfor all T < To 0 E g ~ (0, T; Lp~er(R2)) and X//-O0tUE t2(0, T; Lp2er(R2)), p -()~ + 2#) divu E L2(0, T; W~er2(R2)), Vtt E Lee(0, T; Lp2er(R2)), vu L2(o,r; %2r (R2)), where 79 denotes the projection on the space of divergence free functions. Moreover, the regularity properties stated above hold as long as ii ii sup ]]O(t) llL~ < (X). tE[0,T] Desjardins [18] proved also that the weak solutions constructed in Theorem 10.1 enjoy the weak-strong uniqueness property well-known from the theory of incompressible flows. Specifically, the above weak solution coincides with a strong one as long as the latter exists (cf. [18, Theorem 3]). The lower bounds on the density represent an equally delicate issue. As we have seen in Section 7.3, one cannot avoid vacuum states provided we accept the isentropic model as an adequate description for the long time behaviour of solutions. On the other hand, the density should remain strictly positive for any finite time t provided its initial distribution enjoys this property. Unfortunately, however, this is not known in the class of weak solutions provided N ~>2. To reveal the pathological character of the problem when vacua are present, we follow Liu et al. [66] and consider the isentropic model in one space dimension where the initial distribution of the density is given as o(O) = Oo(x - 2r) + Oo(x + 2r), where O0 is a compactly supported smooth function with support contained in the ball {Ix l < r }. Should the model correspond to physical intuition, one would expect, at least on a short time interval, the solution to be given as O(t, x) = ~(t, x - 2r) + ~(t, x + 2r), u(t, x) = fi(t, x - 2r) + fi(t, x + 2r), where ~, fi solve the problem for the initial data ~(0) = O0. However, as shown in [66], this is not the case. Of course, this apparent difficulty is due to the discrepancy between
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- 371. CHAPTER 4 Dynamic Flows with Liquid/Vapor Phase Transitions Haitao Fan Department ofMathematics, Georgetown University, WashingtonDC 20057, USA and Marshall Slemrod Department ofMathematics, Universityof Wisconsin-Madison, Madison, W153706, USA Contents 1. Introduction .................................................. 375 2. The equations of motion ............................................ 378 3. Initial value problems of the inviscid system (1.3) and admissibility criteria ............... 380 4. Existence of solutions of the Riemann problem (3.1) ............................ 391 4.1. Existence of solutions of the Riemann problem (1.3) ......................... 392 4.2. A-priori estimates ............................................ 401 4.3. Solutions constructed by vanishing similarity viscosity are also admissible by traveling wave criterion .................................................. 409 References ..................................................... 417 HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME I Edited by S.J. Friedlander and D. Serre 9 2002 Elsevier Science B.V. All rights reserved 373
- 372. Preface The motion of fluids has intrigued scientists since antiquity and we may say that the field of mathematical fluid dynamics originated more than two centuries ago. In 1755 Euler [2] gave a mathematical formulation of the principle of conservation of mass in terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear and angular momentum by PDEs. Material symmetry and frame invariance were used by Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint- Venant [11] and Stokes [12]. In the 19th century no sharp distinction was drawn between mathematicians and physicists as we sometime see in more recent times. The formulation of the equations of fluid motion could be considered as either mathematics or physics. The first work in fluid dynamics that has a "modern" mathematical flavor may have been done by Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous problem. Riemann recognized the mathematical nature of the entropy. This notion led him to his duality method for solving the non-characteristic Cauchy problem for linear hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his contemporaries. What we now call the Cauchy problem for a PDE and the search for its solution did not have the significance that it is accorded nowadays. Even Poincar6 did not raise that kind of question in his Th~orie des tourbillons. For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28. Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much as Riemann, Leray developed new mathematical tools which proved to have independent interest: e.g., weak solutions (that we now call Leray's solutions in this context) and topological degree (a joint work with Schauder [4]). Since the 1930s, the interest that mathematicians devote to fluid dynamics has unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic displays such a variety of models and questions that thousands of scientists, among them many mathematicians, focus their research on fluid dynamics.
- 373. Dynamicflowswithliquid~vaporphasetransitions 375 1. Introduction The purpose of this paper is to review some recent results on Navier-Stokes equations with van der Waals type constitutive relation for the pressure: Ow Ou = (conservation of mass), Ot Ox Ou 0 { Ou 62Aa2W} -a~ - ~ -p~w, o~ + ~-~ - (conservation of linear momentum), = u -p+e---e2A at ax ax ~x 2 _+_e2A(aU aW) aO } ~xx~ + x e ~xx (conservation of energy), (1.1) where w is the specific volume, u the fluid velocity, 0 the fluid temperature, s the viscosity, K the coefficient of thermal conductivity, A an assumed constant capillarity coefficient. The constitutive relations for the the pressure p and the specific entropy r/are derived from the thermodynamic relationship ~f Ow ~f r/-- 00' where f (w, O) is the specific free energy given by the relationship f -- RO ln(w - b) - a/w + F(O), where F is in general an arbitrary function of O. Since the specific internal energy e must satisfy e-f +O0, it follows that e---a/w+ F(O)-OF'(O). As a simplifying assumption, we set F (0) -- -cvO In 0 + some constant,
- 374. 376 H. Fan and M. Slemrod where co is the assumed constant specific heat at constant volume. Then the specific internal energy e is given by e = -a/w = cvO + constant and the specific total energy E in (1.1) is 2/2. E = u2/2 -k- e + Aw x That is, the total energy is the sum of contributions from kinetic energy, internal energy, and interfacial energy. The isothermal version of (1.1) is clearly Ow Ou Ot Ox ' Ou mO{ at = Ox -p(w) + s-- - OU 62A 02113 Ox -ff~x2'l (1.2) while its inviscid case is covered by Ow Ou Ot Ox =0, Ou 0 + ----p(w) = o. O---f Ox - - (1.3) The constitutive equation for a van der Waals fluid at fixed temperature below the critical temperature 8a Oc -- 27Rb has the shape depicted in Figure 1. In the isothermal case, we are interested in subcritical temperatures and hence assume p E C 1(R) and p'(w) <0 ifw ~t [a, fl], p'(w)>O ifw~(a, fl). (1.4) The regions w < ot and (w > r) correspond, for van der Waals fluids, liquid and vapor phase region respectively. The line joining (m, p(m, T)) and (M, p(M, T)) is called Maxwell line where two equilibrium phases can coexist. The region ot < w < fl is called the spinodal region. If the fluid ever enters the spinodal region, the fluid will quickly decompose to liquid or vapor or their mixture. In other words, the spinodal region is a highly unstable region. To see this intuitively, we consider a ball of such fluid with w in spinodal region, see Figure 2. Pressures in the ball and its surrounding medium are set
- 375. Dynamicflows with liquid~vaporphase transitions 377 X" "y m o~ ~ M Fig. 1. W wo, p-- p(wo) Fig. 2. If the fluid inside the ball is in the spinodal region, w* 6 (or,/3) then the ball is unstable. equal, so that the system is in equilibrium mechanically. We perturb the fluid in the ball by decreasing the pressure of the surrounding medium a little bit. Then the fluid inside the ball will expand. If the fluid inside the ball is regular, in the sense that an increase in the volume results in the decrease in pressure, the ball will expand a little bit and the pressure inside the ball will drop to the level of that of the surrounding medium and the system will settle down to a new equilibrium close to the one before perturbation. However, when the liquid inside the ball is in spinodal region, such a little increase in w will result in an increase in pressure in the ball and hence the ball will further expand. Beside the instability in the spinodal region, there is another phenomena associated to phase transitions in a typical van der Waals type: metastability. For example, suppose vapor is initially set at rest, and we start to compress it with some w > M. When we reach w = M, the vapor should start to condensate in an ideal equilibrium world. But in the real world, the condensation will not start until we continue to compress so that the vapor enters into the region/3 < w < M. The vapor in this region can stay as vapor for long time until enough many nuclei of liquid are created and then rapid condensation takes place.
- 376. 378 H. FanandM. Slemrod Systems (1.1), (1.2), (1.3) coupled with (1.4) not only serve as prototype models for studying the dynamics of phase transitions, but also are interesting mathematical objects in its own fight. For example, system (1.2) are of hyperbolic-elliptic mixed type with ot < w < 13as its elliptic region. It is well known that the initial value problems for elliptic systems are ill-posed. Systems of hyperbolic type have been extensively studied. The presence of both hyperbolic and elliptic region in (1.2) certainly leads to new phenomena and new issues. The isothermal system (1.2), having physical background and being one of the simplest systems of conservation laws of hyperbolic-elliptic mixed type, certainly qualifies to be a prototype model for studying such systems. In this paper, we shall review some recent results related to (1.1), (1.2), (1.3) with (1.4). Although we tried our best to cover as much related results as possible, it is possible that we missed some. The rest of this paper is arranged as follows: in Section 2, we derive Equation (1.1). In Section 3, we review some results on the initial value problem of (1.3) and related admissibility criteria. In Section 4, we recall the proof of the existence of solutions of (1.3) satisfying the traveling wave criterion via the vanishing similarity-viscosity approach. Although these results and proofs appeared in our earlier works, we present here a revised version which is more readable. 2. The equations of motion We consider the one-dimensional motion of fluid processing a free energy 62A(Ow) 2 f (w, O)= fo(w, O) + --~ -~x " (2.1) Here w is the specific volume, 0 the absolute temperature, A > 0 a constant, and x the Lagrangian coordinate. The term where e > 0 is a small parameter, is the specific interfacial energy introduced by Korteweg [52]. The graph of f0 as a function of w for fixed 0 will vary smoothly from a single well potential for 0 > 0crit to double well potential for 0 < 0crit. The 0crit is called the critical temperature. Discussions of such free energy formulations may be found in [3, 10,13,14,12,25,35,66,67,78]. The stress corresponding to the free energy (2.1) is given by Of Ofo (W, O) -- ,f,2AO2W (2.2) T- Ow = Ow Ox ---g"
- 377. Dynamicflowswithliquid~vaporphasetransitions 379 Note that there is no viscous force in (2.2). Addition of a viscous stress term gives us the stress of the form OU 02W T = -p(w, O) + e Ox eZA OX2 (2.3) suggested by Korteweg's theory of capillarity [52]. In (2.3), u(x, t) denotes the velocity of the fluid, e > 0 is the viscosity and p - Ofo/Ow is the pressure. The one-dimensional balance laws of mass and linear momentum are easily written down: Ow Ou = (mass balance), (2.4a) Ot Ox Ou OT = (linear momentum balance). (2.4b) Ot Ox The equation for balance of energy is more subtle. While a thorough examination of the energy equation appears in Dunn and Serrin [25] it is the conceptually simple approach of [32] we recall here. Let e(w, O) denote the internal energy. Felderhof's postulate is that the internal energy is influenced only by the component of internal stress r = au i.e., the balance of energy is given by -p(w,O) + s-~, De Ou Oh = r t , (2.4c) Ot Ox Ox where h is the heat flux. Unlike Equations (2.4a, b), Equation (2.4c) is not in divergence form. To alleviate this difficulty we consider the specific total energy E -~ -F e (w , O) + ---f- -~x made up the specific kinetic, internal, and interfacial energy. Now compute the time rate of change of E" DE Ot OU 2W 02W =u-- -Fet + e2A ~ ~ Ot Ox OxOt OT Ou 22W OU OX OX OX2 = U F T---- + eZA Ox Oh e2 O(OwOu) F~x+ A~x -~-xOx ' where we have used the relation T -- r - e 2 A 02w We easily see that the balance of energy o-T" can be written as OE 0 (uT)+eeA O (OuOw) Oh (2.5) at =0-7 +0-7
- 378. 380 H.FanandM.Slemrod Ou Ow represents the "interstitial working" [25]. For simplicity we constitute The term eZA ~ 00 where xe > 0 is the (assumed constant) thermal conductivity. h by Fourier's law: h - x e~ Then we may collect the balance laws and write them as Ow Ou = (mass), (2.6a) Ot Ox OU 0[ OU 021/3] Ot = O---X --p(w, O) + e O---X-- ezA ~X 2 (linear momentum), (2.6b) =~ u-p+e~-e2A Ot Ox / L Ox OxZJ + ~2A ~~ +~~ (energy). (2.6c) The isothermal case of (2.6) is Ow Ou = (2.7a) 0t 0x' Ou 0 { Ou e2A O2to} Ot = O---x -p(w) + e O---~- ~x 2 . (2.7b) 3. Initial value problems of the inviscid system (1.3) and admissibility criteria In this section, we recall recent results on the initial value problems of inviscid system (1.3). Most results on the initial value problem of (1.3) are on Riemann problems. The Riemann problem of (1.3) is the initial value problem tot -- Ux -- O, ut+p(W)x--O, (u-, v-l, (u(x, 0), v(x, 0)) = (u+, v+), ifx <0, if x>0. (3.1) Through the study of the Riemann problem, we gain understanding on the behavior of solutions of (1.3). Based on knowledge about solutions to Riemann problems, Glimm's scheme can be used to construct solutions of (1.3) for general initial data. Compare to the viscous system (1.2), the inviscid system (1.3), as an approximation of (1.2), offer the following advantages: the structure of solutions are clearer. Solutions of Riemann problems may be constructed by solving a few algebraic equations. However, these advantages come with a price to pay: solutions of initial value problems of the
- 379. Dynamic flows with liquid~vapor phase transitions 381 inviscid system (1.3) are usually weak solutions with jump discontinuities. Such solutions are nonunique unless further restrictions on weak solutions are applied. These restrictions are called admissibility criteria. The admissibility criterion should pick "good" solutions suitable for the problem under consideration: here we are considering phase transitions modeled by (1.2). The inviscid system (1.3) is used as an approximation of (1.2). Thus, to make solutions of (1.3) to mimic those of (1.2), it is natural to require that admissible solutions of (1.3) to be e --+ 0+ limits of solutions of (1.2) with the same initial value. This is called the vanishing viscosity criterion. However, enforcing the vanishing viscosity criterion is usually very difficult and expensive. For example, to implement the vanishing viscosity criterion, one have to be able to (a) prove that solutions of (1.3) satisfying the criterion exist and (b) verify whether a given solution of (1.3) satisfy the criterion or not. These tasks are usually very difficult and expensive. So far, the part (a) is carried out for strictly hyperbolic systems of conservation laws [11,24,23]. For the (b), some results and techniques are given in [34] for piece-wise smooth solutions, with small shocks, of stricly hyperbolic systems. Thus, simpler admissibility criteria are called for. An internal layer asymptotic analysis on solutions of the viscous system (1.2) indicates that jump discontinuities of solutions of (1.3) must have traveling wave profiles in order for the solution of (1.3) to approximate that of (1.2). Traveling waves of (1.2) are solutions which are functions of the form g(x - st), where the constant s is the speed of the traveling wave. The traveling wave equations corresponding to (1.1) with ~ = (x - st)/e, w = w(~), u = u(~), 0 =0(~) are dw d~ dl) A~ d~ dO K m d~ m - - - l } ~ -- --S2(tO -- 113_) -- p(to, O) -+- p(w_, 0_) - sv, -- --s{(e(w,O) -e(w_,O_)) s 2 Asv 2 - --(to - to_) 2 2 2 (w,u,O)(-~)=(w_,u_,O_), (w, u, 0)(+c~) - (w+, u+, 0+), p(w_,O_)(w-w_)}, (3.2) where s is the speed of the traveling wave. For the isothermal case (1.2), above becomes d//3 d~ dl) d~ -- --S2(tO -- l/3_) -+- p(w_) - p(w) - sv, (w, u)(-~) - (w_, u_), (w, u)(+~) = (w+, u+). (3.3)
- 380. P t I I 382 H. Fan and M. Slemrod w_ ~2(w_,~) w3(W_,~) ~ (w_,s) Fig. 3. ~--- W A shock solution of (1.3) [ (u+, w+), (u w)(x t)-- I ' ' I (u_, w_), if x-st >0, if x - st < 0, (3.4) where s is the speed of the shock, is said to have a traveling wave profile if the traveling wave equation (3.3) has a solution. This leads to the traveling wave admissibility criterion: Traveling wave criterion states that a shock (3.4) is admissible if the system of traveling wave equations (3.3) has a solution. When (3.3) has a solution, we also say that there is a connection between (w_, u_) and (w+, u+). If all singular points of a solution of (1.3) are jump discontinuities and these jump discontinuities are admissible by the traveling wave criterion, we say that the solution is admissible by the traveling wave criterion. The solvability of the traveling wave equation (3.3). We are particularly interested in the case w_ < or, w+ >/~, since this data involves phase changes. Indeed, solutions of Riemann problems of (1.3) cannot take values inside the spinodal region (or,/~) and hence must have a shock jumping over the spinodal region, at least for the case A >~ 1/4 [27, Lemma 2.3(i)]. The solvability of the connecting orbit problems (3.3) were studied by Slemrod [74-76] and Hagan and Slemrod [40], Hagan and Serrin [39] and Shearer [70- 72]. Let w_ ~<ot and s >~0. For simplicity, we assume the ray starting from (w_, p(w_)), with slope -s 2, to the right can intersect the graph of p at most at three points (cf. Figure 3). We denote the w-coordinates of these points by w2(w-,s), w3(w-,s) and w4(w-,s),
- 381. Dynamicflows with liquid~vaporphase transitions 383 respectively. Points w_ and wk(w_, s), k = 2, 3, 4, are equilibrium points of (3.2). w_ and w3 (w_, s) are saddle points of (3.2) while w4(w-, s) is a node of (3.2). By [27], Riemann solvers of (1.3) cannot have values in the spinodal region (or, 13) at least for the case A >~ 1/4, thus, traveling waves connecting w_ and wz(w-, s) is ofno use in this case. Now we consider the existence of a solution of (3.2) connecting w_ and w3(w-, s), i.e., w(-oo) = w_, w(+oo) = w3(w_,s). For w_ E [y,m], if there is a g ~> 0 such that the signed area between the graph of p and the chord connecting (w_, p(w_)) and (w3(w_, s), p(w3(w_, s))) is 0 (cf. Figure 3), then there is a speed s*/> 0 such that 0 <~s* ~<g (3.5) and the problem (3.2) with s = s*, W2 = W3(W-,S*) has a solution, which satisfies ~ (~) > 0 and is a saddle-saddle connection, i.e., 0 ~<s* < v/-P'(W_), s* < v/-P'(W3(W_, s*)). (3.6) We note that this saddle-saddle connection accounts for the usual liquid-vapor phase transitions, including the coexistence of two phase equilibria in the case s = 0. In (3.4), equality holds if and only if g = 0. Furthermore, for any 0 < s < s* the trajectory of (3.2) emanating from (w_, 0) will overshoot w3 (w_, s) and flow to (w4 (w_, s), 0) as ~ --+ cx~. In other words, for all w2 > w4(w-, s*), there is a traveling wave solution of (3.2). Furthermore, this traveling wave solution is a saddle-node connection, i.e., v/--p'(w_) > s > v/--p'(w4(w_,s)). (3.7) These statements were proved in Hagan and Slemrod's paper [40]. Grinfeld [36] proved that if g exists, then for any N 6 Z+, there is a number AN > 0 such that for all A ~> AN, the system (3.2) has N saddle-saddle connection solutions, wj (~), j = O, 1, 2..... N - 1, such that wj (~) intersects w~= 0-axis traversely j times. If g does not exist but so := min(s: w3(s, w_))> O, then there is at least one saddle-saddle connection for all A > 0. If p(w) further satisfies p"(w) (w - w0) > 0 for w 5~ w0, (3.8) for some w0 E (c~,fl) then, for Y ~< w_ ~< m, there is a unique speed s* /> 0 such that w_ can be connected to w3(w_,s*) by a traveling wave solution of (3.2) with w2 = w3 (w_, s*), which is a saddle-saddle connection [69-71 ]. We notice that when (3.8) holds there is no w4(w_, s) for w_ 6 (y, m]. In fact, uniqueness of s* holds for all w_ < ~. In the above paragraph, w_ is fixed. However, if we fix w+ = w3 (w_, s), there can be two w_ < ot such that there are connections with s > 0 between w_ and w3 [8].
