![Introduction xv
of the characteristic cone. In some cases, it happens that E is supported by
Γ only; the open set co(Γ) Γ, on which E vanishes, is then called a lacuna.
For example, the wave equation in dimension 1 + d with d odd and d ≥ 3, has a
lacuna: its fundamental solution is supported by the dual characteristic cone itself
(this explains, for instance, the fact that light rays have no tail). The systematic
study of lacunæ is related to the topology of real algebraic sets.
Compared to linear ones, non-linear problems display fascinating new fea-
tures. In particular, several kinds of non-linear waves arise (shocks, rarefaction
waves, as well as contact discontinuities). They are present already in one space
dimension. The occurrence of shock waves is connected with a loss of regularity
in the solutions in finite time, which can be roughly explained as follows:
non-linearity implies that wave speeds depend on the state; therefore, a non-
constant solution experiences a wave overtaking, which results in the creation of
discontinuities in the derivatives of order m − 1, if m is the order of the system;
such discontinuities are called shock waves, or simply shocks. After blow-up, that
is after creation of shock(s), solutions cannot be smooth any longer. This yields
many questions: what is the meaning of the PDEs for non-smooth solutions;
can we solve the system in terms of weak enough solutions, and if possible in a
unique, physically relevant way? The answer to the first question has been given
by the theory of distributions, which is somehow the mathematical counterpart of
conservation principles in physics: conservation of mass, momentum and energy,
for instance (or Ampère’s and Faraday’s laws in electromagnetism) make sense
indeed as long as fields remain locally bounded. The drawback is – as has long
been known – that weak solutions are by no means unique, and this seems to hurt
the common belief that PDE models in physics describe deterministic processes.
This apparent contradiction may be resolved by the use of a suitable entropy
condition, most often reminiscent of the second principle of thermodynamics.
In one space dimension, entropy conditions have been widely used in the last
decades to prove global well-posedness results in the space of Bounded Vari-
ations (BV) functions – a space known to be inappropriate in several space
dimensions, because of the obstruction on the Lp
norms (see below for a few
more details). Entropy conditions are expected to ensure also multidimensional
well-posedness, even though we do not know yet what would be an appropriate
space: one of the goals of this book is to present a starting point in this direc-
tion, namely (local in time) well-posedness within classes of piecewise smooth
solutions.
Finally, the concept of time reversibility is quite intriguing in the framework
of hyperbolic PDEs. On the one hand, as far as smooth solutions are concerned,
many hyperbolic problems are time reversible, and this seems incompatible with
the decay (already mentioned above) of Lp
norms for p 2 in several space
dimensions. This paradox was actually resolved by Brenner [22,23], who proved
that multidimensional hyperbolic problems are ill-posed, in Hadamard’s sense,
in Lp
for p = 2. Incidentally, Brenner’s result shows that the space BV , which is
built upon the space of bounded measures, itself close to L1
, cannot be appropri-
ate for multidimensional problems. On the other hand, time reversibility is lost](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-19-320.jpg)
![xvi Introduction
(as a mathematical counterpart of the second principle of thermodynamics) once
shocks develop, whence a loss of information, the backward problem becoming ill-
posed. As a matter of fact, shocks may be viewed as free boundaries and as such
they can be sought as solutions of (non-standard) hyperbolic Initial Boundary
Value Problems (IBVP): it turns out that most of the well-posed hyperbolic
IBVPs are irreversible, as will be made clear in particular in this book – a large
part of this volume is indeed dedicated to a systematic study of IBVPs, either
for themselves, or in view of applications to well-posedness in the presence of
shock waves.
Difficulty. Even when a functional framework is available, a rigorous analysis
of hyperbolic problems often requires much more elaborate (or at least more
technical) tools than for elliptic or parabolic problems, notably to cope with the
lack of smoothing effects. The situation is even worse in the non-linear context,
where functional analysis has been useless in the study of weak entropy solutions
so far (except for first-order scalar equations). This is why our knowledge of
global-in-time solutions is so poor, despite tremendous efforts by talented math-
ematicians. Speaking only about the Cauchy problem for quasilinear systems of
first-order conservation laws, in space dimension d with n scalar unknowns, we
know about well-posedness only in the following cases.
r Scalar problems (n = 1), thanks to Kruzkhov’s theory [105].
r One space dimension (d = 1) and small data of bounded variation: existence
results date back to Glimm’s seminal work [70]; uniqueness and continuous
dependence have been obtained by Bressan and coworkers (see, for instance,
[25–27]).
r Small smooth data and large enough space dimension (for then dispersion
can compete with non-linearity and prevent shock formation): most results
from this point of view have been established by Klainerman and coworkers.
See, for instance, Hörmander’s book [88].
Amazingly enough, none of these results apply to such basic systems as the full
gas dynamics equations in one space dimension (n = 3, d = 1) or the isentropic
gas dynamics equations (n = 2) in dimension d ≥ 2.
Other results solve only one part of the problem:
r Global existence for general data when d = 1 and n = 2 (under a genuine
non-linearity assumption) by means of compensated compactness. This was
achieved by DiPerna [49], following an idea by Tartar [202]. Solutions are
then found in L∞
. Unfortunately, no uniqueness proof in such a large space
has been given so far, except for weak–strong uniqueness (uniqueness in L∞
of a classical solution).
r Local existence of smooth solutions for smooth data. This is quite a
good result since it shows at least local well-posedness. It is attributed
to several people (Friedrichs, Gårding, Kato, Leray, and possibly others),](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-20-320.jpg)
![Introduction xvii
depending on specific assumptions that were made. Unfortunately, its
practical implications are limited by the smallness of the existence
time – recall that shock formation precludes, in general, global existence
results within smooth functions.
Having this (modest) state-of-the-art in mind, we can foresee a compromise
regarding multidimensional weak solutions and non-linear problems: it will con-
sist of the analysis of piecewise smooth solutions (involving a finite number
of singularities like shock waves, rarefaction waves or contact discontinuities),
tractable by ‘classical’ tools. This is the point of view we have adopted here, which
defines the scope of this book: we shall consider either (possibly weak) solutions
of linear problems with smooth coefficients or piecewise smooth solutions of
non-linear problems – Cauchy problems and also of Initial Boundary Value
Problems – to multidimensional hyperbolic PDEs. We now present a more
detailed description of the contents.
We have chosen a presentation involving gradually increasing degrees of
difficulty: this is the case for the ordering of the three main ‘theoretical’ parts
of the book – the first one being devoted to linear Cauchy problems, the second
one to linear Initial Boundary Value Problems, and the third one to non-linear
problems; this is also the case inside those parts – the first two parts starting
with constant coefficients before going to variable coefficients, and the third one
starting from Cauchy problems, then going to IBVPs, and culminating with the
shock waves stability analysis. As a consequence, readers should be able to find
the information they need without having to enter overcomplicated frameworks:
most chapters are indeed (almost) self-contained (and as a drawback, the book
is not free from repetitions).
Another deliberate choice of ours has been to concentrate on first-order
systems, even though we are very much aware that higher-order hyperbolic PDEs
are also of great interest. This is mainly a matter of taste, because we come from
the community of conservation laws. In addition, we think that the understanding
of either one of those classes (first-order systems or higher-order scalar equations)
basically provides the understanding of the other class (see, for instance, the book
by Chazarain and Piriou [31], Chapter VII). Consistently with that choice, the
main application we have considered is the first-order system of Euler equations
in gas dynamics, to which the fourth part of the book is entirely devoted. We
have tried to temperate this ‘monomaniac’ attitude by referring from place to
place to higher-order equations, and in particular to the wave equation, which is
the source of several examples throughout the theoretical chapters.
Finally, to keep the length of this book reasonable, we have decided not to
speak of (nevertheless important) questions that are too far away from the shock
waves theory. Thus the reader will not find anything about the propagation
of singularities as developed by Egorov, Hörmander and Taylor. Likewise, non-
local boundary operators as they appear, for instance, in absorbing or trans-
parent boundary conditions will not be considered, and all numerical aspects of](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-21-320.jpg)
![xviii Introduction
hyperbolic IBVPs will be omitted, despite their great theoretical and practical
importance.
First part. The theory of linear Cauchy problems is most classical, even
though some results are not that well-known. The chapter on constant-coefficient
problems is the occasion of pointing out important definitions: Friedrichs sym-
metrizability; directions of hyperbolicity; strict hyperbolicity and more generally
what we call constant hyperbolicity – the eigenvalues of the symbol of a so-
called constantly hyperbolic operator are semisimple and of constant multiplicity,
instead of being simple in the case of strict hyperbolicity. Throughout the
book, all hyperbolic operators will be assumed either Friedrichs symmetrizable
or constantly hyperbolic (or both), as is the case for most operators coming
from physics. The chapter on variable-coefficients Cauchy problems presents,
in more generality, the symmetrizers technique, and in particular introduces
the notion of symbolic symmetrizers, thus illustrating the power of pseudo-
differential calculus (for infinitely smooth coefficients) and even para-differential
calculus (for coefficients of limited regularity).
Second part. The theory of Initial Boundary Value Problems (IBVP) is
inspired from, but tremendously more complicated than, the theory of Cauchy
problems. A kind of introductory chapter is devoted to the easier case of
symmetric dissipative IBVPs. The second chapter addresses constant-coefficients
IBVPs in a half-space, in which a central concept arises, namely the (uniform)
Lopatinskiı̆ condition. This stability condition dates back to the 1970s: simultane-
ously with a work by Lopatinskiı̆ ( [122], unnoticed in the West, Lopatinskiı̆ being
more famous for his older work on elliptic boundary value problems [121]), it was
worked out by Kreiss [103], and independently by Sakamoto [174] for higher-order
equations; in acknowledgement of Kreiss’ work on first-order hyperbolic systems
we shall rather call it the (uniform) Kreiss–Lopatinskiı̆ condition, and we shall
also speak of Kreiss’ symmetrizers, which are symbolic symmetrizers adapted to
IBVPs. The necessity of Kreiss’ symmetrizers shows up indeed when a Laplace–
Fourier transform is applied to the equations (Laplace in the time direction
and Fourier in the spatial boundary direction): to obtain an a priori estimate
without loss of derivatives we need to multiply the equations by a suitable
matrix-valued function, depending homogeneously on space–time frequencies –
thus being a symbol – in place of the energy tensor of the symmetric dissipative
case; that matrix-valued symbol is what we call a Kreiss symmetrizer. The
actual construction of Kreiss’ symmetrizers is quite involved, and requires a
good knowledge of linear algebra and real algebraic geometry. For this reason,
a separate chapter is devoted to the construction of Kreiss’ symmetrizers. The
interplay with algebraic geometry (formerly developed by Petrovskiı̆, Oleinik
and their school) is a deep reason why we need a structural assumption such
as constant hyperbolicity: even with this, there remain tricky points to deal
with, namely the so-called glancing points, where eigenvalues lack regularity.
The chapter on variable-coefficient IBVPs focuses more on the calculus aspects](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-22-320.jpg)
![Introduction xix
of the theory: it shows how to extend well-posedness results to more general
situations – variable coefficients with either infinite or limited regularity, non-
planar boundaries – by means of pseudo- or para-differential calculus.
The remaining chapters of the second part are devoted to more peculiar
topics: characteristic boundaries (which yield involved additional difficulties);
homogeneous IBVPs (which turn out to require only a weakened version of the
uniform Lopatinskiı̆ condition); the so-called class WR, which consists of certain
C ∞
-well-posed problems and is generic in the sense that it is stable under small
disturbances of the operators, but displays estimates with a loss of regularity.
These topical chapters may be skipped by the reader insterested only in the
applications to multidimensional shock stability.
Third part. We must admit that the current knowledge of non-linear multi-
dimensional hyperbolic problems is very much limited: all well-posedness results
presented in this part are short-time results; nevertheless, their proofs are not
that easy. A first chapter reviews Cauchy problems: symmetric (or Friedrichs-
symmetrizable) ones, but also those with symbolic symmetrizers (at is the
case for constantly hyperbolic systems), for which well-posedness was not much
known up to now (the only reference we are aware of is a proceedings paper
by Métivier [132]). Well-posedness is to be understood in Sobolev spaces of
sufficiently high index, or to be more precise, in Hs
(Rd
) with s d/2 + 1 (the
condition ensuring that Hs
(Rd
) is an algebra, whose elements are at least
continuously differentiable, by Sobolev’s theorem). In other words, we speak in
that chapter only of smooth, or classical solutions, except in the very last section,
where we recall the weak–strong uniqueness result of Dafermos and prepare the
way for piecewise smooth solutions considered in the chapter on shock waves.
Then ‘standard’ non-linear IBVPs are considered in a separate chapter, which
is the occasion to see a simplified version of what is going on for shocks. The
chapter on the persistence (or existence and stability) of single shock solutions
was one of the main motivations to write this book. The idea was to give a
comprehensive account of the work done by Majda in the 1980s [124–126], after
it was revisited by Métivier and coworkers [56, 131, 133, 134, 136, 140]. Initially,
we intended to cover also non-classical (multidimensional) shocks, as considered
by Freistühler [58, 59] and Coulombel [40]. But for clarity we have preferred to
concentrate on Lax shocks, while avoiding as much as possible to use their specific
properties so that interested readers could either guess what happens for non-
classical shocks or refer more easily to [40] for instance. We have also deliberately
omitted the most recent developments on characteristic and/or non-constantly
hyperbolic problems.
Fourth part. This concerns one of the most important applications of hyper-
bolic PDEs: gas dynamics. In fact, the theory of hyperbolic conservation laws
was developed, in particular by Peter Lax in the 1950s, by analogy with gas
dynamics: terms like ‘entropy’, ‘compressive’ (or ‘undercompressive’) shock are
reminiscent of this analogy, and the so-called Rankine–Hugoniot jump conditions](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-23-320.jpg)
![xx Introduction
were initially derived (in the late nineteenth century) by these two engineers
(Rankine and Hugoniot) in the framework of gas dynamics. There is a huge
literature on gas dynamics, by engineers, by physicists and by mathematicians.
In recent decades, the latter have had a marked preference for a familiar pressure
law, usually referred to as the γ-law, for it simplifies, to some extent (depending
on the explicit value of γ), the analysis of the Euler equations of gas dynamics.
We have chosen here to consider more general pressure laws, which apply to
so-called real – at least more realistic – fluids and not only perfect gases (as was
the case in earlier mathematical papers, by Weyl [218], Gilbarg [69], etc.).
In a first chapter we address several basic questions, regarding hyperbolicity
and symmetrizability. The second chapter is devoted to boundary conditions
for real fluids, a very important topic for engineers, which has (surprisingly) not
received much attention from mathematicians (see, however, the very nice review
paper by Higdon [84]).
This applied part culminates with the shock-waves analysis for real fluids, in
the last chapter. Even though it seems to belong to ‘folklore’ in the shock-waves
community, the complete investigation of the Kreiss–Lopatinskiı̆ condition for
the Euler equations is hard to find in the literature: in particular, Majda gave
the complete stability conditions in [126] but showed how to derive them only for
isentropic gas dynamics; a complete, analytic proof was published only recently
by Jenssen and Lyng [92]. By contrast, our approach is mostly algebraic, and
works fine for full gas dynamics (of which the isentropic gas dynamics appear
as a special, easier case). In addition, we give an explicit construction of Kreiss
symmetrizers, which (to our knowledge) cannot be found elsewhere, and is fully
elementary (compared to the sophisticated tools used for abstract systems).
Fifth part. This is only a (huge) appendix, collecting useful tools and tech-
niques. The main topics are the Laplace transform – including Paley–Wiener
theorems – pseudo-differential calculus, and its refinement called para-differential
calculus. Less space demanding (or more classical) tools are merely introduced
in the Notations section below.](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-24-320.jpg)
![NOTATIONS
The set of matrices with n rows and p columns, with entries in a field K, is denoted
by Mn×p(K). If p = n, we simply write Mn(K). The latter is an algebra, whose
neutral elements under addition and multiplication are denoted by 0n and In,
respectively. The space Mn×p(K) may be identified to the set of linear maps
from Kp
to Kn
. The transpose matrix is written MT
. The group of invertible
n × n matrices is GLn(K). If p = 1, Mn×1(K) is identified with Kn
.