- 382. 384 H. Fan and M. Slemrod We note when p(w) is a cubic polynomial, we can have an explicit solution for (3.2). Let m+M) p(w) = Po - Pl (w - m)(w - M) w 2 ' (3.9) where m and M are Maxwell constants. Then a solution of (3.2) is (cf. [79,80]) w_-q- w+ w+-- w_ tanh(v/pl w+-w_ ) w(~) = 2 -I- 2 2A 2 (~ - ~o) 9 (3.10) For each w_ fixed, w+ in (2.11) is determined by equations: 3(1 -6A)(2y - z + 1)2 -I-Z2 -- 1, OSy=(m-w+)/(m-m), z--(w+-w_)/(m-m). (3.11) The number of solutions of (3.11) ranges from zero to two. When (3.11) has two solutions, we get two solutions of (3.2) of the form (3.10); one of them has positive speed and the other negative. This is, of course, consistent with Theorem 3.1. In fact, the nonuniqueness of traveling waves connecting a fixed w3 to some w_ is true in general [8]. In addition, Grinfeld [37] and Mischaikow [59] conducted studies on the full sys- tem (2.6) using Conley's index theory. Stability of traveling waves is an important topic for (1.2). In fact, having a stable or metastable shock profile, which is a traveling wave solution of (1.2), is a necessary condition for the shock of (1.3) to be admissible. Hoff and Khodja [47] proved the dynamic stability of certain steady-state solutions of the Navier-Stokes equations for compressible van der Waals fluids Vt -- ttx = 0, ut + p(v e)x = [e(X)Ux/V] X ~ (u2/2 + e)t + [up(v, e)]x -- [e(x)uux/V + 1.(x)T(v, e)x/V]x. (3.12) The steady-state solutions consist of two constant states, corresponding to different phases, separated by a convecting phase boundary. They showed that such solutions are non- linearly stable in the sense that, for nearby, perturbed initial data, the Navier-Stokes system has a global solution that tends to the steady-state solution uniformly as time goes to infinity. Benzoni-Gavage [7] studied the linear stability of planar phase jumps satisfying the traveling wave criterion (3.3) in Eulerian coordinates with viscosity neglected, called capilarity admissibility criterion. She showed that the such phase boundaries are linearly stable. Although neglecting the viscosity is unphysical, such a result served as the base from which she studied the case when the viscosity is small [9] to yield similar results.
- 383. Dynamic flows with liquid~vapor phase transitions 385 Zumbrun [85] proved the linear stability of slow heteroclinic traveling waves of (2.7) under localized perturbation. He also showed that homoclinic traveling waves near Maxwell line involving multiple phase transitions are exponentially unstable. This implies that the slow heteroclinic traveling waves of (2.7) are stable if they are monotone. The method used are spectrum analysis framework [33,86] and some energy estimates. Motion of phase boundary under perturbation and the effect of boundary conditions of (1.2) was studied by Chen and Wang [15]. The initial data is a perturbation of the stationary phase boundary, the Maxwell line. They found the ordinary differential equations describing the motion of the phase boundary under perturbation by an asymptotic expansion and a matching analysis. They conclude that the phase boundary will approach a well defined location as time goes to infinity. Existence of solutions of the Riemann problem for (1.3) satisfying the traveling wave criterion. One method for solving (3.1) is construction of wave and shock curves that are admissible according to some criteria and then construct a wave fan of centered waves and shocks that matches the initial data. Being constructive, this approach yields very detailed structure of the solutions if successfully carried out. The difficulty is that it is hard to know all the admissible shocks to enable such a construction. When A --0 in (3.3), James [49] considered the Riemann problem. Shearer [69], Hsiao [48] proved the existence of solutions of the Riemann problem. In this case, a phase boundary is admissible if and only if the speed of the phase boundary is 0. The uniqueness of such Riemann solutions is proved by Hsiao [48]. See also [50,51 ]. In the case A > 0, the only stationary phase boundary is the one connecting (m, 0) and (M, 0) [74]. This is in perfect agreement with the Maxwell equal area rule. When the Riemann data are in different phase region, e.g., w_ < ot and w+ >/3, Shearer [72] proved that solutions of Riemann problem exist if Iw- - ml + Iw+ - MI + [u+ - u-I is small, where m, M are the Maxwell line constants. He first studied the behavior of traveling waves near Maxwell line, then constructed the Riemann solvers accordingly. To extend his approach to more general Riemann data, one will have to know explicitly, for any given wl, what w2 can be connected to Wl by a traveling wave. This is almost impossible in general. Another approach is to construct the solution of (1.3) as the e ~ 0+ limit of the viscous system (1.2), or simply that of the solutions of Wt ~ Ux -- 8Wxx, (3.13) ut + p(w)x = eUxx, with the same initial data. Although this approach has been carried out successfully for strictly hyperbolic 2 • 2 systems [24,23] for hyperbolic-elliptic mixed type system (1.3), this approach seems quite difficult at present. Thus, Slemrod [77] and Fan [26,28,29] used the vanishing similarity viscosity approach pioneered by Dafermos [20] and Tupciev [81].
- 384. 386 H. Fan and M. Slemrod iT Metastablevapor i, (a) tT x/t= x/t=c Liquid ~ Metastablevapor Metastablevapor (u_,v_) ~u+<u_,v+=v_) (b) Fig. 4. The idea of this approach is to construct the weak solution of (3.1) as the e ~ 0+ limit of the solution of tOt -- Ux = etWxx, ut + p(w)x = etUxx, (u(x, o), v(x, o)) - / (u_, l)_), (u+, v+), / ifx <0, ifx >0. (3.14) This approach enables us to establish the existence of weak solutions of (3.1) for general Riemann data not in the spinodal region. The condition required is p(w) ~ -4-c~ as w --+ qxc~. Although the form of the viscosity used in (3.12) is often criticized as unphysical, it turns out that solutions constructed through e ~ 0+ limiting process of (3.12) are also admissible by the traveling wave criterion derived from (3.11) [26], see Section 4.3. Furthermore, when p" (w)(w - w0) < 0 for w # w0 6 (or, 13) and with w_ < c~ < 13 < w+ (or w+ < ot < 13 < w_), the solution of the Riemann problem admissible by the traveling wave criterion is unique. Thus, under above condition, if one obtains a solution via the vanishing viscosity method (3.12), it would be the same as what we obtained by the vanishing similarity viscosity approach (3.12). When the Riemann data are on the same side of the spinodal region, w+ < m (or w+ > M), Shearer [71] showed that when the Riemann data Iw+ - m[ (or Iw+ - MI) are small, then for some u+, there are at least two solutions for the Riemann problem. An example of the nonuniqueness is illustrated by Figure 3. Shearer's result raised the question that which of the two solutions is physically "good". We think that both solutions are good, but at different times: consider the shock tube experiment corresponding to the Riemann data (w+, u• depicted in Figure 3. We note that -u+ = u_ > u. = 0 and fl < w_ = w+ < M. This data describe the shock tube experiment where vapor moves from both sides towards the center x = 0 where the fluid is at rest. The pressure at the center part increases due to the compression from both sides. The data in Figure 3 are such that the pressure at the center part is above the equilibrium pressure and hence the vapor in the middle part is metastable vapor. The metastable vapor will stay as vapor until enough liquid drops are initiated due to random fluctuation. Thus, in the early stage, the first solution in Figure 4(a) is the good solution. As time increases,
- 385. Dynamicflows withliquid~vaporphase transitions 387 enough many liquid drops are initiated in metastable vapor, more likely in the center part where pressure is higher. Then phase changes occur rapidly via the growth of these liquid drops. In this late stage, we expect to see the second solution, shown in Figure 4(b). We note that the center of the wave in the early time of the second stage may be different from that of the first solution due to the randomness in location of liquid drop initiated. From the above consideration, we see that both solutions in Figure 3 are "good", but at different times. In fact, the two-phase-boundary solution depicted in Figure 4(b) is visually quite stable in numerical simulations once it is initiated [73]. As to when the solution in Figure 4(a) changes to that in Figure 4(b), the viscosity method does not provide answer. This is because system (3.1) with higher-order derivative terms, such as (1.2) is based on interfacial energy. The Maxwell equal area rule is derived from the consideration of interfacial energy which is the reason why viscosity-capillarity type of high-order derivative terms in (1.2) agrees with the Maxwell equal area rule. However, such high-order derivative terms do not cover the mechanism for creating of liquid drops due to random fluctuations. Thus, (1.2) is designed to describe the motion of phase boundaries, not the initiation of new phases. This is why some selection criteria, called initiation criteria, are used to help to decide when new phases are initiated in many classical theory of quasi- static problems, cf. [38]. We imagine that the transition from the solution without phase boundary to the one with two phase boundary is a dynamic process that takes some time for enough many nuclei of new phase to form and grow to complete the phase change. Typically, nucleation process are slow in metastable states unless near the Wilson line, or spinodal limits, which is why we have metastable states. Proper choice of initiation criteria used in (1.3) to correctly describe the physical process is an open problem. Initiation criteria should reflect (a) that if the fluid is in metastable state, then after a sufficiently long time, nucleation process will initiate enough many fluid drops of the stable phase and eventually, the fluid change to the stable phase, and (b) that the further away the metastable fluid is from the equilibrium, the faster the new, stable phase will be initiated. From above consideration, we see that initiation criteria must involve the time spend in metastable state and the distance of the pressure from the equilibrium pressure. So far, initiation criteria used in most works on (3.1) do not include above factors. Rather, these criteria typically specify that if the distance of the pressure from the equilibrium pressure is less than a fixed barrier, then the one-phase solution is picked, otherwise the two-phase solution is picked. This ignored the fact that no matter how close the pressure is to the equilibrium pressure, transition to the two-phase solution will happen later, even though slower. Kinetic relation admissibility criteria. Above considerations demonstrate that the sys- tem (3.1) is not complete by itself. It must be augmented by some selection criteria that helps to find the solution relevant to the problem under consideration. For phase transi- tion problems, such criteria may depend on materials in a complicated way. This leads to another point of view on admissibility criteria for phase boundaries: rather than trac- ing the admissibility criteria for phase boundaries to something more elementary such as viscosity and capillarity, one can consider admissibility criteria as constitutive relations controlling the speed of the phase boundary, determined by the materials and to be mea- sured in laboratory. Such restriction are considered by Truskinovskii [79] and Abeyaratne
- 386. 388 H. Fan and M. Slemrod and Knowles [1] and are called kinetic relations, [1]. Consider an interval [x1, X2] of fluids in the Lagrangian coordinate. The total mechanical energy associated with the interval is E(t) -- -e(w)(x, t) + -~U2(X,t) dx, 1 (3.15) where P = fo p(r/)dr/. A calculation based on (1.3) shows that d p(Wl)Ul -- p(w2)u2 -;TE(t) -- f (wl, w2)s(t) (3.16) at if (u, w)(x, t) is a shock solution of (3.1) and X2 > s(t) > Xl. Here in (3.16), 2 1 f(Wl, W2) -- -- p(r/) dr/-+- ~ (p(w2) -F p(wl))(w2 - Wl). 1 (3.17) We can see that the left-hand side of (3.16) is the excess of rate of work of the external forces over the rate of increase of mechanical energy. Since the motion (3.1) described is isothermal, the well known Clausius-Duhem inequality requires that the instantaneous rate of mechanical dissipation to satisfy f (wl, W2)S ~ O, (3.18) which is the classical entropy criterion. A subsonic phase boundary {(Ul, W2), (U2, //)2); S} is called admissible by kinetic relation criterion if, besides the entropy criterion (3.18) and Rankine-Hugoniot condition, the function f defined by (3.17), also satisfies f(wl, W2) -- qg(S) (3.19) for a function ~0predetermined by the material. The kinetic relation criterion (3.19) actually is a stricter version of the entropy criterion: OtE(u, w) + OxF(u, w) = lZ <~O, (3.20) where/~ is a given nonpositive measure [55]. Abeyaratne and Knowles [1] implemented the kinetic relation criterion onto trilinear materials, i.e., the function p(w) is three-piecewise linear and of the shape depicted in Figure 1. They proved that the Riemann solver satisfying their kinetic relation, the initiation criterion, and the entropy inequality is unique. Later in [2], they extended above result to nonisothermal case with heat conduction taken into consideration. The speed of the phase boundary is not constant. They found an integro-differential equation for the speed. LeFloch [55] proved the L 1 continuous dependence of the Riemann solver, admissible by kinetic relation criterion, on the Riemann data. Abeyaratne and Knowles [1] also showed that, at least for trilinear materials, the traveling wave criterion and the entropy rate criterion, etc., when applied to subsonic phase boundaries, is a kind of kinetic relation
- 387. Dynamicflows with liquid~vaporphase transitions 389 criterion. Later, Fan [30] showed that traveling wave criteria are kinetic relation criteria if p(w) is symmetric around the point (w* = (or +/3)/2, p(w*)). Natalini and Tang [60] considered some discrete kinetic models with the objective of providing a practical tool encompassing various kinetic relations for the phase boundaries. Similar to Shearer's results on nonuniqueness of the Riemann solver, the Riemann problem (3.1) has two solutions admissible by kinetic relation criterion for some Riemann data. One solution entirely lies in one of the phase region {(u, w); w < c~} (or {(u, w); w > t3}) with no phase boundary while the other solution has two phase boundaries and hence takes values in both phase regions. For convenience, we shall call the first solution the one-phase solution and the latter the two-phase solution. In fact, this nonuniqueness phenomenon is common for a large class of local admissibility criteria which are local restrictions on points of discontinuities of solutions of (3.1) [30]. To handle this nonuniqueness, Abeyaratne and Knowles [1] used an initiation criterion which specifies a critical value for a function h and assert that phase transitions will occur and hence the two-phase solution is "good" if the value of h exceeds the critical value. Otherwise, the one-phase solution should be picked. LeFloch [55] considered the (1.3) with initial data being a BV perturbation of an admissible phase boundary. He constructed solutions of (3.1) 1 and (3.1)2 with p being trilinear. The selection criteria used are the kinetic relation criterion, initiation criterion and entropy inequality. He constructed solutions by Glimm's scheme and proved these solutions are admissible by the above selection criteria. In recent papers Bedjaoui and LeFloch has investigated the relation between the kinetic relation and viscosity- capillarity [5,6]. The instability from Glimm scheme when applied to (1.3) is discussed by Pego and Serre [63]. Asakura [4] studied the Cauchy problem for (1.3) with initial data being Maxwell stationary phase boundary plus a small perturbation. He showed that there exists a global in time propagating phase boundary which is admissible in the sense that it satisfies the kinetic relation criterion; the states outside the phase boundary tend to the Maxwell states as time goes to infinity. Colombo and Corli [16] constructed Riemann semigroup of (1.3) admissible by aS-relation, a generalized kinetic relation criterion. In particular, this result allows the authors to build a complete theory of existence (via front tracking) and continuous dependence on the initial data of the solutions. Related papers have been done by Corli [17-19]. Entropy rate criteria. Another interesting admissibility criterion is the entropy rate criterion proposed by Dafermos [21]. This criterion asserts that the weak solution of (3.1) which dissipates the total entropy the fastest is the admissible solution. The total entropy is typically the total mechanical energy: E(t) .= -p(w) + ~u +/'(wo)- u2 dx, (3.21)