Given two matrices M, N ∈ Mn(C), their commutator MN − NM is denoted
by [M, N].
If K = C, the adjoint matrix is written M∗
. It is equal to M
T
, where M
denotes the conjugate of M. We equip Cm
and Rm
with the canonical Hermitian
norm
x =
j
| xj |2 = (x∗
x)1/2
.
This norm is associated to the scalar product
(x, y) =
j
xjyj = y∗
x.
The norm will sometimes be denoted |x|, especially when x is a space variable
or a frequency vector (used in Fourier transform.)
A complex square matrix U is unitary if U∗
U = In, or equivalently UU∗
= In.
The set Un of unitary matrices is a compact subgroup of GLn(C). Its intersection
On with Mn(R) is the set of real orthogonal matrices. The special orthogonal
group SOn is the subgroup defined by the constraint det M = 1.
As usual, Mn×p(C) is equipped with the induced norm
M = sup
Mx
x
.
When the product makes sense, one knows that MN ≤ M N. When p =
n, Mn(C) is thus a normed algebra, and we have Mk
≤ Mk
. If Q is a unitary
(for instance real orthogonal) matrix, one has Q = 1. More generally, the norm
is unitary invariant, which means that M = PMQ whenever P and Q are
unitary.
If M ∈ Mn(C), the set of eigenvalues of M, denoted by SpM, is called the
spectrum of M. The largest modulus of eigenvalues of M is called the spectral
radius of M, and denoted by ρ(M). It is less than or equal to M, and such](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-25-320.jpg)
![Notations xxiii
general, but it does when AB = BA. In particular, exp A is always invertible,
with inverse exp(−A). Other useful formulæ are
exp(MT
) = (exp M)
T
, exp M = exp M, exp(M∗
) = (exp M)∗
.
The exponential of a Hermitian matrix is Hermitian, positive-definite. The map
exp : Hn → HPDn
is actually an homeomorphism. The exponential of a skew-Hermitian matrix is
unitary.
Let A ∈ Mn(C) be given. The space Cn
splits, in a unique way, as the
direct sum of three invariant subspaces, namely the stable, unstable and central
subspaces of A, denoted, respectively, Es(A), Eu(A) and Ec(A). Their invariance
properties read
AEs(A) = Es(A), AEu(A) = Eu(A) and AEc(A) ⊂ Ec(A).
The stable invariant subspace is formed of vectors x such that (exp tA)x tends
to zero as t → +∞, and then the decay is exponentially fast. The unstable
subspace is formed of vectors x such that (exp tA)x tends to zero (exponentially
fast) as t → −∞. The central subspace consists of vectors such that (exp tA)x is
polynomially bounded on R. Since these spaces are invariant under A, this matrix
operates on each one as an endomorphism, say As, Au, Ac. The spectrum of As
(respectively, Au, Ac) has negative (respectively, positive, zero) real part. The
union of these spectra is the whole spectrum of A, with the correct multiplicities.
Hence the dimension of Es(A) is the number of eigenvalues of A of negative real
part (these are called ‘stable eigenvalues’), counted with multiplicities. When
Ec(A) is trivial, meaning that there is no pure imaginary eigenvalue, A is called
hyperbolic (in the sense of Dynamical Systems).
Dunford–Taylor formula. Let γ be a Jordan curve, oriented in the trigono-
metric way, disjoint from SpA. Let σ be the part of SpA that γ enclose. Then
the Cauchy integral
Pσ :=
1
2iπ
γ
(zIn − A)−1
dz
defines a projector (that is P2
σ = Pσ) whose range and kernel are invariant under
A. (Moreover, A commutes with Pσ). The spectrum of the restriction of A to
the range of Pσ is exactly the part of the spectrum of A that belongs to σ. In
other words, R(Pσ) is the direct sum of the generalized eigenspaces associated
to those eigenvalues in σ.
More information about matrices and norms may be found in [187].
Functional spaces
Given an open subset Ω of Rn
, the set of infinitely differentiable functions (with
values in C) that are bounded as well as all their derivatives on Ω is denoted](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-27-320.jpg)
![Notations xxv
refer the reader to Chapter II in [31]; for more details in more general situations,
see for instance the classical monograph by Adams [1]. The notation Hs
w will
stand for the Sobolev space Hs
equipped with the weak topology instead of the
(strong) Hilbert topology.
The Schwartz space of rapidly decreasing functions S (Rn
) will simply be
denoted by S when no confusion can occur as concerns the space dimension. And
similarly, its dual space, consisting of temperate distributions, will be denoted
by S
.
Other tools
We have collected in the appendix various additional tools, ranging from standard
calculus and Fourier–Laplace analysis to pseudo-differential and para-differential
calculus: we hope it will be helpful to the reader.](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-29-320.jpg)
![10 Linear Cauchy Problem with Constant Coefficients
in η, locally in time. Denoting by Xα the derivative with respect to ηα, we
have
dXα
dt
= (B − iA(η))Xα − iAα
X,
and therefore
d(X−1
Xα)
dt
= −iX−1
Aα
X.
Using Duhamel’s formula, as in the proof of Proposition 1.2, we see that
X(t) ≤ c(1 + B |t|),
from which we deduce
X−1
Xα ≤
(1 + B |t|)3
− 1
3B
c2
Aα
.
Finally, we obtain
Xα ≤
(1 + B |t|)4
3B
c3
Aα
.
We leave the reader to estimate the higher derivatives and complete the proof of
the following statement. The case of data in the Schwartz class is done in exactly
the same way, since the Fourier transform is an automorphism of S and that S
is stable under multplication by C ∞
functions with slow growth.
Proposition 1.3 If L is hyperbolic, then the Cauchy problem for (1.1.3) is
well-posed in both S and S
.
1.2.3 The Kreiss’ matrix Theorem
Of course, since L2
-well-posedness implies (S , S
)-well-posedness, hyperbolicity
implies that the spectrum of A(ξ) is real for all ξ in Rd
. It implies even more,
that all A(ξ) are diagonalizable. Though these two facts have a rather simple
proof here, they do not characterize completely hyperbolic systems. We shall
therefore describe the characterization obtained by Kreiss [102,104]. This is an
application of a deeper result that deals with strong well-posedness of general
constant-coefficient evolution problems. However, since we focus only on first-
order systems, we content ourselves with a statement with a simpler proof, due
to Strang [199].
Theorem 1.2 Let ξ → A(ξ) be a linear map from Rd
to Mn(C). Then the
following properties are equivalent to each other:
i) Every A(ξ) is diagonalizable with pure imaginary eigenvalues, uniformly
with respect to ξ:
A(ξ) = P(ξ)−1
diag(iρ1, . . . , iρn)P(ξ), (ρ1(ξ), . . . , ρn(ξ) ∈ R),](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-40-320.jpg)
![12 Linear Cauchy Problem with Constant Coefficients
Actually, A(ξ) is diagonalizable, for if there were a non-trivial Jordan part,
then (zIn − A(ξ))−1
would have a pole of order two or more, contradicting
(1.2.15). Therefore, A(ξ) admits a spectral decomposition
A(ξ) = i
j
ρjEj,
where ρj is real and Ej = Ej(ξ) is a projector (E2
j = Ej), with
EjEk = 0n, (k = j),
j
Ej = In.
Let us define
H = H(ξ) :=
j
EjE∗
j ,
which is a positive-definite Hermitian matrix. Since A(ξ)∗
= −
j ρjE∗
j , it holds
that
H(ξ)A(ξ) = −A(ξ)∗
H(ξ),
from which it follows that H(ξ)1/2
A(ξ)H(ξ)−1/2
is skew-Hermitian. As such,
it is diagonalizable through a unitary transformation. Therefore A(ξ) =
P(ξ)−1
D(ξ)P(ξ), where D(ξ) is diagonal with pure imaginary eigenvalues, and
P(ξ) = U(ξ)H1/2
, where U(ξ) is a unitary matrix.
We finish by proving that P(ξ) is uniformly conditioned. Since P±1
=
H±1/2
= H±1/2
, this amounts to proving that H · H−1
is uniformly
bounded. On the one hand, it holds that
|v|2
=
j
Ejv
2
≤ n
j
|Ejv|2
= n|H1/2
v|2
,
so that H−1/2
≤
√
n. On the other hand, applying (1.2.15) to + iρk, we have
j
( + iρk − iρj)−1
Ej ≤
C
||
.
Letting → 0, we deduce that Ej ≤ C, independently of ξ. It follows that
H ≤ nC2
.
Remarks
i) A more explicit characterization of hyperbolic symbols has been estab-
lished by Mencherini and Spagnolo when n = 2 or n = 3; see [129].
ii) The following example (n = 3 and d = 2), known as Petrowski’s example,
shows that the well-conditioning can fail for systems in which all matrices](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-42-320.jpg)
![Strong well-posedness 13
A(ξ) are diagonalizable with real eigenvalues. Let us take
A1
=
0 1 1
0 0 0
1 0 0
, A2
=
0 0 0
0 0 0
0 0 1
.
One checks easily that the eigenvalues of A(ξ) are real and distinct for
ξ1 = 0, while A2
is already diagonal. Hence, A(ξ) is always diagonalizable
over R. However, as ξ1 tends to zero, one eigenvalue is identically zero,
associated to the eigenvector (ξ2, ξ1, −ξ1)T
, while another one is small,
λ ∼ −ξ2
1ξ−1
2 , associated to (ξ2, 0, λξ2ξ−1
1 )T
. Both eigenvectors have the
same limit (ξ2, 0, 0)T
, which shows that P(ξ) is unbounded as ξ1 tends
to zero. See a similar example in [108]. Oshime [155] has shown that
Petrowsky’s example is somehow canonical when d = 3. On the other hand,
Strang [199] showed that when n = 2, the diagonalizability of every A(ξ) is
equivalent to hyperbolicity, and that such operators are actually Friedrichs
symmetrizable in the sense of the next section.
iii) Uniform diagonalizability of A(ξ) within real matrices has been shown by
Kasahara and Yamaguti [93, 221] to be necessary and sufficient in order
that the Cauchy problem for
∂tu +
α
Aα
∂αu = Bu
be C∞
-well-posed for every matrix B ∈ Mn(R). Of course, the sufficiency
follows from Theorem 1.3 and Proposition 1.2. The necessity statement is
even stronger than the one suggested by the example given in Section 1.1,
since the diagonalizability within R is not sufficient. For instance, if A(ξ)
is given as in the Petrowski example, there are matrices B for which the
Cauchy problem is ill-posed in the Hadamard sense.
1.2.4 Two important classes of hyperbolic systems
We now distinguish two important classes of hyperbolic systems.
Definition 1.2 An operator
L = ∂t +
α
Aα
∂α
is said to be symmetric in Friedrichs’ sense [63], or simply Friedrichs symmetric,
if all matrices Aα
are symmetric; one may also say symmetric hyperbolic. More
generally, it is Friedrichs symmetrizable if there exists a symmetric positive-
definite matrix S0 such that every S0Aα
is symmetric.
An operator M as above is said to be constantly3
hyperbolic if the matrices
A(ξ) are diagonalizable with real eigenvalues and, moreover, as ξ ranges along
3We employ this shortcut in lieu of hyperbolic with characteristic fields of constant multiplic-
ities.](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-43-320.jpg)
![14 Linear Cauchy Problem with Constant Coefficients
Sd−1
, the multiplicities of eigenvalues remain constant. In the special case where
all eigenvalues are real and simple for every ξ ∈ Sd−1
, we say that the operator
is strictly hyperbolic.
Let us point out that in a constantly hyperbolic operator, the eigenvalues may
have non-equal multiplicities, but the set of multiplicities remains constant as
ξ varies. This implies in particular that the eigenspaces depend analytically on
ξ for ξ = 0. This fact easily follows from the construction of eigenprojectors as
Cauchy integrals (see the section ‘Notations’.) To a large extent, the theory of
constantly hyperbolic systems does not differ from the one of strictly hyperbolic
systems. But the analysis is technically simpler in the latter case. This is why
the theory of strictly hyperbolic operators was developed much further in the
first few decades.
Theorem 1.4 If an operator is Friedrichs symmetrizable, or if it is constantly
hyperbolic, then it is hyperbolic.
Proof Let the operator be Friedrichs symmetrizable by S0. Then S−1
0 is
positive-definite and admits a (unique) square root R symmetric positive-definite
(see [187], page 78). Let us denote S0Aα
by Sα
, and S(ξ) =
α ξαSα
as usual.
Then
A(ξ) = S−1
0 S(ξ) = R(RS(ξ)R)R−1
.
The matrix RS(ξ)R is real symmetric and thus may be written as
Q(ξ)T
D(ξ)Q(ξ), where Q is orthogonal. Then A(ξ) is conjugated to D(ξ), A(ξ) =
P(ξ)−1
D(ξ)P(ξ), with P(ξ) = Q(ξ)R−1
and P(ξ)−1
= RQ(ξ)T
. Since our matrix
norm is invariant under left or right multiplication by unitary matrices, we have
P(ξ) P(ξ)−1
= R R−1
=
ρ(S0)ρ(S−1
0 ),
a number independent of ξ. The diagonalization is thus well-conditioned.
Let us instead assume that the system is constantly hyperbolic. The
eigenspaces are continuous functions of ξ in Sd−1
. Choosing continuously a
basis of each eigenspace, we find locally an eigenbasis of A(ξ), which depends
continuously on ξ. This amounts to saying that, along every contractible subset
of Sd−1
, the matrices A(ξ) may be diagonalized by a matrix P(ξ), which depends
continuously on ξ. If the set is, moreover, compact (for instance, a half-sphere), we
obtain that A(ξ) is diagonalizable with a uniformly bounded condition number.
We now cover the sphere by two half-spheres and obtain a diagonalization of A(ξ)
that is well-conditioned on Sd−1
(though possibly not continuously diagonalizable
on the sphere).
In the following example, though a symmetric as well as a strictly hyperbolic
one, the diagonalization of the matrices A(ξ) cannot be done continuously for all](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-44-320.jpg)
![Friedrichs-symmetrizable systems 17
complex conjugation, it satisfies the same property. Therefore, F : Lp
(Rd
) →
Lp
(Rd
) is not an isomorphism for p 2, since p
2. From this remark, we
cannot find a well-posedness result in Lp
for p = 2 by following the above strategy.
It has been proved actually by Brenner [22,23] that, for hyperbolic systems,
the Cauchy problem is ill-posed in Lp
for every p = 2, except in the case where
the matrices Aα
commute to each other. In this particular case, system (1.2.12)
actually decouples into a list of scalar equations, for which (1.2.19) shows the well-
posedness in every Lp
. To see the decoupling, we recall that commuting matrices
that are diagonalizable may be diagonalized in a common basis B = {r1, . . . , rn}
: Aα
rj = λα
j rj. Let us decompose the unknown on the eigenbasis:
u(x, t) =
n
1
wj(x, t)rj.
Then each wj solves a scalar equation:
∂twj +
α
λα
j ∂αwj = 0.
From the well-posedness of (1.2.12) and Duhamel’s formula, we conclude that,
for commuting matrices Aα
, the hyperbolic Cauchy problem for (1.1.3) is also
well-posed in every Lp
. The matrices Aα
do not need to commute with B.
See Section 1.5.2 for an interpretation of the ill-posedness in Lp
(p = 2), in
terms of dispersion and so-called Stricharz estimates.
1.3 Friedrichs-symmetrizable systems
A system in Friedrichs-symmetric form
S0∂tu +
α
Sα
∂αu = 0
may always be transformed into a symmetric system with S0 = In, using the
new unknown ũ := S
1/2
0 u. For the rest of this section, we shall only consider
symmetric systems of the form (1.1.3).
A symmetric system admits an additional conservation law4
in the form
∂t|u|2
+
α
∂α(Aα
u, u) = 0, (1.3.20)
where (·, ·) denotes the canonical scalar product and |u|2
:= (u, u). Equation
(1.3.20) is satisfied at least for C 1
solutions of the system, when5
B = 0. It can be
viewed as an energy identity. Since classical solutions are dense in C (R; L2
(Rd
)n
),
4By conservation law, we mean an equality of the form Divx,t
F = 0 that derives formally from
the equation or system under consideration.
5Otherwise, the right-hand side of (1.3.20) should be 2(Bu, u). In the non-homogeneous case, we
add also 2(f, u).](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-47-320.jpg)
![Directions of hyperbolicity 23
We may apply (1.3.28) to φ , which gives
T
0
θ(t/)u(t), L∗
ψ(t) dt =
1
T
0
θ
(t/)u(t), ψ(t) dt.
Using the continuity in time, we may pass to the limit as → 0+
, and obtain
T
0
u(t), L∗
ψ(t) dt = u(0), ψ(0).
Therefore, assuming u(0) = 0, we see that (1.3.28) is valid for ψ as well, that is
to test functions in D(Rd
× (−∞, T))n
.
We now choose an arbitrary test function f ∈ D(Rd
× (0, T))n
. Obviously,
L∗
is a hyperbolic operator and we can solve the backward Cauchy problem
L∗
χ = f, χ(T) = 0.
Extending f by zero for t ≤ 0, we obtain a unique solution χ ∈ C ∞
(−∞, T; S ).
Applying (1.3.26) to this backward problem, we see that χ(t) has compact
support for each time, with Supp χ(t) included in a ball of the form Bρ(T −t), for a
suitable constant ρ. Also, χ vanishes identically for t close enough to T (because
f does). Truncating, we apply (1.3.28) to ψ(x, t) = θ(t + 1)χ(x, t). This gives
u, f = 0 for all test functions, that is u = 0. Therefore the Cauchy problem
for a Friedrichs-symmetric operator has the uniqueness property in the class
C (0, T; D
).
1.4 Directions of hyperbolicity
The situation for general (weakly) hyperbolic operators is not as neat as that for
Friedrichs-symmetrizable ones. Non-symmetrizable operators do exist, as soon
as d = 2 and n = 3, as shown by Lax [110]. The class of constantly hyperbolic
operators provides a valuable and flexible alternative to Friedrichs-symmetrizable
ones. Their analysis will lead us to several new and useful notions.
In this section, we shall not address the problem of propagation of the support
(with finite velocity), which we solved in the symmetric case. This propagation
holds true for constantly hyperbolic systems, but a rigorous proof needs a theory
of the Cauchy problem for systems with variable coefficients. Such a theory will
be done in Chapter 2, where we shall prove an accurate result.
1.4.1 Properties of the eigenvalues
The results in this section are essentially those of Lax [110], and the arguments
follow Weinberger [217], though we give a more detailed proof of the claim below.
We begin by considering a subspace E in Mn(R), with the property that every
matrix in E has a real spectrum. Without loss of generality, we may assume that
In belongs to E. If M ∈ E, we denote by λ1(M) ≤ · · · ≤ λn(M) the spectrum
of M, counting with multiplicities. The functions λj are positively homogeneous
of order one. They are continuous, but could be non-differentiable at crossing](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-53-320.jpg)
![24 Linear Cauchy Problem with Constant Coefficients
points. In the constantly hyperbolic case, however, they are analytic away from
the origin.
Lemma 1.1 Let A and B be matrices in E, with λ1(B) 0. Then the eigen-
values of B−1
(λIn − A) are real.
Proof From the assumption, we know that B is non-singular. Define a poly-
nomial
P(X, Y ) := det(XIn − A − Y B),
which has degree n with respect to X as well as to Y . Define continuous functions
φj(µ) = λj(A + µB). From homogeneity and continuity, we have
φj(µ) ∼
µλj(B), as µ → +∞,
µλn+1−j(B) as µ → −∞.
Hence φj(µ) tends to ±∞ with µ. By the Intermediate Value Theorem, it must
take any prescribed real value λ at least once.
Thus, let λ∗
be given and µj ∈ R be a root of φj(µj) = λ∗
for each j. Given
one of these roots, µ∗
, let J be the number of indices such that µj = µ∗
. Then
λ∗
is a root of P(·, µ∗
), of order J at least.
Claim 1.1 The multiplicity of µ∗
as a root of P(λ∗
, ·) is larger than or equal
to J.
This claim readily implies the lemma. Its proof is fairly simple when the φjs are
differentiable, for instance in the constantly hyperbolic case. But in the general
case, one must use once more the assumption. To simplify the notations, we
assume without loss of generality that λ∗
= µ∗
= 0, by translating A to A +
µ∗
B − λ∗
In. Let N (N ≥ J) be the multiplicity of the null root of P(·, 0). The
Newton’s polygon of the polynomial P admits the vertices (N, 0) and (0, K).
Let δ be the edge of the Newton’s polygon with vertex (N, 0). We denote its
other vertex by (j, k). Retaining only those monomials of P whose degrees (a, b)
belong to δ, we obtain a polynomial Xj
Q with the following homogeneity:
Q(ak
X, aN−j
Y ) = ak(N−j)
Q(X, Y ).
It is a basic fact in algebraic geometry (see [35], Section 2.8) that, in the vicinity
of the origin, the algebraic curve P(x, y) = 0 is described by simpler curves corre-
sponding to the edges of the Newton polygon, up to analytic diffeomorphisms. In
the present case, these diffeomorphisms have real coefficients (i.e. they preserve
real vectors) since P has real coefficients. The ‘simple’ curve γ associated to
δ is just that with equation Q(x, y) = 0. Hence, points (x, y) in γ with a real
co-ordinate y must be real (because this is so in the curve P = 0.)
Let ω be a root of unity, of order 2(N − j), that is ωN−j
= −1. Because of
the homogeneity, the map (x, y) → (ωk
x, −y) preserves γ. If y is real, the map](https://image.slidesharecdn.com/442241-250518142223-4b2c0970/85/Multidimensional-Hyperbolic-Partial-Differential-Equations-Firstorder-Systems-And-Applications-Sylvie-Benzonigavage-54-320.jpg)
Multidimensional Hyperbolic Partial Differential Equations Firstorder Systems And Applications Sylvie Benzonigavage
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- 5. OXFORD MATHEMATICAL MONOGRAPHS Series Editors J.M. BALL W.T. GOWERS N.J. HITCHIN L. NIRENBERG R. PENROSE A. WILES
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- 7. Multidimensional Hyperbolic Partial Differential Equations First-order Systems and Applications Sylvie Benzoni-Gavage Université Claude Bernard Lyon I Lyon, France Denis Serre ENS de Lyon Lyon, France CLARENDON PRESS • OXFORD 2007
- 8. 3 Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Sylvie Benzoni-Gavage and Denis Serre, 2007 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2007 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by SPI Publisher Services, Pondicherry, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 0-19-921123-X 978-0-19-921123-4 1 3 5 7 9 10 8 6 4 2
- 9. PREFACE Hyperbolic Partial Differential Equations (PDEs), and in particular first-order systems of conservation laws, have been a fashionable topic for over half a century. Many books have been written, but few of them deal with genuinely multidimensional hyperbolic problems: in this respect the most classical, though not so well-known, references are the books by Reiko Sakamoto, by Jacques Chazarain and Alain Piriou, and by Andrew Majda. Quoting Majda from his 1984 book, “the rigorous theory of multi-D conservation laws is a field in its infancy”. We dare say it is still the case today. However, some advances have been made by various authors. To speak only of the stability of shock waves, we may think in particular of: Métivier and coworkers, who continued Majda’s work in several interesting directions – weak shocks, lessening the regularity of the data, elucidation of the ‘block structure’ assumption in the case of characteristics with constant multiplicities (we shall speak here of constantly hyperbolic operators); Freistühler, who extended Majda’s approach to undercompressive shocks, of which an important example is given by phase boundaries in van der Waals fluids, as treated by Benzoni-Gavage; Coulombel and Secchi, who dealt very recently with neutrally stable discontinuities (2D-vortex sheets), thanks to Nash–Moser techniques. Even though it does not pretend to cover the most recent results, this book aims at presenting a comprehensive view of the state-of-the-art, with particular emphasis on problems in which modern tools of analysis have proved useful. A large part of the book is indeed devoted to initial boundary value problems (IBVP), which can only be dealt with by using symbolic symmetrizers, and thus necessitate pseudo-differential calculus (for smooth, non-constant coefficients) or even para-differential calculus (for rough coefficients and therefore also non- linear problems). In addition, the construction of symbolic symmetrizers conceals intriguing questions related to algebraic geometry, which were somewhat hidden in Kreiss’ original paper and in the book by Chazarain and Piriou. In this respect we propose here new insight, in connection with constant coefficient IBVPs. Furthermore, the analysis of (linear) IBVPs, which are important in themselves, enables us to prepare the way for the (non-linear) stability analysis of shock waves. In the matter of complexity, stability of shocks is the culminating topic in this book, which we hope will contribute to make more accessible some of the finest results currently known on multi-D conservation laws. Finally, quoting Constantin Dafermos from his 2000 book, ‘hyperbolic conservation laws and gas dynamics have been traveling hand-by-hand over the past one hundred and fifty years’. Therefore it is not a surprise that we devote a significant part of this book
- 10. vi Preface to that specific and still important application. The idea of dealing with ‘real’ gases was inspired by the PhD thesis of Stéphane Jaouen after Sylvie Benzoni- Gavage was asked by his advisor, Pierre-Arnaud Raviart to act as a referee in the defense. This volume contains enough material for several graduate courses – which were actually taught by either one of the authors in the past few years – depend- ing on the topic one is willing to emphasize: hyperbolic Cauchy problem and IBVP, non-linear waves, or gas dynamics. It provides an extensive bibliography, including classical papers and very recent ones, both in PDE analysis and in applications (mainly to gas dynamics). From place to place, we have adopted an original approach compared to the existing literature, proposed new results, and filled gaps in proofs of important theorems. For some highly technical results, we have preferred to point out the main tools and ideas, together with precise references to original papers, rather than giving extended proofs. We hope that this book will fulfill the expectations of researchers in both hyperbolic PDEs and compressible fluid dynamics, while being accessible to beginners in those fields. We have tried our best to make it self-contained and to proceed as gradually as possible (at the price of some repetition), so that the reader should not be discouraged by her/his first reading. We warmly thank Jean-François Coulombel, whose PhD thesis (under the supervision of Benzoni-Gavage and with the kind help of Guy Métivier) provided the energy necessary to complete the writing of the most technical parts, for his careful reading of the manuscript and numerous useful suggestions. We also thank our respective families for their patience and support. Lyon, April 2006 Sylvie Benzoni-Gavage Denis Serre
- 11. CONTENTS Preface v Introduction xiii Notations xxi PART I. THE LINEAR CAUCHY PROBLEM 1 Linear Cauchy Problem with Constant Coefficients 3 1.1 Very weak well-posedness 4 1.2 Strong well-posedness 7 1.2.1 Hyperbolicity 7 1.2.2 Distributional solutions 9 1.2.3 The Kreiss’ matrix Theorem 10 1.2.4 Two important classes of hyperbolic systems 13 1.2.5 The adjoint operator 15 1.2.6 Classical solutions 15 1.2.7 Well-posedness in Lebesgue spaces 16 1.3 Friedrichs-symmetrizable systems 17 1.3.1 Dependence and influence cone 18 1.3.2 Non-decaying data 20 1.3.3 Uniqueness for non-decaying data 21 1.4 Directions of hyperbolicity 23 1.4.1 Properties of the eigenvalues 23 1.4.2 The characteristic and forward cones 26 1.4.3 Change of variables 27 1.4.4 Homogeneous hyperbolic polynomials 30 1.5 Miscellaneous 32 1.5.1 Hyperbolicity of subsystems 32 1.5.2 Strichartz estimates 36 1.5.3 Systems with differential constraints 41 1.5.4 Splitting of the characteristic polynomial 45 1.5.5 Dimensional restrictions for strictly hyperbolic systems 47 1.5.6 Realization of hyperbolic polynomial 48
- 12. viii Contents 2 Linear Cauchy problem with variable coefficients 50 2.1 Well-posedness in Sobolev spaces 51 2.1.1 Energy estimates in the scalar case 51 2.1.2 Symmetrizers and energy estimates 52 2.1.3 Energy estimates for less-smooth coefficients 58 2.1.4 How energy estimates imply well-posedness 63 2.2 Local uniqueness and finite-speed propagation 72 2.3 Non-decaying infinitely smooth data 80 2.4 Weighted in time estimates 81 PART II. THE LINEAR INITIAL BOUNDARY VALUE PROBLEM 3 Friedrichs-symmetric dissipative IBVPs 85 3.1 The weakly dissipative case 85 3.1.1 Traces 88 3.1.2 Monotonicity of A 89 3.1.3 Maximality of A 90 3.2 Strictly dissipative symmetric IBVPs 93 3.2.1 The a priori estimate 95 3.2.2 Construction of û and u 96 4 Initial boundary value problem in a half-space with constant coefficients 99 4.1 Position of the problem 99 4.1.1 The number of scalar boundary conditions 100 4.1.2 Normal IBVP 102 4.2 The Kreiss–Lopatinskiı̆ condition 102 4.2.1 The non-characteristic case 103 4.2.2 Well-posedness in Sobolev spaces 106 4.2.3 The characteristic case 107 4.3 The uniform Kreiss–Lopatinskiı̆ condition 109 4.3.1 A necessary condition for strong well-posedness 109 4.3.2 The characteristic IBVP 111 4.3.3 An equivalent formulation of (UKL) 112 4.3.4 Example: The dissipative symmetric case 113 4.4 The adjoint IBVP 114 4.5 Main results in the non-characteristic case 118 4.5.1 Kreiss’ symmetrizers 119 4.5.2 Fundamental estimates 120 4.5.3 Existence and uniqueness for the boundary value problem in L2 γ 123 4.5.4 Improved estimates 125
- 13. Contents ix 4.5.5 Existence for the initial boundary value problem 126 4.5.6 Proof of Theorem 4.3 128 4.5.7 Summary 129 4.5.8 Comments 129 4.6 A practical tool 130 4.6.1 The Lopatinskiı̆ determinant 130 4.6.2 ‘Algebraicity’ of the Lopatinskiı̆ determinant 133 4.6.3 A geometrical view of (UKL) condition 136 4.6.4 The Lopatinskiı̆ determinant of the adjoint IBVP 137 5 Construction of a symmetrizer under (UKL) 139 5.1 The block structure at boundary points 139 5.1.1 Proof of Lemma 4.5 139 5.1.2 The block structure 141 5.2 Construction of a Kreiss symmetrizer under (UKL) 144 6 The characteristic IBVP 158 6.1 Facts about the characteristic case 158 6.1.1 A necessary condition for strong well-posedness 159 6.1.2 The case of a linear eigenvalue 162 6.1.3 Facts in two space dimensions 167 6.1.4 The space E−(0, η) 169 6.1.5 Conclusion 174 6.1.6 Ohkubo’s case 175 6.2 Construction of the symmetrizer; characteristic case 176 7 The homogeneous IBVP 182 7.1 Necessary conditions for strong well-posedness 184 7.1.1 An illustration: the wave equation 189 7.2 Weakly dissipative symmetrizer 191 7.3 Surface waves of finite energy 196 8 A classification of linear IBVPs 201 8.1 Some obvious robust classes 202 8.2 Frequency boundary points 202 8.2.1 Hyperbolic boundary points 203 8.2.2 On the continuation of E−(τ, η) 205 8.2.3 Glancing points 207 8.2.4 The Lopatinskiı̆ determinant along the boundary 208 8.3 Weakly well-posed IBVPs of real type 208 8.3.1 The adjoint problem of a BVP of class WR 210 8.4 Well-posedness of unsual type for BVPs of class WR 211
- 14. x Contents 8.4.1 A priori estimates (I) 211 8.4.2 A priori estimates (II) 214 8.4.3 The estimate for the adjoint BVP 216 8.4.4 Existence result for the BVP 217 8.4.5 Propagation property 218 9 Variable-coefficients initial boundary value problems 220 9.1 Energy estimates 222 9.1.1 Functional boundary symmetrizers 225 9.1.2 Local/global Kreiss’ symmetrizers 229 9.1.3 Construction of local Kreiss’ symmetrizers 233 9.1.4 Non-planar boundaries 242 9.1.5 Less-smooth coefficients 245 9.2 How energy estimates imply well-posedness 255 9.2.1 The Boundary Value Problem 255 9.2.2 The homogeneous IBVP 264 9.2.3 The general IBVP (smooth coefficients) 267 9.2.4 Rough coefficients 271 9.2.5 Coefficients of limited regularity 281 PART III. NON-LINEAR PROBLEMS 10 The Cauchy problem for quasilinear systems 291 10.1 Smooth solutions 292 10.1.1 Local well-posedness 292 10.1.2 Continuation of solutions 302 10.2 Weak solutions 304 10.2.1 Entropy solutions 305 10.2.2 Piecewise smooth solutions 311 11 The mixed problem for quasilinear systems 315 11.1 Main results 316 11.1.1 Structural and stability assumptions 316 11.1.2 Conditions on the data 318 11.1.3 Local solutions of the mixed problem 319 11.1.4 Well-posedness of the mixed problem 320 11.2 Proofs 321 11.2.1 Technical material 321 11.2.2 Proof of Theorem 11.1 326 12 Persistence of multidimensional shocks 329 12.1 From FBP to IBVP 331
- 15. Contents xi 12.1.1 The non-linear problem 331 12.1.2 Fixing the boundary 332 12.1.3 Linearized problems 334 12.2 Normal modes analysis 337 12.2.1 Comparison with standard IBVP 337 12.2.2 Nature of shocks 341 12.2.3 The generalized Kreiss–Lopatinskiı̆ condition 344 12.3 Well-posedness of linearized problems 345 12.3.1 Energy estimates for the BVP 345 12.3.2 Adjoint BVP 355 12.3.3 Well-posedness of the BVP 360 12.3.4 The IBVP with zero initial data 366 12.4 Resolution of non-linear IBVP 368 12.4.1 Planar reference shocks 368 12.4.2 Compact shock fronts 374 PART IV. APPLICATIONS TO GAS DYNAMICS 13 The Euler equations for real fluids 385 13.1 Thermodynamics 385 13.2 The Euler equations 391 13.2.1 Derivation and comments 391 13.2.2 Hyperbolicity 392 13.2.3 Symmetrizability 394 13.3 The Cauchy problem 399 13.4 Shock waves 399 13.4.1 The Rankine–Hugoniot condition 399 13.4.2 The Hugoniot adiabats 401 13.4.3 Admissibility criteria 401 14 Boundary conditions for Euler equations 411 14.1 Classification of fluids IBVPs 411 14.2 Dissipative initial boundary value problems 412 14.3 Normal modes analysis 414 14.3.1 The stable subspace of interior equations 414 14.3.2 Derivation of the Lopatinskiı̆ determinant 416 14.4 Construction of a Kreiss symmetrizer 419 15 Shock stability in gas dynamics 424 15.1 Normal modes analysis 424 15.1.1 The stable subspace for interior equations 425 15.1.2 The linearized jump conditions 426 15.1.3 The Lopatinskiı̆ determinant 427