- 388. 390 H. Fan and M. Slemrod where (w0, u0) is the initial value. The dissipation of the total energy when the solutions are piecewise smooth is measured by dE(t) dt - shocks (3.22) where f is given in (3.17). Dafermos [22] further justified this admissibility criterion by proving that in strictly hyperbolic systems, wave fans satisfying Liu's shock admissibility criterion [56] consisting of rarefaction waves and shocks of moderate strength do maximize the rate of entropy production. In elastodynamics, this statement holds for arbitrary shock strength. Hattori [41,42,44] and Pence [64] applied this criterion in their study of Riemann problems of systems of conservation laws of mixed type. Hattori [43] further studied initial value problems of (1.3) using the entropy rate admissibility criterion and Glimm's scheme. He proved the existence of weak solutions when initial data is a BV perturbation of Riemann data and the perturbation is compactly supported. Among admissibility criteria mentioned in the above, the vanishing viscosity crite- ria (1.3) and (3.13) and the entropy rate criterion are of global nature. Others, such as traveling wave criteria and the kinetic relation criterion are local restrictions at points of jump discontinuity. It is a hope that these local restrictions can characterize completely admissible solutions of (3.1). If this hope fails, more conditions, probably conditions of global nature, should be imposed. Thus it is important to experiment with various criteria with global authority, especially those motivated by physics. For example, when the Rie- mann solvers admissible by a local admissibility criteria are not unique, which one does a global admissibility criteria pick? A comparison of the effect of vanishing viscosity crite- rion (3.13), entropy rate criterion and traveling wave criterion derived from (3.13) in the context of (3.1) is made in [30]. In [30], initial data is chosen such that there are two solu- tions of (3.1), admissible by the traveling wave criterion, one being the one-phase solution and the other the two-phase solution. It is found that only the one-phase solution is admis- sible by the vanishing viscosity criterion (3.13), at least for the special pressure function p given in [30]. However, the entropy rate criterion picks the two-phase solution. We note that by the vanishing viscosity criterion, the two-phase solution cannot be initiated from the initial data in (3.1). Once the two-phase solution is initiated, it is quite stable, at least visually in numerical simulations [73]. More results on the viscous system (1.2). boundary value problem Hattori and Mischaikow [46] studied the initial Utt = O'(Ux)x -t- 13Uxxt - 17Uxxxx, u(0, t) -- 0, cr(Ux(1, t)) + VUxt(1, t) - rlUxxx(1, t) = P, Uxx(1,t) --Uxx(O,t) =0. (3.23) Note that Equation (3.23)1 can be tranformed to (1.2) under the variable change v = ut, w = Ux. They proved the existence and uniqueness of the global solution of (4.1) under certain growth condition on or(q). They also proved the existence of global compact
- 389. Dynamicflows with liquid~vaporphase transitions 391 attractor. They obtained the complete bifurcation diagram. The existence and large time behavior of initial boundary value problem of (4.1)l with r/-- 0 is studied by Pego [62]. He showed that discontinuities of the solution are stationary, that the energy of the system cannot be minimized as t --+ oo, among other things. Hattori and Li [45] studied the initial value problems of a fluid dynamic model for materials of Korteweg type in two-dimension: fit -I- (pU)x + (pV)y = O, (pu)t + (puZ)x + (pUV)y + p(p)x = (Tll)x + (T12)y, (pv)t + (puv)x + (pvZ)y + p(p)y = (T21)x + (T22)y. (3.24) They proved the existence of the unique local solution. Their proof does not depend on the monotonicity of the pressure function and hence can be used for van der Waals pressure. Nicolaenko [61] showed the existence of inertial manifolds for (1.2). The proof used the slightly dissipative Hamiltonian structure of the system. Milani, Eden and Nicolaenko [58] established the existence of local attractors and of exponential attractors of finite fractal dimension. This showed that even in regions of mixed type, the initial value problem exhibits finite-dimensional dynamical behavior. Serre [68] used a very interesting approach computing the formal oscillatory limit, in the spirit of Whitham [84], of the thermo-visco-capillarity system as the small parameters tends to zero. He then obtained a system of modulation equations for the limiting motion. 4. Existence of solutions of the Riemann problem (3.1) In this section, we recall our proofs for the following results on the existence of solutions of (3.1) in [77,28]. Their approach is to use the vanishing similarity viscosity (3.14) to establish the existence of weak solutions of (3.1) that are also admissible by traveling wave criterion derived from (3.13). Reasonable shock admissibility criteria should be compatible with translations and dilations of coordinates, under which the system is invariant. Dafermos [22] argued that admissibility should be tested in the framework of Riemann problem, i.e., in the context of solutions of the form U (x, t) = V(x/t) which represent wave fans emanating from the origin at time t = 0. Thus we utilize the system (3.14), which is invariant under translations and dilations of coordinates, to handle the Riemann problem (3.1). This approach has been pursued by many authors in their stydies of Riemann problems [20,54,27,53,81,77,82]. For convenience, we shall call solutions of (3.1)constructed in this way admissible according to the similarity viscosity criterion. The main results are as follows: THEOREM 4.1. Assume in (3.1) that w+ q~[c~,3] and p(w) ~ -+-oo as w --+ Too. (4.1)
- 390. 392 H. Fan and M. Slemrod Then there is a sequence {en}, en --+ O+ as n --+ cxz such that the solution of (3.14) with initial data (3.1)3 converges almost everywhere to a weak solution u(x, t) of (3.1). Furthermore this solution also satisfies the traveling wave criterion derived from (3.13). The structure of solutions of (3.1) when w_ < c~ </5 < w+ constructed in Slemrod [77] and [26,28,29] by the similarity viscosity approach is as follows: each of these solutions can be embedded on a continuous curve in (u, w) the phase plane. Solutions must have a phase boundary, i.e., w(~) r (or,/5) for any ~ e I~. Solutions consist of two wave fans: < 0 the first kind wave fan and ~ > 0 the second kind wave fan. A first (second) kind wave fan consists of 1-shocks and (2-shocks) and 1-simple waves (2-simple waves) and possibly the phase boundary and constant states. ~ = 0 is either a constant state or the phase boundary (cf. [26]). Most of above results are generalized by Lee [54] to the system ut -- f (V)x = O, vt - g(u)x =0, (4.2) where f is strictly increasing and convex, and g is increasing (and either concave or con- vex) except in a finite interval where it is decreasing and so the system is hyperbolic-elliptic mixed type. Under stricter restrictions on p(w), we have the following uniqueness results: THEOREM 4.2. Assume conditions in Theorem 4.1. If w_ < ot < ~ < w+ (or w+ < ot < < w_) and that p" (w) > O for w < ot, and p" (w) < O for w > fl, (4.3) then (i) the solution of (1.3) satisfying the traveling wave criterion based on (1.2) is unique, and (ii) the solution (u ~, wE)(x, t) of (3.14) converges almost everywhere to the unique solution of (1.3)as e --+ 0+. The statement (i) of Theorem 4.2 is proved in [27]. The statement (ii) follows immediately from (i) and Theorem 4.1. The rest of Section 4 is devoted to the proof of Theorem 4.1. 4.1. Existence of solutions of the Riemann problem (1.3) To take the advantage of the invariance of (3.14) under dilatation of coordinates, we make variable change ~ = x/t in (3.14). A simple computation shows that (3.14) reduces to the
- 391. Dynamic flows with liquid~vapor phase transitions 393 following system eu"=-~u' + p(w)', 8W tt -- __~ W f m U', (U, 113)(--00) = (U_, W_), (u, w)(+oo) = (u+, w+). (4.1.0) Our program for proving Theorem 4.1 is to show that there is a solution of (4.1.1) with total variation bounded uniformly in e > 0. Then the first statement in Theorem 4.1 follows. The proof for the second statement of Theorem 4.1 will be given in Section 4.3. To this end, we consider, instead of (4.1.0), the following altered system eu"= -~u' + #p(w)', 8W II -- --~ W I _ l,U I, (u(+L), w(+L))- (u• w• (4.1.1) where L > 1, 0 ~ # ~<1. LEMMA 4.1.1 [77]. Let (ue(~), we(~)) be the solution of (4.1.0). Then one of the following holds on any subinterval (a, b) for which p' (we(~)) < 0. (1) Both ue (~) and we (~) are monotone on (a, b). (2) One of the ue(~) and we(~) is a strictly increasing (decreasing)function with no critical point on (a, b) while the other has at most one critical point that is necessarily a local maximum (minimum)point. (3) If the criticalpoint in (2) is of w(~), then the condition p'(w(~)) < 0 can be relaxed to p'(w(~)) <, Ofor ~ E (a, b). Now, we rewrite Lemma 2.2 of [77] which describes the shape of a solution of (4.1.0) in the elliptic region {(u, w) 6 IR2: w E (or,fl)}. LEMMA 4.1.2. Let (u(~), w(~)) be a solution of (4.1.1)with lZ > O. Then on any interval (11,12) C (-L,L) for which p'(w(~)) > 0 the graph of u(~) versus w(~) is convex at points where w' (~) > 0 and concave at points where w~(~) < O. By considering (4.1.1), the existence of the connecting orbit problem (4.1.0) can be proved, as shown in the following theorem. THEOREM 4.1.3. Suppose u_ < u+ and w+ < or. Then there is a solution of (4.1.0) satisfying that (U(~l), W(~l)) -7(=(u(~2), to(~2)) for any ~l, ~2 E (--00, -+-(~), ~1 zik ~2 and w(s~l)/> tb := max(w_, w+) (4.1.2)
- 392. 394 H. Fan and M. Slemrod and that there are at most two disjoint open intervals (a, b) such that (4.1.3) w(~) ~ (if), ~) for ~ ~ (a, b) (4.1.3a) and either w(a) -- if;, w(b) -- ot or w(a) -- or, w(b) -- (v, (4.1.3b) provided that the possible solution of (4.1.1) satisfying (4.1.2) and (4.1.3) is bounded in CI([-L,+L]), for some M > 0 independent of tx ~ [0, 1] and L > 1. (4.1.3c) PROOF. We rewrite (4.1.1) as ey"(~) = lzf (y)' - ~y'(~), (4.1.4) where y(~) _ (u(~)) w(~) ' -u(~) " Multiplying (4.1.4) by the factor exp(-~2/(2e)) and integrate twice, we can rewrite (4.1.4) as the integral equation: y(~) = y(-L) + z(y) exp dr + -- f(y(r)) dr L -~8 8 L f_f ,, tx ~ r rfty(r)~ex p drd(, 8 L L 2e (4.1.5a) where z(r) = 1 { --f f(Y(r)) dr f-LLexp(@2)d~ y(+L)_ y(_L)_ # L 8 L +lz f_L f_ L L rf(Y(r)) exp( r dr d( 2e = zl (Y) -k-lZz2(Y). (4.1.5b) Choose r/6 (~b,or). (4.1.6)
- 393. Dynamic flows with liquid~vaporphase transitions 395 We are interested in those functions (u(~), w(~)) e C 1([-L, +L]; R 2) satisfying and w(~l) ~>7/ for any ~1, ~2 G [-L, +L], ~1 r ~2 (4.1.7) and that there are at most two disjoint open intervals (a, b) w(~) 6 (rl, or) for ~ ~ (a, b), and either w(a) - rl, w(b) - ~ or w(a) = c~, such that w(b) -~. (4.1.8) We note that (4.1.7) and (4.1.8) is invariant under small C 1 perturbations. The subset in CI ([-L, +L]; IR2) and (4.1.7) and (4.1.8) are satisfied} (4.1.9) is open. We define an integral operator T's x [0, II --+ C 1([-L, L]; R 2) by l; ,f; T(Y, Ix)(~) = y(-L) + z(Y) exp d( + -- f (Y(()) d( L -~e 6 L f f Ix ~ C r/(Y(r)) exp dr d(, s L L 2S (4.1.1o) where z(Y) is given by (4.1.5b). It is clear that a fixed point of T(Y, Ix) is a solution of (4.1.1). It is a matter of routine analysis to show that T maps s x [0, 1] continuously into C 1([-L, L]; R2). Furthermore, we can verify, by taking d/d~ twice on (4.1.10), that T maps ~ x [0, 1] into a bounded, with bound independent of Ix, subset of C 2([-L, L]; R2). Thus T is a compact operator from C 1([-L, L]; R 2) x [0, 1] into C 1([-L, L]; R2). We recall the following fixed point theorem ([57], Theorem IV. 1). PROPOSITION 4.1.4. Let X be a real normed vector space and [2 a bounded open subset of X. Let T 9 x [0, 1] --+ X be a compact operator. If (i) T (x, #) ~ x for x E 0s tx ~ [0, 1], and (ii) the Leray-Shauderdegree DI(T(., O) - I, ~) 7/=O, where I is the identity operator, then T (x, 1) -- x has at least one solution in s
- 394. 396 H. Fan and M. Slemrod To solve our problem, we take X = C 1([-L, +L]; I1~2). We can see that (ii) is satisfied. Indeed, T(Y,O)- Y= Yo- Y, (4.1.11) where ~:= y(L) - y(-L) f~, _~2 J_ exp f_LL exp(w-) d ( L (-~e 2 ) d( + y(-L). We note that T(Y, O) = Yo ~ $2 is the solution of (4.1.4) when/z = 0. It is a fixed function, independent of Y and/z. Then we have DI(T(., 0) - I, S2)= DI(Yo- I, f2) = 1, as desired. Now, we preceed to verify (i) of Proposition 4.1.4. We assume, for contradiction, that there is a fixed point of T (Y,/z), Y = (u, w)(~) 6 052. (4.1.12) Then one of the following cases must hold: Case A. II(u(~), w(~))llcl<t-L,c~;R2) = M + 1. This case is impossible to occur under the condition (4.1.3c). Case B. The condition (4.1.7) is violated. In this case, there are ~1,~2 6 [-L,+L], ~1 < ~2, such that (U(~l), W(~l))= (U(~2), tO(~2)) and W(~I) ~ r/. The curve (u(~), w(~)) in (u, w)-plane near ~ = ~1 and ~ = ~2 cannot go across each other. 1 This is because if otherwise, the curve (u(~), w(~)) in (u, w)-plane plus a CI([-L, +L]; R 2) perturbation still intersects itself and hence is not in S2. Thus, (u(~), w(~)) is not in 052 which yields a contradiction. From Lemma 4.1.1, we know that if (u(~), w(~)) stays inside the region w ~< or, the curve (u(~), w(~)) cannot intersect itself. Thus, w(~3) > c~ for some ~3 E [-L, +L]. 1 Here, we clarify the meaning of "two curves go across each other": For two curves, (Ul, Wl)(~) and (u2, w2)((), to cross each other in (u, w)-plane, they have to intersect each other first: (Ul, Wl)(~l) = (U2, W2)((2) at some points ~1, (2. For convenience, we parameterize the two curves by the length of curve s with s = 0 denoting above point of intersection. If the two curves coincide with each other near s = 0, then the orientation of the parameterization should be such that (u 1, Wl)(S) -- (U2, W2)(S) (1) over [s_, s+] with 0 6 [s_, s+]. In particular, if no such coincidence is present, then s_ = s+ = 0. We further let the interval [s_, s+] be the largest on which (1) holds. We use the following notations: Tj(s) is the tangential direction of the j-th curve, j -- 1, 2, k the normal direction of the (u, w)-plane, which is a constant vector in R3. We say that curves (Ul, Wl)(S) and (u2, w2)(s) go across each other if (1) holds and (T1(s) • T2(s)) 9k does not change sign on an open interval containing [s_, s+ ].