- 16. xii Contents 15.2 Stability conditions 430 15.2.1 General result 430 15.2.2 Notable cases 437 15.2.3 Kreiss symmetrizers 438 15.2.4 Weak stability 440 PART V. APPENDIX A Basic calculus results 443 B Fourier and Laplace analysis 446 B.1 Fourier transform 446 B.2 Laplace transform 447 B.3 Fourier–Laplace transform 448 C Pseudo-/para-differential calculus 449 C.1 Pseudo-differential calculus 450 C.1.1 Symbols and approximate symbols 450 C.1.2 Definition of pseudo-differential operators 452 C.1.3 Basic properties of pseudo-differential operators 453 C.2 Pseudo-differential calculus with a parameter 455 C.3 Littlewood–Paley decomposition 459 C.3.1 Introduction 459 C.3.2 Basic estimates concerning Sobolev spaces 461 C.3.3 Para-products 465 C.3.4 Para-linearization 473 C.3.5 Further estimates 478 C.4 Para-differential calculus 481 C.4.1 Construction of para-differential operators 481 C.4.2 Basic results 486 C.5 Para-differential calculus with a parameter 487 Bibliography 492 Index 505
- 17. INTRODUCTION Within the field of Partial Differential Equations (PDEs), the hyperbolic class is one of the most diversely applicable, mathematically interesting and technically difficult: these (certainly biased) qualifying terms may serve as milestones along an overview of the field, which we propose prior to entering the bulk of this book. Applicability. Hyperbolic PDEs arise as basic models in many applications, and especially in various branches of physics in which finite-speed propagation and/or conservation laws are involved. To quote a few, and nonetheless funda- mental examples, let us start with linear hyperbolic PDEs. The most ancient one is undoubtedly the wave equation – also known in one space dimension as the equation of vibrating strings – dating back to the work of d’Alembert in the eighteenth century, which is closely related to the transport equation. We also have in mind the Maxwell system of electromagnetism, as well as the equation associated with the Dirac operator. Theoretical physics is a source of several semilinear equations and systems – semilinearity being characterized by a linear principal part and non-linear terms in the subprincipal part – for example, the Klein/sine–Gordon equations, the Yang–Mills equations, the Maxwell system for polarized media, etc. The non-linear models – often quasilinear – are even more numerous. The most basic one is provided by the so-called Euler equations of gas dynamics, which opened the way (controversially) in the late nineteenth century to the shock waves theory (later revived, in the 1940s, by the atomic bomb research, and still of interest nowadays for more peaceful applications, in medicine for instance). Speaking of flows, a prototype of scalar, one-dimensional conser- vation law was introduced in the 1950s in traffic flow modelling (under some heuristic assumptions on the drivers’ behaviour), which is nowadays referred to as the Lighthill–Whitham–Richards model. Other non-linear hyperbolic models include: the equations of elastodynamics (of which a linear version is widely used, in the modelling of earthquakes as well as in engineering problems with small deformations); the equations of chemical separation (chromatography, electrophoresis); the magnetohydrodynamics (MHD) equations – the coupling between fluid dynamics and electromagnetism being quite relevant for planets and other astrophysical systems – the Einstein equations of general relativity; non-linear versions of the Maxwell system for strong fields, for example the Born–Infeld model. Hyperbolic equations may also arise as a byproduct of an elaborate piece of analysis, as in the modulation theory of integrable Hamiltonian PDEs (like the Korteweg–de Vries equation and some non-linear Schrödinger
- 18. xiv Introduction equations), in which the envelopes of oscillating solutions are described by solutions of the (hopefully hyperbolic) Whitham equations. This list of hyperbolic PDEs is by no means exhaustive. Of course most of them are to some extent approximate: more realistic models should also involve dissipation processes (for instance in continuum mechanics) or higher-order phenomena, and thus be (at least partially) parabolic or dispersive. However, large-scale phenomena are usually governed by the hyperbolic part: the relevance of hyperbolic PDEs in many applications is in no doubt. Mathematical interest. For both mathematical reasons and physical rele- vance, hyperbolicity is associated with a space–time reference frame, in the sense that there exists a co-ordinate (most often the physical time) playing a special role compared to the other co-ordinates (usually spatial ones). Of course, changes of co-ordinates are always possible and we may speak of time-like co-ordinates and of space-like hypersurfaces: this terminology is familiar to people used to general relativity, and is also relevant in every situation where a hyperbolic operator is given. Except in one-dimensional frameworks, it is by no means possible to interchange the role of space and time variables: the distinction between time and space is a crucial feature of multidimensional hyperbolic PDEs, as we shall see in the analysis of Initial Boundary Value Problems. Multidimensional hyperbolic PDEs constrast with one-dimensional ones from several points of view, in particular in connection with the important notion of dispersion. Indeed, recall that the most visible feature of hyperbolic PDEs is finite-speed propagation. In several space dimensions, when the information is propagated not merely by pure transport, it gets dispersed: this dispersion of signals is itself responsible for a damping phenomenon in all Lp norms with p 2 (by contrast with what usually happens with the L2 norm, independent of time by a conservation of energy principle), and is associated with special, space– time estimates called Strichartz estimates, obtained by fractional integration – Strichartz estimates have been proved much fruitful in particular in the analysis of semilinear hyperbolic Cauchy problems. Another point worth mentioning is the diversity of mathematical tools that have been found useful to the theory of (linear) multidimensional hyperbolic PDEs, ranging from microlocal analysis to algebraic topology (not to mention those that still need to be invented, as we shall suggest below!). The former has been widely used to study the propagation of singularities in wave-like equations. In the same spirit, pseudo- (or even para-) differential calculus is of great help to study linear hyperbolic problems with variable coefficients, as we shall see in the third and fourth parts of this book. The link to algebraic topology might seem less obvious to unaware readers and deserves a little explanation. When studying constant-coefficients hyperbolic operators we are led to consider, in the frequency space, algebraic manifolds called characteristic cones – which are by definition zero sets of symbols, and are linked to finite-speed propagation. The fundamental solution, say E, of a constant-coefficients hyperbolic PDE is indeed known to be supported by the convex hull of Γ, the forward part of the dual
- 19. Introduction xv of the characteristic cone. In some cases, it happens that E is supported by Γ only; the open set co(Γ) Γ, on which E vanishes, is then called a lacuna. For example, the wave equation in dimension 1 + d with d odd and d ≥ 3, has a lacuna: its fundamental solution is supported by the dual characteristic cone itself (this explains, for instance, the fact that light rays have no tail). The systematic study of lacunæ is related to the topology of real algebraic sets. Compared to linear ones, non-linear problems display fascinating new fea- tures. In particular, several kinds of non-linear waves arise (shocks, rarefaction waves, as well as contact discontinuities). They are present already in one space dimension. The occurrence of shock waves is connected with a loss of regularity in the solutions in finite time, which can be roughly explained as follows: non-linearity implies that wave speeds depend on the state; therefore, a non- constant solution experiences a wave overtaking, which results in the creation of discontinuities in the derivatives of order m − 1, if m is the order of the system; such discontinuities are called shock waves, or simply shocks. After blow-up, that is after creation of shock(s), solutions cannot be smooth any longer. This yields many questions: what is the meaning of the PDEs for non-smooth solutions; can we solve the system in terms of weak enough solutions, and if possible in a unique, physically relevant way? The answer to the first question has been given by the theory of distributions, which is somehow the mathematical counterpart of conservation principles in physics: conservation of mass, momentum and energy, for instance (or Ampère’s and Faraday’s laws in electromagnetism) make sense indeed as long as fields remain locally bounded. The drawback is – as has long been known – that weak solutions are by no means unique, and this seems to hurt the common belief that PDE models in physics describe deterministic processes. This apparent contradiction may be resolved by the use of a suitable entropy condition, most often reminiscent of the second principle of thermodynamics. In one space dimension, entropy conditions have been widely used in the last decades to prove global well-posedness results in the space of Bounded Vari- ations (BV) functions – a space known to be inappropriate in several space dimensions, because of the obstruction on the Lp norms (see below for a few more details). Entropy conditions are expected to ensure also multidimensional well-posedness, even though we do not know yet what would be an appropriate space: one of the goals of this book is to present a starting point in this direc- tion, namely (local in time) well-posedness within classes of piecewise smooth solutions. Finally, the concept of time reversibility is quite intriguing in the framework of hyperbolic PDEs. On the one hand, as far as smooth solutions are concerned, many hyperbolic problems are time reversible, and this seems incompatible with the decay (already mentioned above) of Lp norms for p 2 in several space dimensions. This paradox was actually resolved by Brenner [22,23], who proved that multidimensional hyperbolic problems are ill-posed, in Hadamard’s sense, in Lp for p = 2. Incidentally, Brenner’s result shows that the space BV , which is built upon the space of bounded measures, itself close to L1 , cannot be appropri- ate for multidimensional problems. On the other hand, time reversibility is lost
- 20. xvi Introduction (as a mathematical counterpart of the second principle of thermodynamics) once shocks develop, whence a loss of information, the backward problem becoming ill- posed. As a matter of fact, shocks may be viewed as free boundaries and as such they can be sought as solutions of (non-standard) hyperbolic Initial Boundary Value Problems (IBVP): it turns out that most of the well-posed hyperbolic IBVPs are irreversible, as will be made clear in particular in this book – a large part of this volume is indeed dedicated to a systematic study of IBVPs, either for themselves, or in view of applications to well-posedness in the presence of shock waves. Difficulty. Even when a functional framework is available, a rigorous analysis of hyperbolic problems often requires much more elaborate (or at least more technical) tools than for elliptic or parabolic problems, notably to cope with the lack of smoothing effects. The situation is even worse in the non-linear context, where functional analysis has been useless in the study of weak entropy solutions so far (except for first-order scalar equations). This is why our knowledge of global-in-time solutions is so poor, despite tremendous efforts by talented math- ematicians. Speaking only about the Cauchy problem for quasilinear systems of first-order conservation laws, in space dimension d with n scalar unknowns, we know about well-posedness only in the following cases. r Scalar problems (n = 1), thanks to Kruzkhov’s theory [105]. r One space dimension (d = 1) and small data of bounded variation: existence results date back to Glimm’s seminal work [70]; uniqueness and continuous dependence have been obtained by Bressan and coworkers (see, for instance, [25–27]). r Small smooth data and large enough space dimension (for then dispersion can compete with non-linearity and prevent shock formation): most results from this point of view have been established by Klainerman and coworkers. See, for instance, Hörmander’s book [88]. Amazingly enough, none of these results apply to such basic systems as the full gas dynamics equations in one space dimension (n = 3, d = 1) or the isentropic gas dynamics equations (n = 2) in dimension d ≥ 2. Other results solve only one part of the problem: r Global existence for general data when d = 1 and n = 2 (under a genuine non-linearity assumption) by means of compensated compactness. This was achieved by DiPerna [49], following an idea by Tartar [202]. Solutions are then found in L∞ . Unfortunately, no uniqueness proof in such a large space has been given so far, except for weak–strong uniqueness (uniqueness in L∞ of a classical solution). r Local existence of smooth solutions for smooth data. This is quite a good result since it shows at least local well-posedness. It is attributed to several people (Friedrichs, Gårding, Kato, Leray, and possibly others),
- 21. Introduction xvii depending on specific assumptions that were made. Unfortunately, its practical implications are limited by the smallness of the existence time – recall that shock formation precludes, in general, global existence results within smooth functions. Having this (modest) state-of-the-art in mind, we can foresee a compromise regarding multidimensional weak solutions and non-linear problems: it will con- sist of the analysis of piecewise smooth solutions (involving a finite number of singularities like shock waves, rarefaction waves or contact discontinuities), tractable by ‘classical’ tools. This is the point of view we have adopted here, which defines the scope of this book: we shall consider either (possibly weak) solutions of linear problems with smooth coefficients or piecewise smooth solutions of non-linear problems – Cauchy problems and also of Initial Boundary Value Problems – to multidimensional hyperbolic PDEs. We now present a more detailed description of the contents. We have chosen a presentation involving gradually increasing degrees of difficulty: this is the case for the ordering of the three main ‘theoretical’ parts of the book – the first one being devoted to linear Cauchy problems, the second one to linear Initial Boundary Value Problems, and the third one to non-linear problems; this is also the case inside those parts – the first two parts starting with constant coefficients before going to variable coefficients, and the third one starting from Cauchy problems, then going to IBVPs, and culminating with the shock waves stability analysis. As a consequence, readers should be able to find the information they need without having to enter overcomplicated frameworks: most chapters are indeed (almost) self-contained (and as a drawback, the book is not free from repetitions). Another deliberate choice of ours has been to concentrate on first-order systems, even though we are very much aware that higher-order hyperbolic PDEs are also of great interest. This is mainly a matter of taste, because we come from the community of conservation laws. In addition, we think that the understanding of either one of those classes (first-order systems or higher-order scalar equations) basically provides the understanding of the other class (see, for instance, the book by Chazarain and Piriou [31], Chapter VII). Consistently with that choice, the main application we have considered is the first-order system of Euler equations in gas dynamics, to which the fourth part of the book is entirely devoted. We have tried to temperate this ‘monomaniac’ attitude by referring from place to place to higher-order equations, and in particular to the wave equation, which is the source of several examples throughout the theoretical chapters. Finally, to keep the length of this book reasonable, we have decided not to speak of (nevertheless important) questions that are too far away from the shock waves theory. Thus the reader will not find anything about the propagation of singularities as developed by Egorov, Hörmander and Taylor. Likewise, non- local boundary operators as they appear, for instance, in absorbing or trans- parent boundary conditions will not be considered, and all numerical aspects of