- 395. Dynamicflows with liquid~vaporphase transitions 397 We can further describe the curve (u(~), w(~)) in the (u, w)-plane as follows. There is an interval [-L, 01] such that w(~) ~<c~and w(01) =ol, and by Lemma 4.1.1, w'(O1) > O. As ~ increases from 01, (u(~), w(~)) moves into the region ot < w < /3. As long as w'(~) > 0 and w(~) 6 (c~,/3), the curve (u(~), w(~)) in the (u, w)-plane is convex with respect to w. Let (01,02) be the largest interval such that w'(~) > 0 and w(~) 6 (or,/~). (w+, u+ (w_, u_) // ~,, ~=o 1 I Fig. 5. W Then either W(02) -- fl or w(02) E (ly, fl) and w' (02) = 0 holds. For definiteness, we assume that w(02)= ~ and w~(02) > 0, since the other case is simpler. In view of Lemma 4.1.1, this interval is followed by another interval [02,03] in which w(~) ~> and u~(~) > 0 while w(~) has one and only one critical point which is a local maximum point and w(03) =/3. This shows that ~2 ~ [--L, 03). Then there is the maximum interval [03, 04) in which w(~) ~ [or,~], w'(~) < 0 and the curve (u(~), w(~)) in the (u, w)-plane is concave with respect to w. We see that at the right end of the interval, either w(04) = ~, w'(04) < 0 or w(04)/> ~, w'(04) = 0 holds. We claim that wf(04) = 0 is impossible because if otherwise the concavity would make w'(~) > 0 for ~ > 04 and near 04. This would force the curve (u, w)(~) to go across itself in the region w ~>c~ in order to reach w(L) = w+. This is contradictory to (u, w)(.) ~ OX2.Thus, w(04) =or and w'(04) < 0 hold. This also shows that if ~2 ~ [03, 04), then ~1 6 [01,02] and W' (~1) >/0, W'(~2) ~< 0, w(~) > w(~l) for ~ e (~1, ~2).
- 396. 398 H. Fan and M. Slemrod ='ql ~ I I ,w,u., , i-,, / (w,u) / n a 3 =r/2 w Fig. 6. This, however, will lead to a contradiction by integrating (4.1.1b): f~ 2 0 < [w(~) - w(~e)] d~ = e[w'(~e) - w'(~,)] ~<0. 1 Above description shows that the point of self intersection ~2 r [01,04]. Following [03, 04] is the interval [04, 05) in which r/< w(~) ~<or. Let [04, 05) be the largest of such interval. Then w(05) = r/, w'(05) < 0 (4.1.13) holds because if otherwise, w(~) would have a local minimum point in [04,05) and w(05) = or, see Figure 6. By Lemma 4.1.1, u(~) would decrease over the interval [04, 05). After ~ = 05, w(~) would enter the w < c~region. Then the curve (u, w)(.) in (u, w)-plane would have to go across itself in order to connect to (u+, w+), which is prohibited. We further claim that over the interval (05, L], w(~) <~. Indeed, if otherwise, either w(~) > ot for some ~ > 05 or w(~) is less than a and has multiple extreme points in [05, L]. The case of multiple extreme points are impossible in view of Lemma 4.1.1. The other case that w(~4) > c~ for some ~4 ~ (05, L] is also impossible since it and (4.1.13) imply that there are at least three disjoint open intervals (a, b), bounded away from each other, such that w(a) = ot (or rl) and w(b) = 7/ (or c~). But this is impossible for a function (u, w)(.) 6 0s This claim implies that the points of self-intersection satisfies ~2 ~ [04, 05], ~1 E [--L, 01] and w(~2) ~ [r/, or]. Thus, we have w'(~2) < 0and w'(~l) > 0, and w(~) ~>w(~2) = W(~l) for~ ~ [~1, ~2]. Integrating (4.1.1b) over [~1, ~2] and using (U(~l), W(~l)) = (u(~2), w(~2)), we obtain f~2 0 < [w(~) - w(~2)] d~ - e[w'(~2) - w'(~,)] < O, 1
- 397. Dynamic flows with liquid~vaporphase transitions 399 which is a contradiction. Thus, Case B cannot happen. Case C. The condition (4.1.8) is violated. That is, there are more than two disjoint open intervals (a, b) such that 77< w(~) < ot for ~ 6 (a, b) and w(a) = rl (or c~), w(b) = ot (or rl). Since (u, w)(~) 6 0S-2, the number of disjoint open connected component intervals (a, b) with w(a) = rl (or or) and w(b) = ot is four or more. Two of such intervals are (al, bl), (a2, b2) with a2 = bl and w'(bl) = 0. If w(bl) = rl and hence w(al) = w(b2) -- or, then = bl is a local minimum point of w(~). According to our discussion of Case B, it is necessary that bl < 04 and it is impossible that w(b2) = c~. This contradiction shows that w(bl) 7~ rl9 We claim that the other possibility w(bl ) -- ot cannot happen either. Indeed, if w(bl) = ol and hence w(al) -- w(b2) = ~ < ot, the point ~ = bl is a local maximum point for w(~). We see that w(~) ~<o~ for all ~ E [-L, L] since if otherwise, w(~) would have multiple extreme points in one of the connected component of {~ ~ [-L, L]: w(~) ~< o~} which is impossible according to Lemma 4.1.1. Then, the number of disjoint open connected component intervals (a, b) with w(a) = rl (or c~) and w(b) = o~ is just two, not four or more. This contradiction shows that Case C cannot occur. Summarizing our analysis for above three cases, we find that if (u(~), w(~)) E 0$2, then Y = (u(~), w(~)) cannot be a fixed point of T(Y, lz) for # E [0, 1]. Applying Proposition 4.1.4, we see that T(Y, 1) has a fixed point. To prove the existence of solutions of (4.1.0), we need to pass to the limit L --+ oo. We follow Dafermos [20] and extend (u(~), w(~)) as follows [ (u+, w+), (u(~'/~) w(~'/~)) = { ' ' ' /(u-,w-), if~ > L, if~ <-L. By the hypothesis (4.1.4), we see that {(u(.; L), w(.; L))} is precompact in C((-c~, cx~); R2). So, there is a sequence Ln --+ cx~ as n --+ cx~ such that (u(~; Ln), w(~; Ln)) --+ (u(~, cx~), w(~, cx~)) uniformly as n --+ cx~. By integrating (4.1.1a, b) with/z = 1 twice from ~0, we can prove the limit (u(~, cx~), w(~, c~)) satisfies (4.1.1a, b). It remains to prove that (u(-+-cx~,~), w(-+-cx~,cx~)) = (u+, w+). To this end, we manipulate (4.1.1a, b) to obtain d(exp(~2/2e)y'(~))-1 [f(y(~))' exp(~e2)l d~ e or if0' exp(~Z/Ze)y'(~) = y'(O) + - V f (y)y'(~)exp -~e d~. 6 (4.1.14) Applying Gronwall's inequality on (4.1.15), we obtain I ly'0 Iex"( ~< M exp 2e ' (4.1.15)
- 398. 400 H. Fan and M. Slemrod where R > 0 depend at most on M, v and e > 0. Inequality (4.1.21) holds for y (~; L) also. Then (u(+~, ~), w(+cc, ~)) = (u+, w+) follows from (4.1.21) easily. It remains to prove that the solution (u(~, cx~),w(~, cxz)) constructed above satisfies (4.1.2) and (4.1.3). Indeed, the same reasoning for Case B and C implies that (u(~, cx~),w(~, e~)) satisfies (4.1.8) and (4.1.9) also. Since r/6 (if),or) is chosen arbitrarily, (4.1.2) and (4.1.3) hold for (u(~, c~), w(~, cx~)). D COROLLARY 4.1.5. Let (u(~), w(~)) be a solution of (4.1.1) or (4.1.0) satisfying (4.1.2), (4.1.3). Then, (i) The subset of [-L, +L] {~ e I-L, +/-,l" w(~) <~o~} has at most two connected components. Furthermore, each components must have -L or +L as one of its endpoints. (ii) The set consists of at most two connected components. (iii) The set {~ e I-L, +L]. w(~) >/r if nonempty, is an interval. PROOF. This is proved in our discussion in the proof of Theorem 4.1.3, Case B. D The assumption (4.1.4) in above theorem can be replaced by a weaker one, as stated in the following theorem. THEOREM 4.1.6. The conclusion of Theorem 4.1.3 remains valid if (4.1.4a) is replaced by sup ([u(~) I nt- Iw(~)l) <~M1, where M1 is independent of tx ~ [0, 1] and L > 1. PROOF. The proof is the same as that of Theorem 1.3 in [77]. IS] Theorems 4.1.3 and 4.1.6 give the conditions under which (4.1.0) has a connecting orbit for w+ < ot and u_ < u+. Slemrod [77] proved the following theorem for the case w+ < c~ and u_ < u+:
- 399. Dynamicflows with liquid~vaporphase transitions 401 THEOREM 4.1.7. Assume that w+ < ~ and u_ < u+. Then, there is a solution of (4.1.0) satisfying w(~) ~<o~, (4.1.16) if every possible solution of (4.1.1) satisfies (4.1.17) for some constant C independent of tt E [0, 1] and L > 1. 4.2. A-priori estimates In this section, we shall prove the a-priori estimates needed in Theorems 4.1.3 and 4.1.7 as well as some e-independent estimates. Let denotes a solution of (4.1.1) with the properties (4.1.2) and (4.1.3). For clarity, we shall use (u(~), w(~)) instead of (ue (~), we (~)) in this section if no confusion is expected. THEOREM 4.2.1. Suppose w+ < ~ and u_ < u+. Let (ue(~), we(~)) be a solution of (4.1.1) with the properties (4.1.2) and (4.1.3). Then, (4.2.1) where C is, throughout this section, a constant independent of e > 0, # E [0, 1] and 1 < L ~<+cx~. PROOF. When # = 0, our assertion can be easily verified. Thus, we assume # > 0 in the rest of the proof. We first prove uE(~) ~> C. Let ~e be a local minimum point of u~(~). Then either we(~e) ~ (or,fl), wte(~e)< 0 (4.2.2) or ! we(~) E (o~,fl), w~(~) > 0 (4.2.3) hold. Case A. (4.2.2)holds. In this case, by Lemma 4.1.1, w(~e) < fl since if otherwise both ue(~) and we(~) would have critical points in the set {~ E [-L, +L]: we(~) ~>fl}, which is an interval by Corollary 4.1.5. Thus, w(~e) <~ot and hence ~ 9 {~ E [--L, +L]" //)e(~) ~ or} -- [-L, 011U [04, 7t-L], (4.2.4)
- 400. 402 H. Fanand M. Slemrod where 01 ~ 04. If 01 < 04 and ~s e [-L, 01], then w'(~s) < 0 implies that ws(~) also has a critical point in [-L, 01] which is prohibited by Lemma 4.1.1. Thus, ~s 6 [04, L]. We can regard the curve (u, w)(~) in the (u, w)-plane as a function u(w). Then we have dus(~) u'(~) dws(~) w'(~) Performing a calculation on (4.1.1), we obtain d (dus(~)) (dus(~) s~ dws(~) -- # dws(~) dus(se x dws(~) v/-P'(We(es ))) +v/-p'(w~(~))). (4.2.5a) This implies that, as ~ increases, dus(es)/dws(es) is decreasing if Idus(~j)/dwe(es)l <~ v/-P'(We(~)) and is increasing if Idue(es)/dwe(es)l >~v/-p'(ws(es)). Thus the "initial" condition dus(~) dws(~) ~=~ =0 (4.2.5b) leads to that for ~ 6 [04, +L] dus(~) dws(~) ~< max (v/-p'(w)) (4.2.6) w+>>. w>~ f and hence us(~) >~u+ + (or - w_) max (v/-p' (w) ). we[w+,~] (4.2.7) Case B. (4.2.3) holds. By Corollary 4.1.5, [-L, L] can be divided as [-L, L] -- [-L, 011 U (01,02) [,-J[02, 03] U (03, 04) U [04, +L], (4.2.8) where, of course 01 ~<02 ~ 03 ~ 04, and {~ e [-L, -+-L]: ws(~) ~ o/} -- [-L, 011U [04, L], {~ e [-L, nt-L]" t/)s(~) e (c~,fl) } -- (01,02) U (03, 04), {~ E[-L,-+-L]" tOs(~)/> fl} =[02,031. (4.2.9a) (4.2.9b) (4.2.9c) It is clear that when (4.2.3) holds, ~s 6 (01,02) 5; f ~ (cf. Figure 5).
- 401. Dynamic flows with liquid~vaporphase transitions 403 According to the sign of ~e, we have two cases: Case B(1). ~e ~>0. (4.2.10) Since ~e is a local minimum point of u(~), ue dx > 0 for ~ E (~e, ~e + 6) for some 6 > 0. Then we can define /71 "-- sup{ ~"> ~e" ue dx > 0 for ~ E (~e, ~')}. (4.2.11) Since we(~e) ~>ot by (4.2.3), and w~e(~e) > 0, there is a local maximum point /72 of We(~) with/72 > ~e. We can further require that/72 is the least of such points, i.e., /72 "-- sup{ ~"> ~e" w; (() > 0}. (4.2.12) Then, by Lemmas 4.1.1 and 4.1.2,/71 ~ (~e,/72) and hence (cf. Figure 5) /71 > /72 > ~e- (4.2.13) By integrating (4.1.1a) on (~e, ~) where ~ E (~e,/72), we obtain 0 < euedx -- -~'u;(~') d~" + #[p(we(~)) - p(we(~e))]. It follows from (4.2.10) and (4.2.11) that -~ue, dx < 0 for ~ 6 (~e,/71). Thus, in view of (4.2.3), we have 0 < eu;(~) <~ #[p(we(~e))- p(we(~ee))] ~<#[p(we(se)) - p(c~)] for ~eE (~ee,/72). (4.2.14) Therefore, O/ < 1/3e(/72) ~< Wl (4.2.15) holds, where WO :-- y, 1131 := V in Figure 1. Equation (4.2.13) also yields a useful inequality 0 < eule(~) ~<#(p(fl) - p(c~)) (4.2.16) for ~ E [~e,/711.