- 22. xviii Introduction hyperbolic IBVPs will be omitted, despite their great theoretical and practical importance. First part. The theory of linear Cauchy problems is most classical, even though some results are not that well-known. The chapter on constant-coefficient problems is the occasion of pointing out important definitions: Friedrichs sym- metrizability; directions of hyperbolicity; strict hyperbolicity and more generally what we call constant hyperbolicity – the eigenvalues of the symbol of a so- called constantly hyperbolic operator are semisimple and of constant multiplicity, instead of being simple in the case of strict hyperbolicity. Throughout the book, all hyperbolic operators will be assumed either Friedrichs symmetrizable or constantly hyperbolic (or both), as is the case for most operators coming from physics. The chapter on variable-coefficients Cauchy problems presents, in more generality, the symmetrizers technique, and in particular introduces the notion of symbolic symmetrizers, thus illustrating the power of pseudo- differential calculus (for infinitely smooth coefficients) and even para-differential calculus (for coefficients of limited regularity). Second part. The theory of Initial Boundary Value Problems (IBVP) is inspired from, but tremendously more complicated than, the theory of Cauchy problems. A kind of introductory chapter is devoted to the easier case of symmetric dissipative IBVPs. The second chapter addresses constant-coefficients IBVPs in a half-space, in which a central concept arises, namely the (uniform) Lopatinskiı̆ condition. This stability condition dates back to the 1970s: simultane- ously with a work by Lopatinskiı̆ ( [122], unnoticed in the West, Lopatinskiı̆ being more famous for his older work on elliptic boundary value problems [121]), it was worked out by Kreiss [103], and independently by Sakamoto [174] for higher-order equations; in acknowledgement of Kreiss’ work on first-order hyperbolic systems we shall rather call it the (uniform) Kreiss–Lopatinskiı̆ condition, and we shall also speak of Kreiss’ symmetrizers, which are symbolic symmetrizers adapted to IBVPs. The necessity of Kreiss’ symmetrizers shows up indeed when a Laplace– Fourier transform is applied to the equations (Laplace in the time direction and Fourier in the spatial boundary direction): to obtain an a priori estimate without loss of derivatives we need to multiply the equations by a suitable matrix-valued function, depending homogeneously on space–time frequencies – thus being a symbol – in place of the energy tensor of the symmetric dissipative case; that matrix-valued symbol is what we call a Kreiss symmetrizer. The actual construction of Kreiss’ symmetrizers is quite involved, and requires a good knowledge of linear algebra and real algebraic geometry. For this reason, a separate chapter is devoted to the construction of Kreiss’ symmetrizers. The interplay with algebraic geometry (formerly developed by Petrovskiı̆, Oleinik and their school) is a deep reason why we need a structural assumption such as constant hyperbolicity: even with this, there remain tricky points to deal with, namely the so-called glancing points, where eigenvalues lack regularity. The chapter on variable-coefficient IBVPs focuses more on the calculus aspects
- 23. Introduction xix of the theory: it shows how to extend well-posedness results to more general situations – variable coefficients with either infinite or limited regularity, non- planar boundaries – by means of pseudo- or para-differential calculus. The remaining chapters of the second part are devoted to more peculiar topics: characteristic boundaries (which yield involved additional difficulties); homogeneous IBVPs (which turn out to require only a weakened version of the uniform Lopatinskiı̆ condition); the so-called class WR, which consists of certain C ∞ -well-posed problems and is generic in the sense that it is stable under small disturbances of the operators, but displays estimates with a loss of regularity. These topical chapters may be skipped by the reader insterested only in the applications to multidimensional shock stability. Third part. We must admit that the current knowledge of non-linear multi- dimensional hyperbolic problems is very much limited: all well-posedness results presented in this part are short-time results; nevertheless, their proofs are not that easy. A first chapter reviews Cauchy problems: symmetric (or Friedrichs- symmetrizable) ones, but also those with symbolic symmetrizers (at is the case for constantly hyperbolic systems), for which well-posedness was not much known up to now (the only reference we are aware of is a proceedings paper by Métivier [132]). Well-posedness is to be understood in Sobolev spaces of sufficiently high index, or to be more precise, in Hs (Rd ) with s d/2 + 1 (the condition ensuring that Hs (Rd ) is an algebra, whose elements are at least continuously differentiable, by Sobolev’s theorem). In other words, we speak in that chapter only of smooth, or classical solutions, except in the very last section, where we recall the weak–strong uniqueness result of Dafermos and prepare the way for piecewise smooth solutions considered in the chapter on shock waves. Then ‘standard’ non-linear IBVPs are considered in a separate chapter, which is the occasion to see a simplified version of what is going on for shocks. The chapter on the persistence (or existence and stability) of single shock solutions was one of the main motivations to write this book. The idea was to give a comprehensive account of the work done by Majda in the 1980s [124–126], after it was revisited by Métivier and coworkers [56, 131, 133, 134, 136, 140]. Initially, we intended to cover also non-classical (multidimensional) shocks, as considered by Freistühler [58, 59] and Coulombel [40]. But for clarity we have preferred to concentrate on Lax shocks, while avoiding as much as possible to use their specific properties so that interested readers could either guess what happens for non- classical shocks or refer more easily to [40] for instance. We have also deliberately omitted the most recent developments on characteristic and/or non-constantly hyperbolic problems. Fourth part. This concerns one of the most important applications of hyper- bolic PDEs: gas dynamics. In fact, the theory of hyperbolic conservation laws was developed, in particular by Peter Lax in the 1950s, by analogy with gas dynamics: terms like ‘entropy’, ‘compressive’ (or ‘undercompressive’) shock are reminiscent of this analogy, and the so-called Rankine–Hugoniot jump conditions
- 24. xx Introduction were initially derived (in the late nineteenth century) by these two engineers (Rankine and Hugoniot) in the framework of gas dynamics. There is a huge literature on gas dynamics, by engineers, by physicists and by mathematicians. In recent decades, the latter have had a marked preference for a familiar pressure law, usually referred to as the γ-law, for it simplifies, to some extent (depending on the explicit value of γ), the analysis of the Euler equations of gas dynamics. We have chosen here to consider more general pressure laws, which apply to so-called real – at least more realistic – fluids and not only perfect gases (as was the case in earlier mathematical papers, by Weyl [218], Gilbarg [69], etc.). In a first chapter we address several basic questions, regarding hyperbolicity and symmetrizability. The second chapter is devoted to boundary conditions for real fluids, a very important topic for engineers, which has (surprisingly) not received much attention from mathematicians (see, however, the very nice review paper by Higdon [84]). This applied part culminates with the shock-waves analysis for real fluids, in the last chapter. Even though it seems to belong to ‘folklore’ in the shock-waves community, the complete investigation of the Kreiss–Lopatinskiı̆ condition for the Euler equations is hard to find in the literature: in particular, Majda gave the complete stability conditions in [126] but showed how to derive them only for isentropic gas dynamics; a complete, analytic proof was published only recently by Jenssen and Lyng [92]. By contrast, our approach is mostly algebraic, and works fine for full gas dynamics (of which the isentropic gas dynamics appear as a special, easier case). In addition, we give an explicit construction of Kreiss symmetrizers, which (to our knowledge) cannot be found elsewhere, and is fully elementary (compared to the sophisticated tools used for abstract systems). Fifth part. This is only a (huge) appendix, collecting useful tools and tech- niques. The main topics are the Laplace transform – including Paley–Wiener theorems – pseudo-differential calculus, and its refinement called para-differential calculus. Less space demanding (or more classical) tools are merely introduced in the Notations section below.
- 25. NOTATIONS The set of matrices with n rows and p columns, with entries in a field K, is denoted by Mn×p(K). If p = n, we simply write Mn(K). The latter is an algebra, whose neutral elements under addition and multiplication are denoted by 0n and In, respectively. The space Mn×p(K) may be identified to the set of linear maps from Kp to Kn . The transpose matrix is written MT . The group of invertible n × n matrices is GLn(K). If p = 1, Mn×1(K) is identified with Kn . Given two matrices M, N ∈ Mn(C), their commutator MN − NM is denoted by [M, N]. If K = C, the adjoint matrix is written M∗ . It is equal to M T , where M denotes the conjugate of M. We equip Cm and Rm with the canonical Hermitian norm x = j | xj |2 = (x∗ x)1/2 . This norm is associated to the scalar product (x, y) = j xjyj = y∗ x. The norm will sometimes be denoted |x|, especially when x is a space variable or a frequency vector (used in Fourier transform.) A complex square matrix U is unitary if U∗ U = In, or equivalently UU∗ = In. The set Un of unitary matrices is a compact subgroup of GLn(C). Its intersection On with Mn(R) is the set of real orthogonal matrices. The special orthogonal group SOn is the subgroup defined by the constraint det M = 1. As usual, Mn×p(C) is equipped with the induced norm M = sup Mx x . When the product makes sense, one knows that MN ≤ M N. When p = n, Mn(C) is thus a normed algebra, and we have Mk ≤ Mk . If Q is a unitary (for instance real orthogonal) matrix, one has Q = 1. More generally, the norm is unitary invariant, which means that M = PMQ whenever P and Q are unitary. If M ∈ Mn(C), the set of eigenvalues of M, denoted by SpM, is called the spectrum of M. The largest modulus of eigenvalues of M is called the spectral radius of M, and denoted by ρ(M). It is less than or equal to M, and such
- 26. xxii Notations that ρ(M) = lim k→+∞ Mk 1/k . The following formula holds, M2 = ρ(M∗ M) = ρ(MM∗ ). Several other norms on Mn(C) are of great interest, among which is the Frobenius norm, defined by MF := j,k | mjk|2. Since M2 F = Tr(M∗ M) = Tr(MM∗ ), we have M ≤ MF. A complex square matrix M is Hermitian if M∗ = M. It is skew-Hermitian if M∗ = −M. The Hermitian n × n matrices form an R-vector space that we denote by Hn. The cone of positive-definite matrices in Hn is denoted by HPDn. When M is Hermitian, we have M = ρ(M). Every Hermitian matrix is diagonalizable with real eigenvalues, its normalized eigenvectors forming an orthonormal basis. The skew-Hermitian matrices with complex entries form an R-vector space that we denote by Skewn. We remark that Mn(C) = Hn ⊕ Skewn and Skewn = iHn. The intersections of Hn, HPDn and Skewn with the subspace Mn(R) of matrices with real entries are denoted by Symn, SPDn and Altn, respectively. We have Mn(R) = Symn ⊕ Altn. Real symmetric matrices have real eigenvalues and are diagonalizable in an orthogonal basis. Given an n × n matrix M, one defines its exponential by exp M = eM := ∞ k=0 1 k! Mk , which is a convergent series. The map t → exp(tM) is the unique solution of the differential equation dA dt = MA, such that A(0) = In. It solves equivalently the ODE dA dt = AM. The exponential behaves well with respect to conjugation, that is exp(PMP−1 ) = P(exp M)P−1 for all invertible matrix P. The eigenvalues of exp A are the exponentials of those of A. In particular, ρ(exp A) is the exponential of the maximal real part Re λ, as λ runs over SpA. The matrix exp(A + B) does not equal (exp A)(exp B) in
- 27. Notations xxiii general, but it does when AB = BA. In particular, exp A is always invertible, with inverse exp(−A). Other useful formulæ are exp(MT ) = (exp M) T , exp M = exp M, exp(M∗ ) = (exp M)∗ . The exponential of a Hermitian matrix is Hermitian, positive-definite. The map exp : Hn → HPDn is actually an homeomorphism. The exponential of a skew-Hermitian matrix is unitary. Let A ∈ Mn(C) be given. The space Cn splits, in a unique way, as the direct sum of three invariant subspaces, namely the stable, unstable and central subspaces of A, denoted, respectively, Es(A), Eu(A) and Ec(A). Their invariance properties read AEs(A) = Es(A), AEu(A) = Eu(A) and AEc(A) ⊂ Ec(A). The stable invariant subspace is formed of vectors x such that (exp tA)x tends to zero as t → +∞, and then the decay is exponentially fast. The unstable subspace is formed of vectors x such that (exp tA)x tends to zero (exponentially fast) as t → −∞. The central subspace consists of vectors such that (exp tA)x is polynomially bounded on R. Since these spaces are invariant under A, this matrix operates on each one as an endomorphism, say As, Au, Ac. The spectrum of As (respectively, Au, Ac) has negative (respectively, positive, zero) real part. The union of these spectra is the whole spectrum of A, with the correct multiplicities. Hence the dimension of Es(A) is the number of eigenvalues of A of negative real part (these are called ‘stable eigenvalues’), counted with multiplicities. When Ec(A) is trivial, meaning that there is no pure imaginary eigenvalue, A is called hyperbolic (in the sense of Dynamical Systems). Dunford–Taylor formula. Let γ be a Jordan curve, oriented in the trigono- metric way, disjoint from SpA. Let σ be the part of SpA that γ enclose. Then the Cauchy integral Pσ := 1 2iπ γ (zIn − A)−1 dz defines a projector (that is P2 σ = Pσ) whose range and kernel are invariant under A. (Moreover, A commutes with Pσ). The spectrum of the restriction of A to the range of Pσ is exactly the part of the spectrum of A that belongs to σ. In other words, R(Pσ) is the direct sum of the generalized eigenspaces associated to those eigenvalues in σ. More information about matrices and norms may be found in [187]. Functional spaces Given an open subset Ω of Rn , the set of infinitely differentiable functions (with values in C) that are bounded as well as all their derivatives on Ω is denoted
- 28. xxiv Notations by C ∞ b (Ω). The set of compactly supported infinitely differentiable functions (also called test functions) is denoted by D(Ω). Its dual D (Ω) is the space of distributions. The derivation ∂j := ∂/∂xj is a bounded linear operator on D(Ω). Its adjoint is therefore bounded on D (Ω). The distributional derivative, still denoted by ∂j, is the adjoint of −∂j. A multi-index α is a finite sequence (α1, . . . , αn) of natural integers. Its length |α| is the sum j αj. The operator ∂α := ∂α1 1 · · · ∂αn n is a derivation of order |α|. We also use the notation ξα = ξα1 1 · · · ξαn n , when ξ ∈ Rn . Given a C 1 function f : Ω → C, the differential of f at point X is the linear form df(X) : ξ → df(X)ξ := n j=1 ξj ∂jf(X). The map X → df(X) (that is the differential of f) is a differential form. The second differential, or Hessian of f at X is the bilinear form D2 f(X) : (ξ, η) → n i,j=1 ξi ηj ∂i∂jf(X) . We may define differentials of higher orders D3 f, . . . Given a Banach space E, the Lebesgue space of measurable functions u : Ω → E whose pth power is integrable, is denoted by Lp (Ω; E). When E = R or E = C, we simply denote Lp (Ω) if there is no ambiguity. The norm in Lp (Ω; E) is uLp := Ω u(x)p Edx 1/p . If m ∈ N, the Sobolev space Wm,p (Ω; E) is the set of functions in Lp (Ω; E) whose distributional derivatives up to order m belong to Lp . Its norm is defined by uW m,p := |α|≤m ∂α up Lp 1/p . If p = 2 and if E is a Hilbert space, Wm,2 (Ω; E) is a Hilbert space and is denoted Hm (Ω; E), or simply Hm (Ω) if E = C or E = R or if there is no ambiguity. Sobolev spaces of order s (instead of m) may be defined for every real number s. The simplest definition occurs when p = 2, Ω = Rn and E = C, where Hs (Rn ) is isomorphic to a weighted space L2 ((1 + |ξ|2 )s dξ) through the Fourier transform. For a crash course on Hs (Ω) (sometimes also denoted Hs (Ω)), we
- 29. Notations xxv refer the reader to Chapter II in [31]; for more details in more general situations, see for instance the classical monograph by Adams [1]. The notation Hs w will stand for the Sobolev space Hs equipped with the weak topology instead of the (strong) Hilbert topology. The Schwartz space of rapidly decreasing functions S (Rn ) will simply be denoted by S when no confusion can occur as concerns the space dimension. And similarly, its dual space, consisting of temperate distributions, will be denoted by S . Other tools We have collected in the appendix various additional tools, ranging from standard calculus and Fourier–Laplace analysis to pseudo-differential and para-differential calculus: we hope it will be helpful to the reader.