- 402. 404 H. Fan and M. Slemrod Using (4.1.1), we can obtain d2we -Ix (~) = ~u~ dx (dwe(~) 2] ~[l + p'(we(~)) due(~)) " Hence, if dwe(~) due(~) 2 maxwe[wo,tO1](V/[ p' (W)l) and ~ 6 [~e, ~1], then d2Wen -lz 1 du 2 (~)~< ~<-- <0. 2eu~dx 2(p(fl) - p(ot)) 8n Thus, as ~ decreases from//2 to ~, dw/du will increases from 0 and eventually dwe (~) [ _ 1 [ i du~(~) ~=,73 2max~e[wo,wll(V/lp'(w)l) for some r/3 6 (//2, ~e)- Let 04 "-- sup{r/3 E (~e,//2): (4.2.19)is satisfied}. Then, 2 maxwe [wo,W1](%/I P! (W)I) du~(~) ~=.4 du~(~) ~=.2 = f uE(r/4)d2wen au~(,T2 d-~e2.(~) d(ue (~)) ue (/72)- ue (r/4) > or 0 ~ Ue(r/2) -- Ue(r/4) p(fi) - p(c~) max~ e[wo,w,](v/Ip ' (w)1) From (4.2.18), we also see that dwe (~) /> due(~) 2 maxwe [wo,wl](v/[ P' (w)l) (4.2.17) (4.2.18) (4.2.19) (4.2.20) (4.2.21)
- 403. Dynamicflows with liquid~vaporphase transitions 405 or du~(~) dw~(~) 2 max (v/lp'(w)l) wE[wO,Wl] for ~ e (~e, 004).Thus, f w~(rl4) due (~) 0 ~ Ue(002) -- Ue(~e) = Ue(002) -- Ue(004) § -5-~-_.--7L~dwe aw~(~) tlw~ L~) p(fl) - p(~) ~< + 2 max max~e[~0, Wl](v/I p'(w) l) we[~0,w~] (v/lp!(w)l)(Wl - too), (4.2.22) where we used (4.2.15) and 004 E (~e, 001)- Similarly, we can prove that 0 ~ Ue(001) --Ue(002) p(r - p(c~) ~< maxwe [w0,w,](v/I p ' (w)l) § max (v/lp'(w)l)(wl - wo). (4.2.23) tOE[tO0,tt)l] Then we obtain Ue(~e) ~ Ue(001)- 2(p(fl) - p(ot)) maxw E[u,0,tO1](V/ ]P! (W)l) -4 max (v/Ip'(w)l)(w,-wo). we[wo,wl] (4.2.24) If ue(001) ~> u+, then, (4.2.24) shows that ue(~) is bounded from below uniformly in e > 0, # ~ [0, 1] and L > 1. Now, we devote our attention to the case when b/e(001) < U-k-. ! Then, 001 < L because ue(L) = u+. By the definition (4.2.11), of 001, Ue(001) -- 0. Then, by Lemma 4.1.1 and 4.1.2, 001 has to be an extreme point for ue(~). Since ue dx > 0 for E (~e, 001), /71 is a local maximum point. Lemmas 4.1.2 and 4.1.1 implies that either Wle(001) > 0 and We(001) ~ (0/, fl) (4.2.25) or ! We(001) < 0 and We(001) E (or, fl). (4.2.26) The case (4.2.25) cannot happen because it implies that 171 E [-L, 01] which violates the known fact that 001 > ~e E [01,02). Then (4.2.26) infers that there is a local minimum point 05 > 004of ue(~) which satisfies ! ue(005) -- 0 and we(005)~<c~. (4.2.27)
- 404. 406 H. Fan and M. Slemrod Then our argument for the Case A applies and gives us lu+ - ue(r/5)l ~<(or - w+) max (v/-p' (w) ). w~[w+,a] (4.2.28) Using (4.2.28) in (4.2.24), we obtain the desired result 2(p(fl) - p(ot)) //~(~) ~ //e(/~l)- maxw e[w0,11) 1](g/I pl(W)l) - 4 max (v/Ip'(w)l)(Wl -- 1/30) we[wO,Wl] 2(p(fl) - p(a)) /> u~(~5)- max~e [~o,~, ](v/I p ' (w)l) -4 max (v/lp'(w)l)(wl-wo) w~[wo,wl] i> u+ -- (c~ -- w+) max (V/--p'(w)) wc[w+,~] 2(p(fl) -- p(ot)) maxwe[wO,Wl](v/I P'(w) l) -4 max (v/lp'(w)l)(Wl-WO), we[wo,wll (4.2.29) which proves that uE(~) is bounded from below uniformly in e > 0, # E [0, 1] and L > 1. Case B(2) ~e < 0. The proof is similar to Case B(1). The only difference is that instead of (4.2.11), we define r/1 --inf{ff <~," u'(~)> Ofor~ e (~,~E)} and change the rest of the proof accordingly. Similarly, we can also prove that ue(~) is bounded from above uniformly in e > 0, # [0, 1] and L > 1. D In the remainder of this section, we adopt the following notation: u* "- sup{ue(~) I~ e I~, e 9 (0, 1)}, u, "- inf{u,(~) I~ e R, e 9 (0, 1)}. (4.2.30a) (4.2.30b) Once we established the a-priori estimates for ue(~), we can proceed to prove the following results for we(~) by using the similar argument used in [20]. THEOREM 4.2.2. Assume w+ < a and u_ < u+. Let (ue(~), wE(~)) be a solution of (4.1.1) satisfying (4.1.2) and (4.1.3). Then
- 405. Dynamicflows with liquid~vaporphase transitions 407 (i) IIw~(~)llf(t-z,+zl;R2)~ C(e) where C(e) is independent of# e [0, 1], L > 1. (ii) /f Ip(w)l ~ ec, as Iwl ~ ~, and if# = 1, then IIw~(~)IIc(t-L,+L~;R2) ~ C where C is independent of L > 1 and e > O. PROOF. We only prove that w(~) is bounded from above uniformly. The other part of the proof is similar and is omitted. (i) Without loss of generality, we assume we(~) has a local maximum point re. We further assume that re ~<0. (4.2.31) The proof for the other case is similar. By Lemmas 4.1.1 and 4.1.2, ! ue(re) > 0. (4.2.32) We define O "-- inf{~ < re" we dx > 0}. (4.2.33) ! It is clear we(q) ~>0. Integrating (4.1.1b), we obtain 0 >/-ew'e(O) = - ~we dx d~ 4-/z(ue(o) - ue (re)). (4.2.34) By the definition (4.2.33), we see that ~we dx ~<0 on (7, re) and hence f Te 0>~ ~wedxd~ >~#(ue(o)-ue(re))>~u,-u*. (4.2.35) If ~ ~<min(-1, re), then fo ~ f0 ~ ~'w'e(~') d~" <~ - w'e(~') d~" -- we(q) - we(~). From the definition (4.2.33), we know that either 0 = -L or 0 is a local minimum point of ue(~), In view of Lemmas 4.1.1 and 4.1.2, we(q) e [min(w_, w+), fl]. Then above inequality yields we (~) ~< - ~w'e(~) d~" 4- we (7) ~<u* - u, 4- fl (4.2.36) for all ~ ~<min(- 1, re). In other words, we(~) is bounded from above uniformly in e > 0, /z e [0, 1] and L > 1 if ~ ~<min(-1, re). For ~ e (-1, re], we have, from (4.1.1b), that f re 0 ~> -ew~ dx - -~'w'~(~') d~" +/z(u~(~) - u~(r~)) ~>u, - u*.
- 406. 408 H. Fan and M. Slemrod This implies that we(re) ~<we(-1) + Cl (e) <. u* - u, + ~ + Cl (e). Thus, the statement (i) is proved. (ii) It remains to consider the case when re 6 (-1,0] and/x = 1. For each e, we can choose 0 6 (-2,-1) such that ue(O) <<.u* - u,. By integrating (4.1.1a), with/x -- 1, on [0, re], we obtain L %'e p(wE(rE)) =eu'E(rE)--eu'e(O)+ p(wE(O))-- ~uEdxd~ fo <~-eulE(O) + p(wE(O)) - ~uE dx d~. (4.2.37) Every term on the fight-hand side of (4.2.33) is bounded uniformly in e > 0 and L > 1. Thus, by virtue of the assumption on p in the theorem, WE(re) are bounded from below uniformly in e > 0 and L > 1. D THEOREM 4.2.3. Assume w+ < ot and u_ > u+. Let (uE(~), wE(~)) be a possible solution of (4.1.1) satisfying wE(~) ~ or. Then (i) [[uE(~)IIc([_L,+L];R2 <~C where C is a constant independent of L > 1, /~ ~ [0, 1] and e > O. (ii) [[we(~)[[C([_L,+L];~2 <~C(e) where C(e) is independent of l~ ~ [0, 1], L > 1. (iii) If [p(w)[--+ 00, as [w[--~ 00, and if l~ = 1, then [[wE(~)[[C([-L,+L];R2 ~<C where C is independent of L > 1 and e > O. PROOF. The proof is almost the same as that of Theorem 4.2.2. V1 Combining Theorems 4.1.3, 4.1.4, 4.2.1 and 4.2.2, we obtain the following result. THEOREM 4.2.4. (i) Assume w+ < ot and u_ < u+. There is a solution (uE(~), wE(~)) of (4.1.0) satisfying for any ~l, ~2 e (-00, +00), ~l r ~2 and wE(~l) ~> w := rain(w_, w+) (4.2.38) and that there are at most two disjoint open intervals (a, b) such that (4.2.39) wE(~) ~ (t~, or), and (4.2.39a) either we (a) = 6v, we (b) = ot or we (a) = ~, we (b) = 6v. (4.2.39b)
- 407. Dynamicflows withliquid~vaporphasetransitions 409 (ii) For the case w+ < ~ and u_ > u+, there is a solution of (4.1.0) satisfying w~(~) <<.~. (iii) There is a subsequence {Sn}, Sn --+ O+ as n --+ ~, such that (us, (~), we, (~)) given in (i) and (ii) converges a.e. to a weak solution (u(~), w(~)) of the Riemannproblem (3.1). Furthermore, the solutions we constructed have at most two phase boundaries. PROOF. (i) and (ii) Theorems 4.1.6, 4.2.1, 4.2.2 and 4.2.3 provide the a-priori estimates needed by Theorems 4.1.3 and 4.1.7 Thus, parts (i), (ii) are established. (iii) From Corollary 4.1.5, we know that the solutions of (4.1.0) provided in (i), ue(~) and ws(~) are piecewise monotone. Thus, ({us(~), Ws(~)) given in (i) has total variation bounded uniformly in e > 0. Then the classical Helly's theorem states that there is a sequence {Sn}, en --+ 0+ as n --+ oe, such that (ue(~), we(~)) converges almost everywhere. Apply this limit to the weak form of (3.14), we see that the limit is a weak solution of (3.1). D 4.3. Solutions constructed by vanishing similarity viscosity are also admissible by traveling wave criterion In this section, we shall prove that solutions constructed by vanishing similarity viscosity is also admissible by traveling wave criterion. Let (us(~), we(~)) denote the solution of (4.1.0). From last section, we see that (ue(~), ws(~)) are bounded uniformly in e > 0. Let us denote the upper and lower bounds of ue(~) by u, and u* respectively. Similarly, the upper and lower bounds of we (~) is denoted by w, and w* respectively. For simplicity of presentation, we restrict ourselves to the case w_ < ot </3 < w+. In this case, solutions (ue(~), we(~)) of (4.1.0) have the following shapes: there are two points ~ = 01 < 02, depending on e, such that we (~) ~<or, for ~ E (-cx~, 01], (4.3. la) ot < ws(~) </3 for ~ E (01,02), Ws(01) = Or, We(02) =/3 (4.3.1b) and ws(~) t>/3 for ~/> 02. (4.3.1c) According to Lemmas 4.1.1 and 4.1.2, over each of the intervals (-co, 01) and (02, OO), there are three possibilities: (i) The function we (~) has one local extreme point and us (~) is monotone. (ii) The function us (~) has one local extreme point and we (~) is monotone. (iii) Both us(~) and ws(~) are monotone. ' d2 /dw 2 > 0 hold. According to Over the interval (01,02), inequalities ws(~) > 0 and us shapes (i)-(iii), there are nine different combinations of shapes for (us (~), we (~)). We also see that over each of the regions w ~<or, c~ < w </3 and w ~>/3, we can consider the curve (us (~), ws (~)) in (u, w)-plane as the curve of the function Us(w) or We(u), depending on which of us(~) and ws(~) is monotone.
- 408. 410 H. Fan and M. Slemrod LEMMA 4.3.1. Let ~ > 0 be some fixed small number so that w_ <~-3 </~+3 < w+. (4.3.2) (a) In the region w <. c~-6 (or w ~ fl +6) either dUE(w)/dw or dWE(u)/du is uniformly bounded in e. (b) In the region ~ - 6 <. w <. fl + 3, dUE(w)/dw is uniformly bounded. PROOF. (a) According to the possibilities for shapes of (uE(~), we (~)) in the region w ~<ot and w/> r, there are three cases: Case A. There is a critical point for uE(~) in the region w ~<oe (or w ~>13). We can calculate from (4.1.0) to get d (dUE(__w))(dUE(w)) e e ~-~ dw = dw + p'(wE(~)). (4.3.3) At the critical point ~ = re of uE(~), dUE(w)/dwl~=r~ = 0. Then Equation (4.3.3) says that dUE(w) dw ~=~ is decreasing as ~ increases from re. But dUE(w)/dw cannot decrease to below -maxw,<~w<~w, v/lff(w)l since d/d~(dUE(w)/dw) will become positive if dUE(w)/dw reaches -maxw,<<w<<w, v/lff(w)l. Similarly, as $ decreases from re, dUE(w)/dw will increase but never reaches maxw, <<w<<. w, v/lff(w)l because if it does, d/d~(dUE(w)/dw) will become negative. This shows that dUE(w) dw ~< max v/lp'(w)l (4.3.4) w, <<. vo<<. w* if uE(~) has a critical point re with WE(re) ~<Otor WE(re) ~>ft. Case B. The function wE(~) has a critical point in the region w ~<ot (or w ~>fl). Then, uE(~) is necessarily monotone in the region. If the critical point is in the region w ~<or, then it is the absolute minimum point of wE(~), according to Lemma 4.1.1. If the critical point re is in the region w ~>r, then it is the absolute maximum point of wE(~). Thus, WE(re) is in the region w ~<c~ - 3 (or w>~f+~). From (4.1.0), we derive that du )--[l+ du )]" (4.3.5)
- 409. Dynamic flows with liquid~vaporphase transitions 411 At the critical point ~ --re, we have d~(u)J~=re --0. Similar to our analysis follow- ing (4.3.3), we can prove that dWe (u) du 1 ~< max (4.3.6) ~I~,,~-~luI~+~,~*~ v/Ip'(w)l for w E [w,, ot - 6] U [fl + 6, w*]. Case C. There is no critical point for ue(~) or we(~) in the region w ~<ot (or w >t/3). For this case, we claim that one of IdUe(w)/dwl and IdWe(u)/dul are bounded uniformly. To prove this claim, it suffices to prove if one of them is not bounded uniformly, then the other is. There are the following three possibilities: Subcase C1. The function dUe(w)/dw is not bounded from below uniformly in e in the region w ~<or. Let the absolute minimum point of dUe (w)/dw in the closed region w ~<ot be ~ -- re. Then there is a sequence en, such that duen] --+ -cx3 dw ~=r~n as n --+ oc. (4.3.7) For simplicity, we denote this sequence by e. In the region w ~<c~, it is necessary that dwe,/d~ > 0 in order to connect to w(ec) = w+ > /3 > or. Therefore, we have due,/d~ < 0 in the region w ~< c~ due to (4.3.7). Evaluating Equation (4.3.5) around the point ~ = re, we find that dUe (w)/dw is increasing at ~ = re, the absolute minimum point of dUe(w)/dw over w_ ~< w ~< or. This implies we (re) -- w_ and hence re = -oc. Now, applying our reasoning from (4.3.5)-(4.3.6) and using (4.3.15): dWe (u) du ~=~n --+0 as n-~ cx~, we can obtain (4.3.6). Subcase C2. The function dUe(w)/dw is not bounded from above uniformly in e in the region w ~< or. Let the absolute maximum point of dUe(w)/dw in the closed region {~: we(~) ~<cr} = [-oc, 01] be ~ = re. Then there is a sequence en, such that dUe(w) dw oc as n ~ oe. (4.3.8) For simplicity, we denote this sequence by e. Similar to Subcase C1, when w ~<or, it is necessary that dwe,/d~ > 0. Therefore, we have duen/d~ > 0 in the region w ~<or. By Equation (4.3.5), we find that dUe(w)/dw is increasing at ~ as long as dUe(w) >i max v/Ip'(w)l. dw ~=~
- 410. 412 H. Fan and M. Slemrod This implies we(re) = or. Recalling that We(O1)--~Ol and We(02)-"fl, we can see that re = 01 and dUe (w) /> dUe (w) dw dw ~=01 for 01 ~ ~ ~ 02 (4.3.9) due to Lemma 4.1.2. Then we have f w(O2) dUe(w) Uen (02) -- Uen (01) -- ,JtO(01) dw duen] (13 -or) ~ oo ~ d w / > ~ ~=01 as n --+ c~. This violates the uniform boundedness of ue(~), Theorem 4.2.1. This contradiction shows that Subcase C2 cannot occur. Subcase C3. The function dUe(w)/dw is not bounded uniformly in e in the region w>r The proof for this case is similar to Subcases C 1 and C2. Combining the Cases A-C, we complete the proof of (a). (b) Since We(u) is convex in the region ot ~< w ~< 13, the absolute extreme values of dWe(u)/du over the region ot - 6 ~< w ~< fl + 6 must occur in the region [or - 6, c~] U [fl, fl + 6]. Our proof for Subcase C2 for (a) shows that dUe (w)/dw is bounded uniformly from above when w 6 [or - 6, or]. Now we shall prove that dUe(w)/dw is also bounded uniformly from below when w ~ [or - 6, or]. To this end, we assume its contrary, i.e., there is a sequence {en} such that dUe(w)] dw ~=r~ --~ -cx~ (4.3.10) as n ~ o0 for some re 6 R with wen(re) 6 [c~- 6, c~]. Equation (4.3.3) implies that dUe(w)/dw is decreasing as ~ decreases when dUe(w) ~<- max V/Ip'(w)l. dw w,<<.w<~w* Let ~ = ~1 ) --OO be the point such that W(~I) -- W- and w_ ~< we(~) ~ we(re) for ~ (~l, re ]. Then, we have _ fw(r~) dUe(w) Ren(7~e)-- Uen(~1) dW(~l) dw du en I ~dw<~---d-~w ~=~(c~ -~ - w_) ~ -oc as n --+ cxz. This violates the uniform boundedness of we(~), Theorem 4.2.2. This contradiction proves that dUe(w)/dw is uniformly bounded from above in the region [ol - a,~]. The same proof can be used to prove that dUe(w)/dw is uniformly bounded when w ~ [f,/~ +a]. D