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- 31. PART I THE LINEAR CAUCHY PROBLEM
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- 33. 1 LINEAR CAUCHY PROBLEM WITH CONSTANT COEFFICIENTS The general Cauchy problem Let d ≥ 1 be the space dimension and x = (x1, . . . , xd) denote the space variable, t being the time variable. The Cauchy problem that we consider in this section is posed in the whole space Rd , while t ranges on an interval, typically (0, T), where T ≤ +∞. A constant-coefficient first-order system is determined by d + 1 matrices A1 , . . . , Ad , B given in Mn(R), where n ≥ 1 is the size of the system. Then the Cauchy problem consists in finding solutions u(x, t) of ∂u ∂t + d α=1 Aα ∂u ∂xα = Bu + f, (1.0.1) where f = f(x, t) and the initial datum u(·, t = 0) = a are given in suitable functional spaces. To shorten the notation, we shall rewrite equivalently ∂tu + α Aα ∂αu = Bu + f. When f ≡ 0, the Cauchy problem is said to be homogeneous. A well-posedness property holds for the homogeneous problem when, given a in a functional space X, there exists one and only one solution u in C (0, T; Y ), for some other functional space Y , the map X → C (0, T; Y ) a → u being continuous. ‘Solution’ is understood here in the distributional sense. Exis- tence and continuity imply X ⊂ Y , since the map a → u(0) must be continuous. We use the general notation X St → Y a → u(t). Since a homogeneous system is, at a formal level, an autonomous differen- tial equation with respect to time, we should like to have the semigroup
- 34. 4 Linear Cauchy Problem with Constant Coefficients property St+s = St ◦ Ss, s, t ≥ 0, this of course requires that Y = X. We then say that the homogeneous Cauchy problem defines a continuous semigroup if for every initial data a ∈ X, there exists a unique distributional solution of class C (R+ ; X). Note that the word ‘continuous’ relies on the continuity with respect to time of the solution, but not on the continuity of t → St in the operator norm. Semigroup theory actually tells us that, if X is a Banach space, the continuity in the operator norm corresponds to ordinary differential equations, a context that does not apply in PDEs. When the homogeneous Cauchy problem defines a continuous semigroup on a functional space X, we expect to solve the non-homogeneous one using Duhamel’s formula: u(t) = Sta + t 0 St−sf(s)ds, (1.0.2) provided that at least f ∈ L1 (0, T; X). For this reason, we focus on the homoge- neous Cauchy problem and content ourselves in constructing the semigroup. Before entering into the theory, let us remark that, since (1.0.1) writes ∂u ∂t = Pu + f, where P is a differential operator of order less than or equal to one, the order with respect to time of this evolution equation, the Cauchy–Kowalevska theory applies. For instance, if f = 0, analytic initial data yield unique analytic solutions. However, these solutions exist only on a short time interval (0, T∗ (a)). Since analytic data are unlikely in real life, and since local solutions are of little interest, we shall not concern ourselves with this result. 1.1 Very weak well-posedness We first look at the necessary conditions for a very weak notion of well-posedness, where X = S (Rd ) (the Schwartz class) and Y = S (Rd ), the set of tempered distributions. Surprisingly, this analysis will provide us with a rather strong necessary condition, sometimes called weak hyperbolicity1 . Let us assume that the homogeneous Cauchy problem is well-posed in this context. Let a be a datum and u be the solution. From the equation ∂u ∂t + d α=1 Aα ∂u ∂xα = Bu, (1.1.3) 1Some authors call it simply hyperbolicity, and use the term strong hyperbolicity for the notion that we shall call hyperbolicity. Thus, depending on the authors, there is either the weak and normal hyperbolicities, or the normal and strong ones.
- 35. Very weak well-posedness 5 we obtain u ∈ C ∞ (0, T; Y ). This allows us to Fourier transform (1.1.3) in the spatial directions. We obtain that (1.1.3) is equivalent to ∂û ∂t + i d α=1 ηαAα û = Bû. Using the notation A(η) := d α=1 ηαAα , we rewrite this equation as an ODE in t, parametrized by η ∂û ∂t = (B − iA(η))û. (1.1.4) Since û(·, 0) = â, the solution of (1.1.4) is explicitly given by û(η, t) = et(B−iA(η)) â(η). (1.1.5) By well-posedness (1.1.5) defines a tempered distribution for every choice of â in the Schwartz class, continuously in time. In other words, the bilinear map (φ, ψ) → Rd ψ(η)∗ et(B−iA(η)) φ(η) dη, (1.1.6) which is well-defined for compactly supported smooth vector fields φ and ψ, is continuous in the Schwartz topology, uniformly for t in compact intervals. Let λ be a simple eigenvalue of A(ξ) for some ξ ∈ Rd . Then, there is a C ∞ map (t, σ) → (µ, r), defined on a neighbourhood W of (0, ξ), such that µ(0, ξ) = −iλ and (t2 B − iA(σ))r(t, σ) = µ(t, σ)r(t, σ), r ≡ 1. Let us choose a non-zero compactly supported smooth function θ : Rd → C with θ(0) = 0. Then, for small enough t 0, the condition η − t−2 ξ ∈ Supp θ implies (t, t2 η) ∈ W. For such a t, we may define two compactly supported smooth vector fields by φt (η) := θ(η − t−2 ξ)r(t, t2 η), ψt (η) := θ(η − t−2 ξ)(t, t2 η), where is an eigenfield of the adjoint matrix (t2 B − iA(σ))∗ , defined and normalized as above. We then apply (1.1.6) to (φt , ψt ). The sequence (φt )t→0 is bounded in the Schwartz topology, and similarly is (ψt )t→0. Therefore Rd (ψt )∗ et(B−iA(η)) φt dη = Rd eµ(t,t2 η)/t ( · r)(t, t2 η)|θ(η − ξ/t2 )|2 dη is bounded as t → 0. Since it behaves like c exp(−iλ/t) for a non-zero constant c, we conclude that Im λ ≤ 0. Applying also this conclusion to the simple eigenvalue λ̄, we find that λ is real.
- 36. 6 Linear Cauchy Problem with Constant Coefficients The case of an eigenvalue of constant multiplicity in some open set of frequencies η can be treated along the same ideas; it must be real too. Finally, the points η at which the multiplicities are not locally constant form an algebraic submanifold, thus a set of void interior. By continuity, the reality must hold everywhere. We have thus proved Proposition 1.1 The (S , S ) well-posedness requires that the spectrum of A(ξ) be real for all ξ in Rd . When (S , S ) well-posedness does not hold, a Hadamard instability occurs: for most (in the Baire sense) data a in S , and for all T 0, the Cauchy problem does not admit any solution of class C (0, T; S ). This is a consequence of the Principle of Uniform Boundedness. Example The Cauchy–Riemann equations provide the simplest system for which this instability holds. One has d = 1, n = 2: ∂tu1 + ∂xu2 = 0, ∂tu2 − ∂xu1 = 0. This example shows that a boundary value problem for a system of partial dif- ferential equations may be well-posed though the corresponding Cauchy problem is ill-posed. The converse of Proposition 1.1 does not hold in general, mainly because of the interaction between non-semisimple eigenvalues of A(ξ) with the mixing induced by B. Let us take again a simple example with d = 1, n = 2, and A = A1 = 0 1 0 0 , B = 0 0 1 0 . Since the matrix exp(−iξA) = I2 − iξA has polynomial growth, the Cauchy problem for the operator ∂t + α Aα ∂α is well-posed in the (S , S ) sense, and even in the (S , S ) sense. Actually, its solution is explicitly given by u1(t) = a1 − ta 2, u2(t) ≡ a2. (We see that there is an immediate loss of regularity.) However, with our non-zero B, the matrix M := t(B − iξA) satisfies M2 = −it2 ξI2, which implies that exp(t(B − iξA)) = cos ω I2 + sin ω ω M, where ω = t(iξ)1/2 . Since Im ω = ±t ξ 2 1/2 ,
- 37. Strong well-posedness 7 we see that offdiagonal coefficients of exp M grow like exp(c|ξ|1/2 ) as ξ tends to infinity, provided t = 0. Then a calculation similar to the one in the proof of Proposition 1.1 shows that this Cauchy problem is ill-posed in the (S , S ) sense. 1.2 Strong well-posedness The previous example suggests that the notion of well-posedness in the (rather weak) (S , S ) sense might not be stable under small disturbance (the instability result would be the same with B instead of B). For this reason, we shall merely consider the well-posedness when Y = X and X is a Banach space. We then speak about strong well-posedness in X (or X-well-posedness). When this holds, the map St : a → u(t) defines a continuous semigroup on X. It can be shown that if X is a Banach space, there exist two constants c, ω, such that StL(X) ≤ ceωt , (1.2.7) Proposition 1.2 Let X be a Banach space. Then well-posedness (with Y = X) for some B ∈ Mn(R) implies the same property for all B. This amounts to saying that well-posedness is a property of (A1 , . . . , Ad ) alone. Proof Assume strong well-posedness for a given matrix B0. The problem ∂u ∂t + d α=1 Aα ∂u ∂xα = B0u (1.2.8) defines a continuous semigroup (St)t≥0. One has (1.2.7) with suitable constants c and ω. From Duhamel’s formula, (1.1.3) with a matrix B = B0 + C instead of B0, is equivalent to u(t) = Sta + t 0 St−sCu(s)ds. (1.2.9) Then we can solve (1.2.9) by a Picard iteration. Let us denote by Ru the right- hand side of (1.2.9), and I = (0, T) (with T 0) a time interval where we look for a solution. Because of (1.2.7), there exists a large enough N so that RN is contractant on C (I; X). Therefore, there exists a unique solution of (1.1.3) in C (I; X). Since T is arbitrary, the solution is global in time. 1.2.1 Hyperbolicity We first consider spaces X where the Fourier transform defines an isomorphism onto some other Banach space Z. Typically, X will be a Sobolev space Hs (Rd )n and Z is a weighted L2 -space: Z = L2 s(Rd )n , L2 s(Rd ) := {v ∈ L2 loc(Rd ) ; (1 + |ξ|2 )s/2 v ∈ L2 (Rd )}.
- 38. 8 Linear Cauchy Problem with Constant Coefficients Because of this example, we shall assume that multiplication by a measurable function g defines a continuous operator from Z to itself if and only if g is bounded. Looking for a solution u ∈ C (I; X) of (1.1.3) is simply looking for a solution v ∈ C (I; Z) of ∂v ∂t = (B − iA(η))v, v(η, 0) = â(η). (1.2.10) Thanks to Proposition 1.2, we may restrict ourselves to the case where B = 0n. Then v must obey the formula v(η, t) = e−itA(η) â(η), where â is given in Z. In order that v(t) belong to Z for all â, it is necessary and sufficient that η → exp(−itA(η)) be bounded. Since tA(η) = A(tη), this is equivalent to writing sup ξ∈Rd exp(iA(ξ)) +∞. (1.2.11) Let us emphasize that this property does not depend on the time t, once t = 0. Definition 1.1 A first-order operator L = ∂t + α Aα ∂α is called hyperbolic if the corresponding symbol ξ → A(ξ) satisfies (1.2.11). More generally, a system (1.0.1) (whatever B is) that satisfies (1.2.11) is called a hyperbolic2 system of first-order PDEs. After having proven that hyperbolicity is a necessary condition, we show that it is sufficient for the Hs -well-posedness. It remains to prove the continuity of t → v(t) with values in Z, when â is given in Z. For that, we write v(τ) − v(t)2 Z = Rd e−iτA(η) − e−itA(η) â(η) 2 (1 + |η|2 )s dη. Thanks to (1.2.11), the integrand is bounded by c|â(η)|2 (1 + |η|2 )s , an integrable function, independent of τ. Likewise, it tends pointwisely to zero, as τ → t. Lebesgue’s Theorem then implies that lim τ→t v(τ) − v(t)Z = 0. Let us summarize the results that we obtained: 2Some authors write strongly hyperbolic in this definition and keep the terminology hyperbolic for those systems that are well-posed in C ∞, that is whose a priori estimates may display a loss of derivatives.
- 39. Strong well-posedness 9 Theorem 1.1 r Let s be a real number. The Cauchy problem for ∂tu + α Aα ∂αu = 0 (1.2.12) is Hs -well-posed if and only if this system is hyperbolic. r If the operator L (defined as above) is hyperbolic, then the Cauchy problem for (1.1.3) is Hs -well-posed for every real number s. r In particular, the Cauchy problem is well-posed in Hs if and only if it is well-posed in L2 . Let us point out that hyperbolicity does not involve the matrix B. Since the well-posedness in a Hilbertian Sobolev space holds or does not, independently of the regularity level s, we feel free to rename this property strong well-posedness. Backward Cauchy problem We considered up to now the forward Cauchy problem, namely the determination of u(t) for times t larger than the initial time. Its well-posedness within L2 was shown to be equivalent to hyperbolicity. Reversing the time arrow amounts to making the change ∂t → −∂t. This has the same effect as changing the matrices Aα into −Aα . The L2 -well-posedness of the Cauchy problem is thus equivalent to the hyperbolicity of the new system ∂su − α Aα ∂αu = −Bu. This writes as sup ξ∈Rd exp(−iA(ξ)) +∞, which is the same as (1.2.11), via the change of dummy variable ξ → −ξ. Finally, the strong well-posedness of backward and forward Cauchy problems are equivalent to each other. For a hyperbolic system and a data a ∈ Hs (Rd )n , there exists a unique solution of (1.1.3) u ∈ C (R; Hs (Rd )n ) such that u(0) = a. Let us emphasize that here, t ranges on the whole line, not only on R+ . 1.2.2 Distributional solutions When (1.1.3) is hyperbolic, one can also solve the Cauchy problem for data in the set S of tempered distribution. For that, we again use the Fourier transform since it is an automorphism of S . We again define û by the formula (1.1.5). We only have to show that this definition makes sense in S for every t, and that u is continuous from Rt to S . For that, we have to show that X(t) := exp(t(B − iA(η))) is a C ∞ function of η, with slow growth at infinity, locally uniformly in time. We shall show that its derivatives are actually bounded with respect to η. The regularity is trivial, and we already know that X(t) is bounded
- 40. 10 Linear Cauchy Problem with Constant Coefficients in η, locally in time. Denoting by Xα the derivative with respect to ηα, we have dXα dt = (B − iA(η))Xα − iAα X, and therefore d(X−1 Xα) dt = −iX−1 Aα X. Using Duhamel’s formula, as in the proof of Proposition 1.2, we see that X(t) ≤ c(1 + B |t|), from which we deduce X−1 Xα ≤ (1 + B |t|)3 − 1 3B c2 Aα . Finally, we obtain Xα ≤ (1 + B |t|)4 3B c3 Aα . We leave the reader to estimate the higher derivatives and complete the proof of the following statement. The case of data in the Schwartz class is done in exactly the same way, since the Fourier transform is an automorphism of S and that S is stable under multplication by C ∞ functions with slow growth. Proposition 1.3 If L is hyperbolic, then the Cauchy problem for (1.1.3) is well-posed in both S and S . 1.2.3 The Kreiss’ matrix Theorem Of course, since L2 -well-posedness implies (S , S )-well-posedness, hyperbolicity implies that the spectrum of A(ξ) is real for all ξ in Rd . It implies even more, that all A(ξ) are diagonalizable. Though these two facts have a rather simple proof here, they do not characterize completely hyperbolic systems. We shall therefore describe the characterization obtained by Kreiss [102,104]. This is an application of a deeper result that deals with strong well-posedness of general constant-coefficient evolution problems. However, since we focus only on first- order systems, we content ourselves with a statement with a simpler proof, due to Strang [199]. Theorem 1.2 Let ξ → A(ξ) be a linear map from Rd to Mn(C). Then the following properties are equivalent to each other: i) Every A(ξ) is diagonalizable with pure imaginary eigenvalues, uniformly with respect to ξ: A(ξ) = P(ξ)−1 diag(iρ1, . . . , iρn)P(ξ), (ρ1(ξ), . . . , ρn(ξ) ∈ R),
- 41. Strong well-posedness 11 with P(ξ)−1 · P(ξ) ≤ C , ∀ξ ∈ Rd . (1.2.13) ii) There exists a constant C 0, such that etA(ξ) ≤ C, ∀ξ ∈ Rd , ∀t ≥ 0. (1.2.14) iii) There exists a constant C 0, such that (zIn − A(ξ))−1 ≤ C Re z , ∀ξ ∈ Rd , ∀Re z 0. (1.2.15) Note that, replacing (z, ξ) by (−z, −ξ), we also obtain (1.2.15) with Re z = 0. Applying Theorem 1.2, we readily obtain the following. Theorem 1.3 The Cauchy problem for a first-order system ∂tu + α Aα ∂αu = 0, x ∈ Rd is Hs -well-posed if and only if the following two properties hold. r The matrices A(ξ) are diagonalizable with real eigenvalues, A(ξ) = P(ξ)−1 diag(ρ1(ξ), . . . , ρn(ξ))P(ξ), (ρ1, . . . , ρn ∈ R). r Their diagonalization is well-conditioned (one may also say that the matri- ces A(ξ) are uniformly diagonalizable) : supξ∈Sd−1 P(ξ)−1 · P(ξ) +∞. Proof The fact that i) implies ii) is proved easily. Actually, etA(ξ) = P−1 etD P ≤ C etD . When D is diagonal with pure imaginary entries, exp(tD) is unitary, and the right-hand side equals C . The fact that ii) implies iii) is easy too. The following equality holds provided the integral involved in it converges in norm (A − zIn) ∞ 0 e−zt etA dt = −In. (1.2.16) Because of (1.2.14), the integral converges for every z ∈ C with positive real part. This gives a bound for the inverse of zIn − A, of the form (zIn − A)−1 ≤ C Re z , Re z 0. It remains to prove that iii) implies i). Thus, let us assume (1.2.15). Replacing (z, ξ) by (−z, −ξ), we see that the bound holds for Re z = 0, with |Re z| in the denominator. Thus the spectrum of A(ξ) is purely imaginary.
- 42. 12 Linear Cauchy Problem with Constant Coefficients Actually, A(ξ) is diagonalizable, for if there were a non-trivial Jordan part, then (zIn − A(ξ))−1 would have a pole of order two or more, contradicting (1.2.15). Therefore, A(ξ) admits a spectral decomposition A(ξ) = i j ρjEj, where ρj is real and Ej = Ej(ξ) is a projector (E2 j = Ej), with EjEk = 0n, (k = j), j Ej = In. Let us define H = H(ξ) := j EjE∗ j , which is a positive-definite Hermitian matrix. Since A(ξ)∗ = − j ρjE∗ j , it holds that H(ξ)A(ξ) = −A(ξ)∗ H(ξ), from which it follows that H(ξ)1/2 A(ξ)H(ξ)−1/2 is skew-Hermitian. As such, it is diagonalizable through a unitary transformation. Therefore A(ξ) = P(ξ)−1 D(ξ)P(ξ), where D(ξ) is diagonal with pure imaginary eigenvalues, and P(ξ) = U(ξ)H1/2 , where U(ξ) is a unitary matrix. We finish by proving that P(ξ) is uniformly conditioned. Since P±1 = H±1/2 = H±1/2 , this amounts to proving that H · H−1 is uniformly bounded. On the one hand, it holds that |v|2 = j Ejv 2 ≤ n j |Ejv|2 = n|H1/2 v|2 , so that H−1/2 ≤ √ n. On the other hand, applying (1.2.15) to + iρk, we have j ( + iρk − iρj)−1 Ej ≤ C || . Letting → 0, we deduce that Ej ≤ C, independently of ξ. It follows that H ≤ nC2 . Remarks i) A more explicit characterization of hyperbolic symbols has been estab- lished by Mencherini and Spagnolo when n = 2 or n = 3; see [129]. ii) The following example (n = 3 and d = 2), known as Petrowski’s example, shows that the well-conditioning can fail for systems in which all matrices
- 43. Strong well-posedness 13 A(ξ) are diagonalizable with real eigenvalues. Let us take A1 = 0 1 1 0 0 0 1 0 0 , A2 = 0 0 0 0 0 0 0 0 1 . One checks easily that the eigenvalues of A(ξ) are real and distinct for ξ1 = 0, while A2 is already diagonal. Hence, A(ξ) is always diagonalizable over R. However, as ξ1 tends to zero, one eigenvalue is identically zero, associated to the eigenvector (ξ2, ξ1, −ξ1)T , while another one is small, λ ∼ −ξ2 1ξ−1 2 , associated to (ξ2, 0, λξ2ξ−1 1 )T . Both eigenvectors have the same limit (ξ2, 0, 0)T , which shows that P(ξ) is unbounded as ξ1 tends to zero. See a similar example in [108]. Oshime [155] has shown that Petrowsky’s example is somehow canonical when d = 3. On the other hand, Strang [199] showed that when n = 2, the diagonalizability of every A(ξ) is equivalent to hyperbolicity, and that such operators are actually Friedrichs symmetrizable in the sense of the next section. iii) Uniform diagonalizability of A(ξ) within real matrices has been shown by Kasahara and Yamaguti [93, 221] to be necessary and sufficient in order that the Cauchy problem for ∂tu + α Aα ∂αu = Bu be C∞ -well-posed for every matrix B ∈ Mn(R). Of course, the sufficiency follows from Theorem 1.3 and Proposition 1.2. The necessity statement is even stronger than the one suggested by the example given in Section 1.1, since the diagonalizability within R is not sufficient. For instance, if A(ξ) is given as in the Petrowski example, there are matrices B for which the Cauchy problem is ill-posed in the Hadamard sense. 1.2.4 Two important classes of hyperbolic systems We now distinguish two important classes of hyperbolic systems. Definition 1.2 An operator L = ∂t + α Aα ∂α is said to be symmetric in Friedrichs’ sense [63], or simply Friedrichs symmetric, if all matrices Aα are symmetric; one may also say symmetric hyperbolic. More generally, it is Friedrichs symmetrizable if there exists a symmetric positive- definite matrix S0 such that every S0Aα is symmetric. An operator M as above is said to be constantly3 hyperbolic if the matrices A(ξ) are diagonalizable with real eigenvalues and, moreover, as ξ ranges along 3We employ this shortcut in lieu of hyperbolic with characteristic fields of constant multiplic- ities.
- 44. 14 Linear Cauchy Problem with Constant Coefficients Sd−1 , the multiplicities of eigenvalues remain constant. In the special case where all eigenvalues are real and simple for every ξ ∈ Sd−1 , we say that the operator is strictly hyperbolic. Let us point out that in a constantly hyperbolic operator, the eigenvalues may have non-equal multiplicities, but the set of multiplicities remains constant as ξ varies. This implies in particular that the eigenspaces depend analytically on ξ for ξ = 0. This fact easily follows from the construction of eigenprojectors as Cauchy integrals (see the section ‘Notations’.) To a large extent, the theory of constantly hyperbolic systems does not differ from the one of strictly hyperbolic systems. But the analysis is technically simpler in the latter case. This is why the theory of strictly hyperbolic operators was developed much further in the first few decades. Theorem 1.4 If an operator is Friedrichs symmetrizable, or if it is constantly hyperbolic, then it is hyperbolic. Proof Let the operator be Friedrichs symmetrizable by S0. Then S−1 0 is positive-definite and admits a (unique) square root R symmetric positive-definite (see [187], page 78). Let us denote S0Aα by Sα , and S(ξ) = α ξαSα as usual. Then A(ξ) = S−1 0 S(ξ) = R(RS(ξ)R)R−1 . The matrix RS(ξ)R is real symmetric and thus may be written as Q(ξ)T D(ξ)Q(ξ), where Q is orthogonal. Then A(ξ) is conjugated to D(ξ), A(ξ) = P(ξ)−1 D(ξ)P(ξ), with P(ξ) = Q(ξ)R−1 and P(ξ)−1 = RQ(ξ)T . Since our matrix norm is invariant under left or right multiplication by unitary matrices, we have P(ξ) P(ξ)−1 = R R−1 = ρ(S0)ρ(S−1 0 ), a number independent of ξ. The diagonalization is thus well-conditioned. Let us instead assume that the system is constantly hyperbolic. The eigenspaces are continuous functions of ξ in Sd−1 . Choosing continuously a basis of each eigenspace, we find locally an eigenbasis of A(ξ), which depends continuously on ξ. This amounts to saying that, along every contractible subset of Sd−1 , the matrices A(ξ) may be diagonalized by a matrix P(ξ), which depends continuously on ξ. If the set is, moreover, compact (for instance, a half-sphere), we obtain that A(ξ) is diagonalizable with a uniformly bounded condition number. We now cover the sphere by two half-spheres and obtain a diagonalization of A(ξ) that is well-conditioned on Sd−1 (though possibly not continuously diagonalizable on the sphere). In the following example, though a symmetric as well as a strictly hyperbolic one, the diagonalization of the matrices A(ξ) cannot be done continuously for all
- 45. Strong well-posedness 15 ξ ∈ S1 : ∂tu + 1 0 0 −1 ∂1u + 0 1 1 0 ∂2u = 0. (1.2.17) Here, Sp(A(ξ)) = {−|ξ|, |ξ|}. Each eigenvector, when followed continuously as ξ varies along S1 , rotates with a speed half of the speed of ξ. For ξ = (cos θ, sin θ)T and θ ∈ [0, 2π), the eigenvectors are cos θ 2 sin θ 2 , − sin θ 2 cos θ 2 . The eigenbasis is reversed after one loop around the origin. This shows that the matrix P(ξ) cannot be chosen continuously. In other words, the eigenbundle is non-trivial. 1.2.5 The adjoint operator Let L be a hyperbolic operator as above. We define as usual the adjoint operator L∗ by the identity +∞ −∞ Rd (v · (Lu) − u · (L∗ v))dx dt = 0, (1.2.18) for every u, v ∈ D(Rd+1 )n . Notice that the scalar product under consideration is the one in the L2 -space in (x, t)-variables. With L = ∂t + α Aα ∂α, an integration by parts gives immediately the formula L∗ = −∂t − α (Aα )T ∂α. The matrix A(ξ)T , being similar to A(ξ), is diagonalizable. Since A(ξ)T is diag- onalized by P(ξ)−T (with the notations of Theorem 1.3), and since the matrix norm is invariant under transposition, we see that −L∗ is hyperbolic too. If L is strictly, or constantly, hyperbolic, so is L∗ . If L is Friedrichs symmetrizable, with S0 ∈ SDPn and Sα := S0 Aα ∈ Symn, then (S0 )−1 symmetrizes −L∗ since it is positive-definite and (S0 )−1 (Aα )T = (S0 )−1 Sα (S0 )−1 is symmetric. Therefore, L∗ is Friedrichs symmetrizable. The adjoint operator will be used in the existence theory of the Cauchy prob- lem (the duality method) or in the uniqueness theory (Holmgren’s argument), the latter being useful even in the quasilinear case. Both aspects are displayed in Chapter 2. 1.2.6 Classical solutions Let the system (1.1.3) be hyperbolic. According to Theorem 1.1, the Cauchy problem is well-posed in Hs . Using the system itself, we find that, whenever
- 46. 16 Linear Cauchy Problem with Constant Coefficients a ∈ Hs (Rd )n , u ∈ C (R; Hs (Rd )n ) ∩ C 1 (R; Hs−1 (Rd )n ). Let us assume that s 1 + d/2. By Sobolev embedding, Hs ⊂ C 1 and Hs−1 ⊂ C hold. We conclude that all distributional first-order derivatives are actually continuous functions of space and time. Therefore, u belongs to C 1 (Rd × R)n and is a classical solution of (1.1.3). More generally, a ∈ Hs (Rd )n with s k + d/2 implies that u is of class C k . Let us consider the non-homogeneous Cauchy problem, with a ∈ Hs (Rd )n and f ∈ L1 (R; Hs (Rd )n ) ∩ C (R; Hs−1 (Rd )n ) for s 1 + d/2. Then Duhamel’s formula immediately gives u ∈ C (R; Hs (Rd )n ), and the equation gives ∂tu ∈ C (R; Hs−1 (Rd )n ). We again conclude that u is C 1 and is a classical solution of (1.0.1). Since Hs (Rd )n is dense in normal functional spaces, as L2 or S , we see that classical solutions are dense in weaker solutions, like those in C (R; L2 (Rd )n ). We shall make use of this observation each time when some identity trivially holds for classical solutions. The scalar case When n = 1, the unknown u(x, t) is scalar-valued and all matrices are real numbers, denoted by a1 , . . . , an , b. The supremum in (1.2.11) is equal to one, so that the equation is hyperbolic. It turns out that the Cauchy problem may be solved explicitly, thanks to the method of characteristics. Let v denote the vector with components aα . Then a classical solution of (1.1.3) satisfies, for all y ∈ Rd , d dt u(y + t v, t) = bu(y + t v, t), which gives u(y + t v, t) = etb a(y), or u(x, t) = etb a(x − t v). (1.2.19) This formula gives the distributional solution for a ∈ S as well. The solution of the Cauchy problem for the non-homogeneous equation (1.0.1) is given by u(x, t) = etb a(x − t v) + t 0 e(t−s)b f(x − (t − s) v, s) ds. 1.2.7 Well-posedness in Lebesgue spaces The theory of the Cauchy problem is intimately related to Fourier analysis, which does not adapt correctly to Lebesgue spaces Lp other than L2 . The procedure followed above requires that F be an isomorphism from some space X to another one Z. It is known that F extends continuously from Lp (Rd ) to its dual Lp (Rd ) when 1 ≤ p ≤ 2, and only in these cases. Since F−1 is conjugated to F through
- 47. Friedrichs-symmetrizable systems 17 complex conjugation, it satisfies the same property. Therefore, F : Lp (Rd ) → Lp (Rd ) is not an isomorphism for p 2, since p 2. From this remark, we cannot find a well-posedness result in Lp for p = 2 by following the above strategy. It has been proved actually by Brenner [22,23] that, for hyperbolic systems, the Cauchy problem is ill-posed in Lp for every p = 2, except in the case where the matrices Aα commute to each other. In this particular case, system (1.2.12) actually decouples into a list of scalar equations, for which (1.2.19) shows the well- posedness in every Lp . To see the decoupling, we recall that commuting matrices that are diagonalizable may be diagonalized in a common basis B = {r1, . . . , rn} : Aα rj = λα j rj. Let us decompose the unknown on the eigenbasis: u(x, t) = n 1 wj(x, t)rj. Then each wj solves a scalar equation: ∂twj + α λα j ∂αwj = 0. From the well-posedness of (1.2.12) and Duhamel’s formula, we conclude that, for commuting matrices Aα , the hyperbolic Cauchy problem for (1.1.3) is also well-posed in every Lp . The matrices Aα do not need to commute with B. See Section 1.5.2 for an interpretation of the ill-posedness in Lp (p = 2), in terms of dispersion and so-called Stricharz estimates. 1.3 Friedrichs-symmetrizable systems A system in Friedrichs-symmetric form S0∂tu + α Sα ∂αu = 0 may always be transformed into a symmetric system with S0 = In, using the new unknown ũ := S 1/2 0 u. For the rest of this section, we shall only consider symmetric systems of the form (1.1.3). A symmetric system admits an additional conservation law4 in the form ∂t|u|2 + α ∂α(Aα u, u) = 0, (1.3.20) where (·, ·) denotes the canonical scalar product and |u|2 := (u, u). Equation (1.3.20) is satisfied at least for C 1 solutions of the system, when5 B = 0. It can be viewed as an energy identity. Since classical solutions are dense in C (R; L2 (Rd )n ), 4By conservation law, we mean an equality of the form Divx,t F = 0 that derives formally from the equation or system under consideration. 5Otherwise, the right-hand side of (1.3.20) should be 2(Bu, u). In the non-homogeneous case, we add also 2(f, u).