- 411. Dynamic flows with liquid~vaporphase transitions 413 Now, we consider the a --+ 0+ limit of a convergent subsequence of (ue(~), we (~)). We denote the convergent subsequence of (ue(~), we (~)) by (uen(~), wen (~)) and the limit by (u(~), w(~)). LEMMA 4.3.2. Let (uen(~), wen(~)) be a convergent sequences of (ue(~), wE(~)). Then, there is a subsequence of {an}, denoted by {an} again, such that {Uen(W)} converges to a locally Lipschitz continuous (in u or w) curve. Furthermore, the limit (u(~), w(~)) lies on this curve for every ~ E R. PROOF. By further extracting subsequences, we can make all duen/dw or all dw,n/du to be bounded uniformly in an over the region w ~<ot - 8. The same can be achieved for the region w >~ 13 4- 8. For definiteness and simplicity of presentation, we consider the case where all dw,n/du are bounded uniformly in the region w ~<ot - 8, and all duen/dw are bounded uniformly in the region w ~> 13 + 8. Then, the curve (u, n(~), wen (~)) in (u, w)- plane, can be regarded as a function of w in the region w* ) w ~> 134- 6 and a function of u in the region w ~<c~ - 8. These two pieces of curves are connected by the part of the curve (u,n (~), wen (~)) over the interval (4.3.11) where 61 > 6 (4.3.12) and satisfies (4.3.2). This middle piece can be considered, by Lemma 4.3.1 (b), as a function of w. Each of these three pieces of curves are uniformly bounded in C 1 over their domains of definition which are intervals bounded uniformly in e. Thus, there is a subsequence of {an}, denoted by {an} again, such that all these three pieces converges as n --+ cx) to Lipschitz continuous (in variable u or w) curves in (u, w)-plane. Due to the overlaps of the middle curve with the other two pieces, (4.3.11)-(4.3.12), the three pieces of the limit curves form a continuous curve in (u, w)-plane. This curve is locally Lipschitz in u or w with Lipschitz constant uniformly bounded in a. We call this curve the base curve. Now, we prove that the limit (u(~), w(~)) is on the base curve. Fix a ~ ~ R. For definiteness, we shall assume that w(~) ~<ot - 6. All other cases can be handled similarly. Either w(~) < ot - 6 or w(~) = ot - 6 > c~ - 61. In either cases, duen/dw is bounded uniformly in an and the base curve is parameterized as U(w) in the region w, ~< w ~< 13. Then we have IU(w(~)) - u(~) I - nli2n IU~, (w(~)) - u~. (~)l = nli2n IU~, (w(~)) - u~ (w,~ (~))l ~< lim CIw( )- ws < )l -0. n---~(x3 (4.3.13) Thus, the point (u(~), w(~)) is on the base curve U(w). D
- 412. 414 H. Fan and M. Slemrod For convenience, we parameterize the base curve (U(s), W(s)) where s is the length of the arc joining points (u_, w_) and (U(s), W(s)). In this kind of parameterization, the s defined by (u(~), w(~)) = (U(s), W(s)) increases when ~ increases. Now, we study the discontinuities of (u(~), w(~)). Let ~0 be a point of discontinuity of (u(~), w(~)). We use C~o to denote the portion of the base curve in the (u, w)-plane that connects points (u(~0-), w(~0-)) and (u(~0+), w(~0+)). We fix (t/, tb) E C~0. Similar to [20], we define, for n large, ~e, (w; t/, tb) to be the branch of the inverse function of w = wen(~) for which (4.3.14) as n ~ oo. We further define, for n large, ~e,, fien, W'enby the relations ~en := ~En(t~) + e(, (4.3.15) fi~n(() "-- ue, (~e,), (4.3.16) w"~n(() "- wen(~en). (4.3.17) LEMMA 4.3.3. Let ~o be a point of discontinuity of (u(~), w(~)). For (Fte n((), wAen (()) defined above, there is a subsequence of {en}, also denoted by {en}, such that (blen(~'), WAen(~'))--->(t~(~'), W(~'))E C1(I~;I~2) asn --+ oo (4.3.18) uniformly for ( in a compact subset of N. (fi(( ), ~(( )) satisfies thefollowing initial value problem" d~(() d( = -~0(t/(()- u(~0-))+ p(@(())- p(w(~o-)), (4.3.19a) d~(() d( (4.3.19b) (0) -- t/ ~'(0) -- ffJ. (4.3.19c) Furthermore, (fi((), ~(()) lies on C~o. PROOF. Clearly, (fien((), w'~n(()) have uniformly bounded total variation since (ue(~), we(~)) do. Thus, there is a subsequence of {en}, again denoted by {en}, such that (fie~ ((), w'~. (()) --+ (fi((), ~(g')) as n --+ (4.3.20) for any ~"6 IR.
- 413. Dynamicflows with liquid~vaporphase transitions 415 By Lemma 4.3.1, we can choose a small neighborhood V~0 of (u(~0-), w (~0-)) in the (u, w)-plane such that due', s e E IR such that (uen (~), we',(s e)) E V~o} (4.3.21a) or dwe', du E N such that (ue,,(~), we,,(~)) E V~o] (4.3.21b) is bounded uniformly in n. Since in each of the region w ~<or, ot ~< w ~< fl, w ~> fl, at least one of ue(~) and we(~) is monotone, in each of above three regions, one of U(s) and W(s) is monotone. We can further choose V~0 small and (ua, wa) 6 C~0 n V~0 such that U(s) or W(s) is monotone in V~0. For definiteness, we can assume, without loss of generality, that (4.3.21a) holds and both we',(~) and W(s) is monotone in V~0.The proof for the other case is similar. There is, for n large, (4.3.22) such that 3 (4.3.23a) 3 Iw;n<O n>l -YnnZg (4.3.23b) From (4.3.7) and -+ it is easily seen that ~e',(wa) -+ ~0 (4.3.24) and hence 0e', --+ ~0. Since W(s) and we',(~) are monotone, the limit liminfn-+ec we',(0e',) lies between w(~0-) and wa. Thus, extracting, if necessary, another subsequence, we deduce Wen(Oe',) --+ tO2 as n --+ oo (4.3.25) for some 11)2 between w(~0-) and wa. Then, by (2.8a), we have that limn-+o~ue', (we', (Oe',)) - g(w2) -- U2, (4.3.26)
- 414. 416 H. Fan and M. Slemrod where (u2, 1/)2) 9 V~o. For simplicity, we shall write e instead of en in the rest of this section. Integrating Equations (1.3) from 0e to re :-- ~e(tb) + e~', we get d~s(~') d~" = -~o[fie(~')- ue(Oe)] + P(We (~')) - p(we(Oe)) f0 ~ + eue' (Oe) - (~ - ~o)u~e(~) d~, (4.3.27a) d~e (~') d~" = -[a~(~- .~(o~] - ~o[~(c~- w~(O~] ! ! + ewe(Oe) - (~ - ~0)we(~) d~. (4.3.27b) By (4.3.23) ew~e(Oe) and eu~e(Oe) approach 0 as e --+ 0 uniformly in ~'. Recalling that 0e --+ ~o, re ~ ~o as n ~ c~, uniformly in ~" for ~"in compact subsets of R, we see that the last term in (4.3.27a, b) vanish, as n ~ cxz,uniformly in ~"in a compact set. A classical theorem of the theory of ordinary differential equations implies that (fie(g), we(~')) --+ (fi(~'), ~(~')), as n --+ ~x~,uniformly on compact subsets of R, and that dfi(~') d~" = --~O(t~(~') -- u2) -t- p(w(~')) -- p(w2), (4.3.28a) d~(~) m d~ [a~- .~] - ~0(~- w~), (4.3.28b) fi(0) = if, ~(0) = if). (4.3.28c) By letting V~o shrink to (u(~o-), w(~o-)) so as to force (u2, W2) ~ (U(~0--), W(~0--)), we obtain (4.3.19). Similar to our proof of Lemma 4.3.2, we can prove that (fi(~'), ~(~')) is on the curve (U(s), W(s)) for all ~" e R. We note that (fi(0) = ~(0)) = 07, tb) e C~o, and that as ~" increases (or decreases) from ~"= 0, the point (fi(~'), ~(~')) moves toward the end point (u(~o+), w(~o+)) ((u(~o-), w(~o-))) along C~o. The point (fi(~'), ~(~')) cannot cross (u(~o+), w(~o+)) and (u(~o-), w(~o-)) to go outside of C~o. This is because if it did go out of C~o, there would be a point ~'l 9 R such that (a(~l~. ~(~l~)- (.(~o-~. w(~o-~) or (.(~o+~. w(~o+~) Then the Rankine-Hugoniot condition, satisfied by any jump solution of (3.1) with speed ~o, -~o(u(~o+)- u(~o-))+ p(w(~o+))- p(w(~o-))= O, (4.3.29a)
- 415. Dynamic flows with liquid~vapor phase transitions 417 - - [. - - o (4.3.29b) yields that at (fi((), ~(()) -- (u(~0-), w(~0-)) or (t~((), ~(()) _= (u(~0+), w(~0+)) by the uniqueness of solutions of initial value problems of systems of ordinary differential equations. This, however, violates the (fi(0), ~(0)) = (t7,tb) ensured by (4.3.14)-(4.3.17). This contradiction proves that (/~((), ~(()) ~ C~0 for all ( E R. F-1 COROLLARY 4.3.4. Let (u(~), w(~)) be a weak solution of (3.1) constructed as the limit of a convergent sequence {Uen(~), Wen(~)} of solutions of (3.14) with the same initial data (3.1)3. Then, (u(~), w(~)) is also admissible by the traveling wave criterion based on (3.13), which is the same as that based on (1.2) when A = 1/4. PROOF. The limit (u(~), w(~)) has bounded total variation. Then points of discontinuity of (u(~), w(~)) are points of jump discontinuity. Let ~0 be a point of jump discontinuity of (u(~), w(~)). Lemma 4.3.3 states that (4.3.19) has a solution. We note that the system (4.3.19)1 and 2 is equivalent to the traveling wave equation (3.13) and the speed s = ~0. Indeed, the speed of the jump discontinuity of (u(~), w(~)) at x/t = ~ -~o is ~0. We note that C~0 in last lemma is the portion of the base curve connecting the points (u(~0-), w(~o-)) and (u(~o+), w(~0+)). As ( increases from 0 to oo, the point (fi((), ~(()) moves monotonically toward (u(~0+), w(~0+)) along the curve C~o. In the ( -+ cxz limit, (fi((), ~(()) must approach to an equilibrium point of (4.3.19) on C~0.Equilibrium points of (4.19) are points (u l, Wl) that satisfies the Rankin-Hugoniot condition (4.3.27) with (u(~0+), w(~0+)) replaced by (ul, Wl). Similarly, as ( decreases from 0 to -c~, the point (/~((), ~(()) will move toward (u (~o-), w(~o-)) and approaches an equilibrium point in the ( -+ cx~ limit. Thus, when there are only finitely many equilibrium points for each fixed speed ~0, the jump discontinuity (u(~0-), w(~o-)), (u (~0+), w(~o+)) can be connected together by finitely many traveling waves of the same speed ~'o. F-l References [1] R. Abeyaratne and J. Knowles, Kinetic relations and the propagation ofphase boundaries in solids, Arch. Rational Mech. Anal. 114 (2) (1991), 119-154. [2] R. Abeyaratne and J. Knowles, Dynamics ofpropagating phase boundaries: thermoelastic solids with heat conduction, Arch. Rational Mech. Anal. 126 (3) (1994), 203-230. [3] E.C. Aifantis and J. Serrin, The mechanical theory of fluid interfaces and Maxwell's rule, J. Colloidal Interface Sci. 96 (1983), 517-529 [4] E Asakura, Large time stability ofpropagating phase boundaries, Hyperbolic Problems: Theory, Numerics, Applications, Vol. I (Ztirich, 1998), Internat. Ser. Numer. Math., Vol. 129, Birkhauser, Basel (1999), 21-29. [5] N. Bedjaoui and EG. LeFloch, Difussive-dispersive traveling waves and kinetic relations: an hyperbolic- elleptic model of phase transitions, Preprint (2000). [6] N. Bedjaoui and EG. LeFloch, Difussive-dispersive traveling waves and kinetic relations, Part I: nonconvex hyperbolic conservation laws, Preprint (2000). [7] S. Benzoni-Gavage, Stability of multi-dimensional phase transitions in a van der Waals fluid, Nonlinear Anal. 31 (1-2) (1998), 243-263. [8] S. Benzoni-Gavage, Nonuniqueness ofphase transitions near the Maxwell line, Proc. Amer. Math. Soc. 127 (4) (1999), 1183-1190.
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- 419. CHAPTER 5 The Cauchy Problem for the Euler Equations for Compressible Fluids Gui-Qiang Chen Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA E-mail: gqchen@math.northwestern,edu and Dehua Wang Department of Mathematics, Universityof Pittsburgh, Pittsburgh, PA 15260, USA E-mail: dwang@math.pitt,edu Contents 1. Introduction .................................................. 423 2. Local well-posedness for smooth solutions ................................. 428 3. Global well-posedness for smooth solutions ................................ 432 4. Formation of singularities in smooth solutions ............................... 436 4.1. One-dimensional Euler equations .................................. 437 4.2. Three-dimensional Euler equations ................................. 438 4.3. Other results .............................................. 441 5. Local well-posedness for discontinuous solutions ............................. 443 6. Global discontinuous solutions I: Riemann solutions ........................... 446 6.1. The Riemann problem and Lax's theorems ............................. 446 6.2. Isothermal Euler equations ...................................... 449 6.3. Isentropic Euler equations ...................................... 451 6.4. Non-isentropic Euler equations .................................... 454 7. Global discontinuous solutions II: Glimm solutions ............................ 458 7.1. The Glimm scheme and existence .................................. 458 7.2. Decay of solutions .......................................... 463 7.3. L 1-stability of Glimm solutions ................................... 468 7.4. Wave-front tracking algorithm and L 1-stability ........................... 470 8. Global discontinuous solutions III: entropy solutions in BV ....................... 479 8.1. Generalized characteristics and decay ................................ 479 8.2. Uniqueness of Riemann solutions .................................. 483 HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME I Edited by S.J. Friedlander and D. Serre 9 2002 Elsevier Science B.V. All rights reserved 421
- 420. 422 G.-Q. Chen and D. Wang 8.3. Large-time stability of entropy solutions .............................. 486 9. Global discontinuous solutions IV: entropy solutions in L~ ....................... 488 9.1. Isentropic Euler equations ...................................... 488 9.2. Entropy-entropy flux pairs ...................................... 489 9.3. Compactness framework ....................................... 492 9.4. Convergence of the Lax-Friedrichs scheme and the Godunov scheme .............. 497 9.5. Existence and compactness of entropy solutions .......................... 504 9.6. Decay of periodic entropy solutions ................................. 505 9.7. Stability of rarefaction waves and vacuum states .......................... 508 9.8. Other results .............................................. 512 10. Global discontinuous solutions V: the multidimensional case ....................... 513 10.1. Multidimensional Euler equations with geometric structure .................... 513 10.2. The multidimensional Riemann problem .............................. 517 11. Euler equations for compressible fluids with source terms ........................ 521 11.1. Euler equations with relaxation ................................... 521 11.2. Euler equations for exothermically reacting fluids ......................... 529 Acknowledgments ................................................. 531 References ..................................................... 531 Abstract Some recent developments in the study of the Cauchy problem for the Euler equations for compressible fluids are reviewed. The local and global well-posedness for smooth solutions is presented, and the formation of singularity is exhibited; then the local and global well- posedness for discontinuous solutions, including the B V theory and the L ~ theory, is extensively discussed. Some recent developments in the study of the Euler equations with source terms are also reviewed.
- 421. The Cauchyproblem for the Euler equationsfor compressiblefluids 423 1. Introduction The Cauchy problem for the Euler equations for compressible fluids in d space dimensions is the initial value problem for the system of d + 2 conservation laws Otp+V.m=O, (mm) 8tm+V. +Vp--O, P (1.1) for (x, t) ~ IRd+l , Rd++l "= ]Rd • (0, OO), with initial data (p, m, E)lt=o : (po, mo, Eo)(x), x ~ ~d, (1.2) where (P0, m0, E0)(x) is a given vector function of x 6 IRd. System (1.1) is closed by the constitutive relations 1 ]ml 2 p = p(p, e), E = + pe. (1.3) 2 p In (1.1) and (1.3), r -- 1/p is the deformation gradient (specific volume for fluids, strain for solids), v -- (Vl ..... Vd) T is the fluid velocity, with pv = m the momentum vector, p is the scalar pressure, and E is the total energy, with e the internal energy which is a given function of (r, p) or (p, p) defined through thermodynamical relations. The notation a | b denotes the tensor product of the vectors a and b. The other two thermodynamic variables are the temperature 0 and the entropy S. If (p, S) are chosen as the independent variables, then the constitutive relations can be written as (e, p, O) = (e(p, S), p(p, S), O(p, S)), (1.4) governed by P 0 dS = de + p dr = de - --T dp. p~ (1.5) For a polytropic gas, R p = RpO, e = cvO, Y = 1 +--, (1.6) Cv and p = p(p, S) = xp z e s/cv, tr pz_ leS/cv e -- ~ , (1.7) y--1
- 422. 424 G.-Q. Chen and D. Wang where R > 0 may be taken to be the universal gas constant divided by the effective molecular weight of the particular gas, Cv > 0 is the specific heat at constant volume, y > 1 the adiabatic exponent, and x > 0 can be any constant under scaling. As will be shown in Section 4, no matter how smooth the Cauchy data (1.2) are, solutions of (1.1) generally develop singularities in a finite time. Hence, system (1.1) is complemented by the Clausius inequality Ot(pa(S)) + V. (ma(S)) ~>0 (1.8) in the sense of distributions for any a(S) ~ C1,at(S) ~ 0, in order to single out physically relevant discontinuous solutions, called entropy solutions. The Euler equations for a compressible fluid that flows isentropically take the following simpler form: Otp+V.m=O, Otm+V. (m| m) (1.9) +Vp=0, where the pressure is regarded as a function of density, p = p(p, So), with constant So. For a polytropic gas, P(P) = xoP• V>I, (1.10) where x0 > 0 is any constant under scaling. This system can be derived as follows. It is well known that, for smooth solutions of (1.1), the entropy S(p, E) is conserved along fluid particle trajectories, i.e., Ot(/9S) + V. (mS) = O. (1.11) If the entropy is initially a uniform constant and the solution remains smooth, then (1.11) implies that the energy equation can be eliminated, and the entropy S keeps the same constant in later time, in comparison with non-smooth solutions (entropy solutions) for which only S(x, t) ~ min S(x, 0) is generally available (see [297]). Thus, under constant initial entropy, a smooth solution of (1.1) satisfies the equations in (1.9). Furthermore, it should be observed that solutions of system (1.9) are also a good approximation to solutions of system (1.1) even after shocks form, since the entropy increases across a shock to third-order in wave strength for solutions of (1.1) (cf. [120]), while in (1.9) the entropy is constant. Moreover, system (1.9) is an excellent model for isothermal fluid flow with F = 1, and for shallow water flow with y = 2. In the one-dimensional case, system (1.1) in Eulerian coordinates is Otp + Oxm =0, Otm + Ox --7 + p =0, OtE + OX(p(E + p)) =0, (1.12)
- 423. The Cauchy problem for the Euler equations for compressible fluids 425 1 m• with E = 2 p + pe. The system above can be rewritten in Lagrangian coordinates in one- to-one correspondence so long as the fluid flow stays away from vacuum p = 0: Otr -- OxV --0, Otv+ Oxp=0, Ot(e + ~) + Ox(pv) = O, (1.13) with v = m/p, where the coordinates (x, t) are the Lagrangian coordinates, which are different from the Eulerian coordinates for (1.12); for simplicity of notations, we do not distinguish them. For the isentropic case, systems (1.12) and (1.13) reduce to: and atp + Oxm= O, Otm+ ox(m-~~+p) =0, (1.14) Otr -- OxV =O, Otv+ Oxp--0, (1.15) respectively, where the pressure p is determined by (1.10) for the polytropic case, p -- p(p) = p(r), r = 1/p. The Cauchy problem for all the systems above fits into the following general conservation form: Otu+V.f(u)=O, u~R n, x~R e , (1.16) with initial data: ul,=o = uo(x), (1.17) where f = (fl ..... fd) :R n --+ (Rn)d is a nonlinear mapping with fi :R n ~ Rn, i = 1..... d. Besides (1.1)-(1.15), many partial differential equations arising in the physical or engineering sciences can also be formulated into the form (1.16) or its variants. The hyperbolicity of system (1.16) requires that, for any co ~ Sd-l, the matrix (Vf(u) 9co)n• have n real eigenvalues )~i(u, co), i = 1, 2..... n, and be diagonalizable. One of the main difficulties in dealing with (1.16) and (1.17) is that solutions of the Cauchy problem (even those starting out from smooth initial data) generally develop singularities in a finite time, because of the physical phenomena of focusing and breaking of waves and the development of shock waves and vortices, among others. For this reason, attention focuses on solutions in the space of discontinuous functions. Therefore, one can not directly use the classical analytic techniques that predominate in the theory of partial differential equations of other types.
- 424. 426 G.-Q. Chen and D. Wang Another main difficulty is nonstrict hyperbolicity or resonance of (1.16), that is, there exist some 090 9 Sd-1 and u0 9 I~n such that ~i(u0, coo) -- )U(u0, coo) for some i ~ j. In particular, for the Euler equations, such a degeneracy occurs at the vacuum states or from the multiplicity of eigenvalues of the system. The correspondence of (1.8) in the context of hyperbolic conservation laws is the Lax entropy inequality: OtO(u) + V. q(u) ~<0 (1.18) in the sense of distributions for any C2 entropy-entropy flux pair (r/, q):]1~n ~ ]1~ • I~d, q = (ql ..... qd), satisfying V2rl(U) ~ O, Vqi(u) : Vr/(u)Vfi(u), i = 1..... d. Most sections in this paper focus on the Cauchy problem for one-dimensional hyperbolic systems of n conservation laws OtU+Oxf(u)=O, ueR n, xeI~, t>O, (1.19) with Cauchy data: ult=0 = u0(x). (1.20) The Euler equations can describe more complicated physical fluid flows by coupling with other physical equations. One of the most important examples is the Euler equations for nonequilibrium thermodynamic fluids. In local thermodynamic equilibrium as we discussed above, system (1.1) is closed by the constitutive relation (1.3). When the temperature varies over a wide range, the gas may not be in local thermodynamic equilibrium, and the pressure p may then be regarded as a function of only a part e of the specific internal energy, while another part q is governed by a rate equation: Ot(pq) 4- Vx . (mq) = Q(p,e) -q es(p, e) (1.21) and Iml2 p = p(p, e), E = 4- p(e 4- q), (1.22) 2p where e > 0 is a parameter measuring the relaxation time, which is small in general, and Q (p, e) and s (p, e) are given functions of (p, e). The equations in (1.1) and (1.21) with (1.22) define the Euler equations for nonequilibrium fluids, which model the nonequilibrium thermodynamical processes.
- 425. The Cauchyproblemfor the Euler equationsfor compressiblefluids 427 Another important example is the inviscid combustion equations that consist of the Euler equations in (1.1) adjoined with the continuum chemistry equation: O,(pZ) 4- V . (mZ) -- -ck(O)pZ, 4>(0)- Ke -~176176 (1.23) where 00 and K are some positive constants, Z denotes the mass fraction of unburnt gas so that 1 - Z is the mass fraction of burnt gas. Here we assume that there are only two species present, the unburnt gas and the burnt gas, and the unburnt gas is converted to the burnt gas through a one-step irreversible exothermic chemical reaction with an Arrhenius kinetic mechanism. As regards the equations in (1.1), a modification of the internal energy e is the only change in these equations. The internal energy of the mixture, e(p, S, Z), is defined within a constant by e(p, S, Z) -- Zeu(p, S) 4- (1 - Z)eb(p, S), with eu and eb the internal energies of the unburnt and burnt gas, respectively. For simplicity, we assume that both of the burnt and unburnt gas are ideal with the same y-law so that eu(p, S) = CvO+ q0, eb = cvO, with q0 > 0 the normalized energy of formation at some reference temperature for the unburnt gas for an exothermic reaction. Then e(p, S, Z) = cvO(p, S) 4- qoZ, p(p, S) 0 (p, S) -- ~ . (1.24) Rp Then the equations in (1.1) and (1.23) with (1.24) define the inviscid combustion equations, which model detonation waves in combustion. This paper is organized as follows. In Section 2, we present a local well-posedness theory for smooth solutions and then in Section 3 a global well-posedness theory for smooth solutions. In Section 4, we exhibit the formation of singularity in smooth solutions, the main feature of the Cauchy problem for the Euler equations. In Section 5, we present a local well-posedness theory for discontinuous entropy solutions. From Section 6 to Section 10, we discuss global well-posedness theories for discontin- uous entropy solutions. In Section 6, we present a global theory for discontinuous entropy solutions of the Riemann problem, the simplest Cauchy problem with discontinuous initial data. First we recall two Lax's theorems for the local behavior of wave curves in the phase space and the existence of global solutions of the Riemann problem, respectively, for general one- dimensional conservation laws with small Riemann data. Then we discuss the construction of global Riemann solutions and their behavior for the isothermal, isentropic, and non- isentropic Euler equations in (1.12)-(1.15)with large Riemann data, respectively. In Section 7, we focus on the global discontinuous solutions obtained from the Glimm scheme [130], called Glimm solutions. We first describe the Glimm scheme for hyperbolic
- 426. 428 G.-Q. Chenand D. Wang conservation laws and a global well-posedness theory for the Glimm solutions, including the existence, decay, and Ll-stability of the Glimm solutions. The Glimm scheme is also applied to the construction of global entropy solutions of the isothermal Euler equations with large initial data. We also present an alternative method, the wave-front tracking method, to construct global discontinuous solutions, which can be identified with a trajectory of the standard Riemann semigroup, and to yield the L 1-stability of the solutions. In Section 8, our focus is on general global discontinuous solutions in L~ A B ~oc satisfying the Lax entropy inequality and without specific reference on the method for construction of the solutions. We first describe a theory of generalized characteristics and its direct applications to the decay problem of the discontinuous solutions under the assumption that the traces of the solutions along any space-like curves are functions of locally bounded variation. Then we study the uniqueness of Riemann solutions and the asymptotic stability of entropy solutions in BV for gas dynamics, without additional a priori information on the solutions besides the natural Lax entropy inequality. In Section 9, our focus is on the one-dimensional system of the isentropic Euler equations and its global discontinuous solutions in L~ satisfying only the weak Lax entropy inequality. We first carefully analyze the system and its entropy-entropy flux pairs. Then we describe a general compactness framework, with a proof for the case y = 5/3, for establishing the existence, compactness, and decay of entropy solutions in L~, and the convergence of finite-difference schemes including the Lax-Friedrichs scheme and the Godunov scheme. We discuss the stability of rarefaction waves and vacuum states even in a broader class of discontinuous entropy solutions in Le~. We also record some related results for the system of elasticity and the non-isentropic Euler equations. In Section 10, we discuss global discontinuous solutions for the multidimensional case. We describe a shock capturing difference scheme and its applications to the multidimensional Euler equations for compressible fluids with geometric structure. Then we present some classifications and phenomena of solution structures of the two- dimensional Riemann problem, especially wave interactions and elementary waves, for the Euler equations and some further results in this direction. In Section 11, we consider the Euler equations for compressible fluids with source terms. Our focus is on two of the most important examples: relaxation effect and combustion effect. Some new phenomena are reviewed. We remark that, in this paper, we focus only on some recent developments in the theoretical study of the Cauchy problem for the Euler equations for compressible fluids. We refer the reader to other papers in these volumes, as well as Glimm and Majda [134], Godlewski and Raviart [138], LeVeque [189], Lions [201], Perthame [255], Tadmor [296], Toro [306], and the references cited therein for related topics including various kinetic formulations and approximate methods for the Cauchy problem for the Euler equations. 2. Local well-posedness for smooth solutions Consider the three-dimensional Euler equations in (1.1) and (1.7) for polytropic compress- ible fluids staying away from the vacuum, which are rewritten in terms of the density p 6 R,
- 427. The Cauchyproblemfor the Euler equationsfor compressiblefluids 429 the velocity v 6 R 3, and the entropy S 6 11~(taking tc = Cv -- 1 without loss of generality) in the form: Ot/9 + V. (pv) = 0, Ot(pv) + V. (pv | v) + Vp = 0, OtS + v. VS-O, (2.1) with the equation of state: p = p(p, S) -- p• es, y > 1. System (2.1) is a 5 • 5 system of conservation laws. It can be written in terms of the variables (p, v, S) in the equivalent form in the region where the solution is smooth: Otp -+-v. Vp + yp V . v=0, p(Otv + v. Vv) + V p- 0, OrS + v. VS =0, (2.2) with p = p(p, S) - pl/• e-S~• The norm of the Sobolev space HS (Ra) is denoted by 12 Ilglls2= E d[Dag dx. Ioll~<s For g 6 L c~([0, T]; HS), define 111g IIIs,r = sup IIg<,t>lls, o~<t~<T For the Cauchy problem of (2.2) with smooth initial data: (p, v, S)lt=0 - (p0, v0, S0)(x), (2.3) the following local existence theorem of smooth solutions holds. THEOREM 2.1. Assume (Po, vo, So) 6 H s N Lc~(R3) with s > 5/2 and po(x) > 0. Then there is a finite time T E (0, cx)), depending on the H s and L ~ norms of the initial data, such that the Cauchy problem (2.2) and (2.3) has a unique bounded smooth solution (p, v, S) E C 1(R3 x [0, T]), with p(x, t) > 0 for all (x, t) E R 3 x [0, T], and (p, v, S) 6 C([0, T]; H s) A C 1([0, T]; HS-1). Consider the Cauchy problem (1.16) and (1.17) for a general hyperbolic system of conservation laws with the values of u lying in the state space G, an open set in 1t~ n. The state space G arises because physical quantities such as the density should be positive. Assume that (1.16) has the following structure of symmetric hyperbolic systems: For all u ~ G, there is a positive definite symmetric matrix Ao(u) that is smooth in u and satisfies CO1In ~<A0 (u) <~C0In (2.4)
- 428. 430 G.-Q. Chen and D. Wang with a constant co uniform for u E G1, for any G1 C G1 C G, such that Ai(u) = A0(u)Vfi (u) is symmetric, where Vfi (u), i = 1..... d, are the n • n Jacobian matrices and In is the n • n identity matrix. A consequence of this structure for (1.16) is that the linearized problem of (1.16) and (1.17) is well-posed (see Majda [223]). The matrix A0(u) is called the symmetrizing matrix of system (1.16). Multiplying (1.16) by the matrix A0(u) and denoting A(u) = (A1 (u) ..... Ad(u)) yield the system: Ao(u)Otu + A(u)Vu = 0. (2.5) An important observation is that almost all equations of classical physics of the form (1.16) admit this structure. For example, the equations in (2.2) for polytropic gases are symmetrized by the 5 • 5 matrix (yp) 1 0 O) Ao(p, S) = 0 p(p, S)I3 0 . 0 0 1 Therefore, Theorem 2.1 is a consequence of the following theorem on the local existence of smooth solutions, with the specific state space G - {(p, v, S)7-" p > 0} C ~5, for the general symmetric hyperbolic system (1.16). THEOREM 2.2. Assume that u0" R d --+ G is in H s fq Loo with s > d/2 + 1. Then, for the Cauchy problem (1.16) and (1.17), there exists a finite time T -- T(llu011s, Ilu011Lo~) (0, oo) such that there is a unique bounded classical solution u 6 C 1(Rd x [0, T]) with u(x, t) ~ G for (x, t) ~ Rd x [0, T] and u 6 C([0, T]; H s) 0 CI([0, T]; HS-1). The proof of this theorem proceeds via a classical iteration scheme. An outline of the proof of Theorem 2.2 (thus Theorem 2.1) is given as follows. To prove the existence of the smooth solution of (1.16) and (1.17), it is equivalent to construct the smooth solution of (2.5) and (1.17) by applying the symmetrizing matrix A0(u). Choose the standard mollifier j (x) 6 C~ (Rd), supp j (x) ___{x" Ixl ~< 1}, j(x) >~ 0, fRdj(x)dx = 1, and set je(x) = e-dj(x/e). For k- 0, 1, 2..... take e~ = 2-~eo, where eo > 0 is a constant, and define u~ 6 Coo (Rd) by u~(x) = Jeku0(x) - f•d je~ (X -- y)u0(y) dy. We construct the solution of (2.5) and (1.17) through the following iteration scheme: Set u~ t) - u~ and define u~+1(x, t), for k - 0, 1, 2..... inductively as the solution of the linear equations: A0(uk)0tu k+l + A(u/~)Vu/~+1 -- O, uk+l It=0 -- u~+1 (x). (2.6) From the well-known properties of the mollification: Ilu~ - u011s ~ 0, as k --+ ~, and Ilu~ - u0 II0 <~ C0ek Ilu0 Ill, for some constant Co, it is evident that u~+1 E C ~ (R d X [0, Tk])
- 429. The Cauchyproblem for the Euler equationsfor compressiblefluids 431 is well-defined on the time interval [0, Tk]. Here Tk > 0 denotes the largest time where the estimate Illu~ - U0111~,Tk~< C1 holds for some constant C1 > 0. Then there is a constant T, > 0 such that Tk >~ T, (T0 = c~) for k -- 0, 1, 2..... which follows from the following estimates" k+l Illu~+l - u~ <- c,, Illu, Ills-l,T, ~<Ce, (2.7) for all k - 0, 1, 2..... with some constant C2 > 0. From (2.6), we obtain A0(uk)0t (u~+l- u ~) + A (u~:)V(u~+1- u~) = E~:, (2.8) where Ek = -(A0(u k) -- Ao(uk-1))Otuk- (A(u k) -- A(uk-1))Vu k- Use the standard energy estimate method for the linearized problem (2.8) to obtain IIIu~+' - u~l110,~ ~ CeCY ([[u~+' - u~ll0 + zl IIE~ I110,T) The property of mollification, (2.7), and Taylor's theorem yield I1-~+' - -~oIio ~<C2-~' ItiE, it10,~ ~ cI IIu~ - u~-' II10,~. For small T such that C2T exp(CT) < 1, one obtains OO El Iluk+l - u l110 T k=l which implies that there exists u E C ([0, T]; L 2(•d)) such that lim IIIu~ - ul110,~ - 0. (2.9) k--+cx~ From (2.7), we have Illu~llls,T + Illut~llls_l,T ~ C, and u~(x, t) belongs to a bounded set of G for (x, t) ~ ~d x [0, T]. Then the interpolation inequalities imply that, for any r with O<.r<s, 1-r/s r/s 1-r/s Illu~ - u' Ill ~ <~CsII1-~ - -' III0,~ II1-~ --' II ,,~ <~c lllu~ --' III r, 0, T (2.10) From (2.9) and (2.10), limk~ IIluk- ulllr, T --0 for any 0 ~< r < s. Thus, choosing r > d/2 + 1, Sobolev's lemma implies .k __~u in C([0, t]; C 1(I~d)). (2.11)
- 430. 432 G.-Q. Chen and D. Wang From (2.8) and (2.11), one can conclude that u~ ~ u in C([0, T]; C(IRd)), u a cl(]l~ d • [0, T]), and u(x, t) is the smooth solution of (1.16) and (1.17). To prove u ~ C([0, T]; H s) A C 1([0, T]; HS-1), it is sufficient to prove u ~ C([0, T]; HS), since it follows from the equations in (2.5) that u ~ C l ([0, T]; Hs-l). The proof can be further reduced to verifying that u(x, t) is strongly fight-continuous at t = 0, since the same argument works for the strong fight-continuity at any other t ~ [0, T), and the strong fight-continuity on [0, T) implies the strong left-continuity on (0, T] because the equations in (2.5) are reversible in time. REMARK 2.1. Theorem 2.2 was established by Majda [223] which relies solely on the elementary linear existence theory for symmetric hyperbolic systems with smooth coefficients (Courant and Hilbert [77]), as we illustrated above. Moreover, a sharp continuation principle was also proved there: For u0 ~ H s, with s > d/2 + 1, the interval [0, T) with T < oe is the maximal interval of the classical H s existence for (1.16) if and only if either II(Ut, Du)IIz~ ~ oo as t --+ T, or, as t ~ T, u(x, t) escapes every compact subset K ~ G. The first catastrophe in this principle is associated with the formation of shock waves in the smooth solutions, and the second is associated with a blow-up phenomenon. Kato also gave a proof of Theorem 2.2, in [164], which uses the abstract semigroup theory of evolution equations to treat appropriate linearized problems. In [165], Kato also formulated and applied this basic idea in an abstract framework which yields the local existence of smooth solutions for many interesting equations of mathematical physics. See Crandall and Souganidis [78] for related discussions. In [226], Makino, Ukai and Kawashima established the local existence of classical solu- tions of the Cauchy problem with compactly supported initial data for the multidimensional Euler equations, with the aid of the theory of quasilinear symmetric hyperbolic systems; in particular, they introduced a symmetrization which works for initial data having compact support or vanishing at infinity. There are also discussions on the local existence of smooth solutions of the three-dimensional Euler equations (2.1) in Chemin [35]. REMARK 2.2. For the one-dimensional Cauchy problem (1.19) and (1.20), it is known from Friedrichs [122], Lax [175], and Li and Yu [195] that, if u0(x) is in C 1 for all x E R with finite C 1 norm, then there is a unique C 1 solution u(x, t), for (x, t) 6 IR x [0, T], with sufficiently small T. As a consequence, the one-dimensional Euler equations in (1.12)- (1.15) admit a unique local C 1 solution provided that the initial data are in C 1 with finite C 1 norm and stay away from the vacuum. 3. Global well-posedness for smooth solutions Consider the Cauchy problem for the one-dimensional isentropic Euler equations of gas dynamics in (1.14), for x ~ IR and t > 0, with initial data: (p, re)It=0 = (p0, mo)(x), (3.1)
- 431. The Cauchyproblemfor the Euler equationsfor compressiblefluids 433 and y-law for pressure: p(p) = p• ~, > 1. (3.2) For the case 1 < y ~< 3, which is of physical significance, system (1.14) is genuinely nonlinear in the sense of Lax [181] in the domain {(x, t)" p(x, t) ~ 0}. For p > 0, consider the velocity v - m/p and vo(x) - mo(x)/po(x). The eigenvalues of (1.14) are 1 --V--C, ~,2--V+C, where c = pO, with 0 - ~ 6 (0, 1], is the sound speed. The Riemann invariants of (1.14) are Wl -- Wl (p, V) :--- V -+- pO pO 0 ' W2 -- w2(p, V) := V 0 Set WlO(X) "-- Wl (po(x), VO(X)), W20(X) "-- W2(Ro(X), VO(X)) as the initial values of the Riemann invariants. With the aid of the method of characteristics (see Lax [178]), the following global existence theorem of smooth solutions of (1.14) and (3.1) can be proved. THEOREM 3.1. Suppose that the initial data (Po, vo)(x), with po(x) > O, are in CI(]R), with finite C 1 norm and ' (x)>~o, Wl0 W20(X ) ~>0, (3.3) for all x ~ IR. Then the Cauchy problem (1.14) and (3.1) has a unique global C 1 solution (p, v)(x, t), with p(x, t) > Ofor all x E R and t > O. PROOF. First we show that, if po(x) > 0, no vacuum will develop at any time t > 0 for the smooth solution. From the first equation of (1.14), d dt p = -pOx v, (3.4) where d = 8t + v(x, t)G dt denotes the directional derivative along the direction dx =v(x,t). dt (3.5)
- 432. 434 G.-Q. ChenandD. Wang For any point (2, D 6 N2 :-- {(x, t)" x 6 N, t 6 N+}, N+ = (0, oe), the integral curve of (3.5) through (2, D is denoted by x = x(t; 2, D. At t = 0, it passes through the point (x0(2, D, 0) "- (x(0; 2, D, 0). Along the curve x = x(t; 2, D, the solution of the ordinary differential equation (3.4) with initial data: Plt=o -- po(xo(~, i)) (f0 ) P(x, D = po(xo(x, D) exp - OxV(X(t; 2, D, t) dt > O. To prove the global existence of the C 1 solution (p, v)(x, t), given the local existence from Remark 2.2, it is sufficient to prove the following uniform a priori estimate: For any fixed T > 0, if the Cauchy problem (1.14) and (3.1) has a unique C 1 solution (p, v)(x, t) for x 6 R and t 6 [0, T), then the C l norm of (p, v)(x, t) is bounded on 1R x [0, T]. For a smooth solution (p, v) of system (1.14), one can verify by straightforward calculations that the derivatives of the Riemann invariants Wl and w2 along the characteristics are zero: t__ 0 w 1 , w2 --0, (3.6) where ~ = Ot + )~20x and ~ = Ot -a t- ~.10x are the differentiation operators along the = 0 in (3.6) with respect to the spatial variable characteristics. Differentiate the equation w 1 x to obtain 02txWl -Jr-~,202xWl @ OWl~,2(OxWl)2 + Ow2~,2OxWlOxt02 =0. ~ - 2COxW2, by setting r = OxW l and noticing Since 0 = w 2 --- w 2 ! 1 +0 1-0 W 2 ~,2 = ~,2(Wl, W2) = 2 Wl + 2 w2, OxW2 2c one has 1-0 1 +Or2 + w~2r =0. r~ + 2 4c Set 0-1 0-1 s - ~ In p = ~ ln(wl - w2). 2 20 Then 1-0 = -- W 2 0to 2 S = Ot02S 4c and s ~ t 1-0 4c ! ~ W 2.
- 433. The Cauchy problemfor the Euler equationsfor compressiblefluids 435 Thus , 1 +Or2 r + ~ +s'r--O. 2 Set g - eSr - - p(O-1)/2OxWl" Then 1-0 ,1-4-0(0 )20 g -------~ [Wl - w2l g2. (3.7) Similarly, for h - - p(O-1)/2OxW2, one has h~ ~ m ~ I-0 1+o(o )20 2 ~[wl - w21 h2. (3.8) Let x -- x(fl, t) be the forward characteristic passing through any fixed point (/3, 0) at t = 0, defined by dx(fl, t) dt x(3, o)= 3. According to (3.6), wl is constant along characteristics, and thus wl(x(fl, t),t)= wl (13,0) = Wl0(fl) and sup [wl (x, t)[ = sup [wlo(x)[. Similarly, w2 is constant along the backward characteristics corresponding to the eigenvalue )~l, and sup [wz(x,t)[ -- sup [wz0(x)[. For any given point (x(fl, t), t) on the forward characteristic x = x(fl, t), there exists a unique c~= c~(fl, t) ~>fl such that wz(x(fl, t), t) = wzo(ot). Therefore, along the characteristic x = x(fl, t), one has from (3.7) that Then where 1-0 1+o(o )20 dt = 2 ~ [w,0(fl)- w2o(ot(fl, t))] 0-1 glt-o - Po(fl)-r- l/.)'10(/~)" g(x(fl, t), t) 2, g(x(fl, t), t) = Po(fl) 052 W'lo(fl) 1 +foK(fl, r)dr K (fl, t) = 1-0 1+o(o )20 2 ~ Iwlo(fl) -- W20(OC(fl, '))l 0-1 po(3)-z- w'lo(3). (3.9) (3.10) (3.11)
- 434. 436 G.-Q. Chen and D. Wang From (3.3), K(/~, t) ~>0. Thus, g(x(/~, t), t) is bounded, and 1-0 g(x(~,t),t) is also bounded. Similarly, OxW2is also bounded from (3.8). As a consequence, the C 1 norms of p = (0(Wl - w2)/2) 1/~ and v = (wl + w2)/2 are bounded on • x [0, T]. The proof is complete. 7q REMARK 3.1. In the proof of Theorem 3.1, the second-order derivatives of the Riemann invariants are formally used. However, the final equality (3.10) does not involve these second-order derivatives. Some appropriate arguments of approximation or weak formulation can be used to show that the conclusion is still valid for C 1 solutions. REMARK 3.2. For the global existence of smooth solutions of general one-dimensional hyperbolic systems of conservation laws, we refer the reader to Li [194] which contains some results and discussions on this subject. Also see Lin [197,198] and the references cited therein for the global existence of Lipschitz continuous solutions for the case that discontinuous initial data may not stay away from the vacuum. For the three-dimensional Euler equations for polytropic gases in (2.1), Serre and Grassin in [141,142,273] studied the existence of global smooth solutions under appropriate assumptions on the initial data for both isentropic and non-isentropic cases. It was proved in [141] that the three- dimensional Euler equations for a polytropic gas in (2.1) have global smooth solutions, provided that the initial entropy So and the initial density P0 are small enough and the initial velocity v0 forces particles to spread out, which are of similar nature to the condition (3.3). 4. Formation of singularities in smooth solutions The formation of shock waves is a fundamental physical phenomenon manifested in solutions of the Euler equations for compressible fluids, which are a prototypical example of hyperbolic systems of conservation laws. This phenomenon can be explained by mathematical analysis by showing the finite-time formation of singularities in the solutions. For nonlinear scalar conservation laws, the development of shock waves can be explained through the intersection of characteristics; see the discussions in Lax [180, 181] and Majda [223]. For systems in one space dimension, this problem has been extensively studied by using the method of characteristics developed in Lax [178], John [161], Liu [206], Klainerman and Majda [170], Dafermos [83], etc. For systems with multidimensional space variables, the method of characteristics has not been proved tractable. An efficient method, involving the use of averaged quantities, was developed in Sideris [282] for hyperbolic systems of conservation laws and was further refined in Sideris [283] for the three-dimensional Euler equations. Also see Majda [223].
- 435. The Cauchyproblemfor theEulerequationsfor compressiblefluids 437 4.1. One-dimensional Euler equations Consider the Cauchy problem (1.14) and (3.1) for the one-dimensional Euler equations of isentropic gas dynamics. With the notations in Section 3, the following result on the formation of singularity in smooth solutions of (1.14) and (3.1) follows. THEOREM 4.1. The lifespan of any smooth solution of (1.14) and (3.1), staying away from the vacuum, is finite, for C 1 initial data (po, vo)(x), with po(x) > 0 and finite C 1 norm satisfying W~lo(fl) < O, or W~2o(fl)< O, (4.1) for some point fl~ N. Furthermore, if there exist two positive constants ~ and s such that min wlo(x) - max W20(X) :"- S > O, (4.2) x x and, for some point fl E N, w~lo(fl) ~<-s, or W~o(/3)<<.-s, (4.3) then the lifespan of any smooth solution of (1.14) and (3.1) does not exceed 0-1 2 (o)2 Ol T, = (1 +0)s 2~ Ilpoll 2 c(R)- (4.4) PROOF. For a smooth solution (p, v)(x, t) of system (1.14), one can verify, as in the proof of Theorem 3.1, that p (x, t) > 0, and 1-0 l+0(0 )20 g'= 2 ~(Wl - w2) g2, with g = --p(O-1)/2OxWl. By defining the characteristic x = x(fl, t) passing through the point (fl, 0), 136 R, as in the proof of Theorem 3.1, we have, as in (3.10) and (3.11), 0--1 ! g(x(~, t), t) = po(~)-z- wlo(~ ) l+foK(fl, r)dr with K (fl, t) - 1-0 1 +0 1-o o-1 2 p(x(~, t), t)-~ po(~)-~ wlo(~).
- 436. 438 G.-Q. Chen and D. Wang If the smooth solution stays away from the vacuum, i.e., the density p has a positive lower bound, then one concludes that g(x(fl, t), t) will blow up at a certain finite time f if W'lo(fl) < 0. Under the condition (4.2) and if Wlo(fl) <~-e in (4.3), g(x(fl, t), t) will ! blow up at some finite time which is less than or equal to T. defined in (4.4). If w20(/3) ~<0 or further Wtzo(fl) <<.-e, similar consequence can be obtained from (3.8). This completes the proof of Theorem 4.1. D REMARK 4.1. The argument was developed in Lax [178] for 2 x 2 hyperbolic systems of conservation laws with genuine nonlinearity. The implication of the result is that the first derivatives of solutions blow up in a finite time, while the solutions stay themselves bounded and away from the vacuum. This is in agreement