- 48. 18 Linear Cauchy Problem with Constant Coefficients and since u → ∂t|u|2 + α ∂α(Aα u, u) is a continuous map from this class into D (Rd+1 ), we conclude that (1.3.20) holds whenever a ∈ L2 (Rd )n . With suitable decay at infinity, (1.3.20) implies d dt Rd |u(x, t)|2 dx = 0, which readily gives u(t)L2 ≡ aL2 . (1.3.21) Again, this identity is true for all data a given in L2 (Rd )n , since r it is trivially true for a ∈ S , where we know that u(t) ∈ S , since such functions decay fast at infinity, r S is a dense subset of L2 . 1.3.1 Dependence and influence cone Actually, we can do a better job from (1.3.20). Let us first consider classical solutions, for some matrix B. The set V of pairs (λ, ν) such that the symmetric matrix λIn + A(ν) is non-negative is a closed convex cone. Given a point (X, T) ∈ Rd × R, we define a set K by K := {(x, t) ; λ(t − T) + (x − X) · ν ≤ 0, ∀(λ, ν) ∈ V}. As an intersection of half-spaces passing through (X, T), K is a convex cone with basis (X, T), and its boundary K has almost everywhere a tangent space, which is one of the hyperplanes λ(t − T) + (x − X) · ν = 0 for some (λ, ν) in the boundary of V. Given times t1 t2 T, we integrate the identity ∂t|u|2 + α ∂α(Aα u, u) = 2(Bu, u) on the truncated cone K(t1, t2) := {(x, t) ∈ K ; t1 t t2}. Using Green’s for- mula, we obtain ∂K(t1,t2) n0|u|2 + α nα(Aα u, u) dS = 2 K(t1,t2) (Bu, u) dx dt, (1.3.22) where dS stands for the area element, while n = (n1, . . . , nd, n0) is the outward unit normal. On the top (t = t2), n = (0, . . . , 0, 1), holds while on the bottom, n = (0, . . . , 0, −1). Denoting ω(t) := {x ; (x, t) ∈ K}, the corresponding contributions
- 49. Friedrichs-symmetrizable systems 19 are thus ω(t2) |u(x, t2)|2 dx − ω(t1) |u(x, t1)|2 dx. On the lateral boundary, one has n = 1 λ2 + |ν|2 (ν, λ) for some (λ, ν) in V, which depends on (x, t). The parenthesis in (1.3.22) becomes 1 λ2 + |ν|2 ((λIn + A(ν))u, u). Thus the corresponding integral is non-negative. Denoting by y(t) the integral of |u(t)|2 over ω(t), it follows that y(t2) − y(t1) ≤ 2 K(t1,t2) (Bu, u) dx dt ≤ 2B t2 t1 y(t) dt. Then, from the Gronwall inequality, we obtain that y(t2) ≤ e2(t2−t1)B y(t1). In particular, for 0 t T, we obtain ω(t) |u(x, t)|2 dx ≤ e2tB ω(0) |a(x)|2 dx. (1.3.23) Because of the density of classical solutions in the set of L2 -solutions, and since its terms are L2 -continuous, we find that (1.3.23) is valid for every L2 - solutions. Inequality (1.3.23) contains the following fact: If a vanishes identically on ω(0), then so does u(t) on ω(t). Equivalently, the value of u at the point (X, T) (assuming that the solution is continuous) depends only on the restriction of the initial data a to the set ω(0). Definition 1.3 The set ω(0) = {x ∈ Rd ; (x − X) · ν ≤ λT, ∀(λ, ν) ∈ V} is the domain of dependence of the point (X, T). Let us illustrate this notion with the system (1.2.17), to which we add a parameter c having the dimension of a velocity: ∂tu + c 1 0 0 −1 ∂1u + c 0 1 1 0 ∂2u = 0.
- 50. 20 Linear Cauchy Problem with Constant Coefficients Since λI2 + A(ν) = λ + cν1 cν2 cν2 λ − cν1 , the cone V is given by the inequality c|ν| ≤ λ. Thus the domain of dependence of (X, T) is the ball centred at X of radius cT. We now fix a point x at initial time and look at those points (X, T) for which x belongs to their domains of dependence. Let us define a convex cone C+ by C+ := {y ∈ Rd ; λ + y · ν ≥ 0, ∀(λ, ν) ∈ V}. Defining y = (X − x)/T, we have 1 + y · ν ≥ 0, that is y ∈ C+ . There- fore X = x + Ty ∈ x + TC+ . We deduce that u vanishes identically outside of Supp a + TC+ , where a = u(·, 0). We have thus proved a propagation property: Proposition 1.4 Let the system (1.1.3) be symmetric. Given a ∈ L2 (Rd )n , let u be the solution of the Cauchy problem. Then, for t1 t2, Supp u(t2) ⊂ Supp u(t1) + (t2 − t1)C+ . (1.3.24) Reversing the time arrow, we likewise have Supp u(t1) ⊂ Supp u(t2) + (t2 − t1)C− , (1.3.25) where C− := {y ∈ Rd ; λ + y · ν ≥ 0, ∀ν ∈ −V}. This result naturally yields the notion: Definition 1.4 Given a domain ω at initial time. The influence domain of ω at time t 0 is the set ω + tC+ . Remark From Duhamel’s formula, we extend the propagation property to the non-homogeneous problem. For instance, the solution for data a ∈ L2 and f ∈ L1 (0, T; L2 ) satisfies Supp u(t) ⊂ (Supp a + tC+ ) ∪ 0st (Supp f(s) + (t − s)C+ ). (1.3.26) 1.3.2 Non-decaying data Though the previous calculation applies only to solutions in C (R; L2 ), where we already know the uniqueness of a solution, it can be used to construct solutions for much more general data than square-integrable ones. First, the inequality (1.3.23) implies a propagation with finite speed: if a ∈ L2 (Rd )n and t 0, the support of u(t) is contained in the sum Supp a + tC+ . We now use the following facts:
- 51. Friedrichs-symmetrizable systems 21 r L2 is dense in S , r for a in S , there exists a unique solution in C (R; S ) (see Proposition 1.3), r the distributions that vanish on a given open subset of Rd form a closed subspace in S . We conclude that (1.3.24) and (1.3.25) hold for a symmetric system when a is a tempered distribution. We use this property to define a solution when the initial data is a (not necessarily tempered) distribution. Let a belong to D (Rd )n . Given a point y ∈ Rd and a positive number R, denote by C(y; R) the set y + RC− . Choose a cut- off φ in D(Rd ), such that φ ≡ 1 on C(y, R). The product φa, being a compactly supported distribution, is a tempered one. Therefore, there exists a unique uφ , solution of (1.1.3) in C (R; S ), with initial data φa. For two choices φ, ψ of cut- off functions, (φ − ψ)a vanishes on C(y; R), so that uφ (t) and uψ (t) coincide on C(y; R − t) for 0 t R. This allows us to define a restriction of uφ on the cone K(y; R) := 0tR {t} × C(y; R − t). As shown above, this restriction, denoted by uy,R does not depend on the choice of the cut-off. It actually depends only on the restriction of a on C(y; R). Now, if a point (z, t) lies in the intersection of two such cones K(y1; R1) and K(y2; R2), it belongs to a third one K(y3; R3), which is included in their intersection. The restrictions of uy1,R1 and uy2,R2 to K(y3; R3) are equal, since they depend only on the restriction of a on C(y3; R3). We obtain in this way a unique distribution u ∈ C (R+ ; D ), whose restriction on every cone K(y; R) coincides with uy,R. It solves (1.1.3) in the distributional sense, and takes the value a as t = 0. Reversing the time arrow, we solve the backward Cauchy problem as well. This construction is relevant, for instance, when a is L2 loc rather than square- integrable. It can be used also when a is in Lp loc for p = 2, even though the cor- responding solutions are not C (R; Lp ) in general, because of Brenner’s theorem. 1.3.3 Uniqueness for non-decaying data The construction made above, though defining a unique distribution, does not tell us about the uniqueness in C (0, T; X) for a ∈ X, when X = D (Rd )n or X = L2 loc(Rd )n for instance. This is because we got uniqueness results through the use of Fourier transform, a tool that does not apply here. We describe below two relevant techniques. Let us begin with X = L2 loc. We assume that u ∈ C (0, T; X) solves (1.1.3) with a = 0. We use the localization method. Let K(y; R) be a cone as in the previous section, and φ ∈ D(Rd ) be such that φ(x) = 1, ∀x ∈ 0tR C(y; R − t),
- 52. 22 Linear Cauchy Problem with Constant Coefficients the latter set being the x-projection of K(y; R). Multiplying (1.1.3) by φ, and denoting v := φu, we obtain ∂tv + α Aα ∂αv = Bv + f, where v ∈ C (0, T; L2 (Rd )n ) and f := (∂tφ + A(∇xφ))u ∈ C (0, T; L2 (Rd )n ). At this point, we are allowed to write the energy estimate ∂t|v|2 + α ∂α(Aα v, v) = 2Re (Bv + f, v), which gives for every 0 ≤ t1 t2 R, after integration, ω(t2) |v(t2)|2 dx ≤ ω(t1) |v(t1)|2 dx + t2 t1 dt ω(t) 2Re ((Bv, v) + (f, v))dx, (1.3.27) where ω(t) := C(y; R − t). However, the equalities v = u, f = 0 hold in K(y; R). Therefore (1.3.27) reduces to ω(t2) |u(t2)|2 dx ≤ ω(t1) |u(t1)|2 dx + 2 t2 t1 dt ω(t) Re (Bu, u) dx. This, with the Gronwall inequality, gives ω(t) |u(x, t)|2 dx ≤ e2tB ω(0) |u(x, 0)|2 dx = 0. Since y and R are arbitrary, we obtain u ≡ 0 almost everywhere, which is the uniqueness property. We now turn to the case X = D (Rd )n , where the former argument does not work. Our main ingredient is the Holmgren principle, a tool that we shall develop more systematically in subsequent chapters. We assume that u ∈ C (0, T; X) solves (1.1.3) in the distributional sense. This means that, for every test function φ ∈ D(Rd × (0, T))n , it holds that u, L∗ φ = 0, L∗ := −∂t − α (Aα )T ∂α − BT . This may be rewritten as T 0 u(t), L∗ φ(t) dt = 0. (1.3.28) Let ψ be a slightly more general test function: ψ ∈ D(Rd × (−∞, T))n . Choosing θ ∈ C ∞ (R) with θ(τ) = 0 for τ 1 and θ(τ) = 1 for τ 2, we define φ (x, t) = θ(t/)ψ(x, t).
- 53. Directions of hyperbolicity 23 We may apply (1.3.28) to φ , which gives T 0 θ(t/)u(t), L∗ ψ(t) dt = 1 T 0 θ (t/)u(t), ψ(t) dt. Using the continuity in time, we may pass to the limit as → 0+ , and obtain T 0 u(t), L∗ ψ(t) dt = u(0), ψ(0). Therefore, assuming u(0) = 0, we see that (1.3.28) is valid for ψ as well, that is to test functions in D(Rd × (−∞, T))n . We now choose an arbitrary test function f ∈ D(Rd × (0, T))n . Obviously, L∗ is a hyperbolic operator and we can solve the backward Cauchy problem L∗ χ = f, χ(T) = 0. Extending f by zero for t ≤ 0, we obtain a unique solution χ ∈ C ∞ (−∞, T; S ). Applying (1.3.26) to this backward problem, we see that χ(t) has compact support for each time, with Supp χ(t) included in a ball of the form Bρ(T −t), for a suitable constant ρ. Also, χ vanishes identically for t close enough to T (because f does). Truncating, we apply (1.3.28) to ψ(x, t) = θ(t + 1)χ(x, t). This gives u, f = 0 for all test functions, that is u = 0. Therefore the Cauchy problem for a Friedrichs-symmetric operator has the uniqueness property in the class C (0, T; D ). 1.4 Directions of hyperbolicity The situation for general (weakly) hyperbolic operators is not as neat as that for Friedrichs-symmetrizable ones. Non-symmetrizable operators do exist, as soon as d = 2 and n = 3, as shown by Lax [110]. The class of constantly hyperbolic operators provides a valuable and flexible alternative to Friedrichs-symmetrizable ones. Their analysis will lead us to several new and useful notions. In this section, we shall not address the problem of propagation of the support (with finite velocity), which we solved in the symmetric case. This propagation holds true for constantly hyperbolic systems, but a rigorous proof needs a theory of the Cauchy problem for systems with variable coefficients. Such a theory will be done in Chapter 2, where we shall prove an accurate result. 1.4.1 Properties of the eigenvalues The results in this section are essentially those of Lax [110], and the arguments follow Weinberger [217], though we give a more detailed proof of the claim below. We begin by considering a subspace E in Mn(R), with the property that every matrix in E has a real spectrum. Without loss of generality, we may assume that In belongs to E. If M ∈ E, we denote by λ1(M) ≤ · · · ≤ λn(M) the spectrum of M, counting with multiplicities. The functions λj are positively homogeneous of order one. They are continuous, but could be non-differentiable at crossing
- 54. 24 Linear Cauchy Problem with Constant Coefficients points. In the constantly hyperbolic case, however, they are analytic away from the origin. Lemma 1.1 Let A and B be matrices in E, with λ1(B) 0. Then the eigen- values of B−1 (λIn − A) are real. Proof From the assumption, we know that B is non-singular. Define a poly- nomial P(X, Y ) := det(XIn − A − Y B), which has degree n with respect to X as well as to Y . Define continuous functions φj(µ) = λj(A + µB). From homogeneity and continuity, we have φj(µ) ∼ µλj(B), as µ → +∞, µλn+1−j(B) as µ → −∞. Hence φj(µ) tends to ±∞ with µ. By the Intermediate Value Theorem, it must take any prescribed real value λ at least once. Thus, let λ∗ be given and µj ∈ R be a root of φj(µj) = λ∗ for each j. Given one of these roots, µ∗ , let J be the number of indices such that µj = µ∗ . Then λ∗ is a root of P(·, µ∗ ), of order J at least. Claim 1.1 The multiplicity of µ∗ as a root of P(λ∗ , ·) is larger than or equal to J. This claim readily implies the lemma. Its proof is fairly simple when the φjs are differentiable, for instance in the constantly hyperbolic case. But in the general case, one must use once more the assumption. To simplify the notations, we assume without loss of generality that λ∗ = µ∗ = 0, by translating A to A + µ∗ B − λ∗ In. Let N (N ≥ J) be the multiplicity of the null root of P(·, 0). The Newton’s polygon of the polynomial P admits the vertices (N, 0) and (0, K). Let δ be the edge of the Newton’s polygon with vertex (N, 0). We denote its other vertex by (j, k). Retaining only those monomials of P whose degrees (a, b) belong to δ, we obtain a polynomial Xj Q with the following homogeneity: Q(ak X, aN−j Y ) = ak(N−j) Q(X, Y ). It is a basic fact in algebraic geometry (see [35], Section 2.8) that, in the vicinity of the origin, the algebraic curve P(x, y) = 0 is described by simpler curves corre- sponding to the edges of the Newton polygon, up to analytic diffeomorphisms. In the present case, these diffeomorphisms have real coefficients (i.e. they preserve real vectors) since P has real coefficients. The ‘simple’ curve γ associated to δ is just that with equation Q(x, y) = 0. Hence, points (x, y) in γ with a real co-ordinate y must be real (because this is so in the curve P = 0.) Let ω be a root of unity, of order 2(N − j), that is ωN−j = −1. Because of the homogeneity, the map (x, y) → (ωk x, −y) preserves γ. If y is real, the map
- 55. Exploring the Variety of Random Documents with Different Content
- 56. and the royal veto, in order to be of effect, must be exercised within three years. To this Grand Council each colony was to send a number of representatives, proportioned to its contributions to the continental military service; yet no colony was to send less than two or more than seven representatives. With the exception of such matters of general concern as were to be managed by the Grand Council, each colony was to retain its powers of legislation intact. On an emergency, any colony might singly defend itself against foreign attack, and the federal government was prohibited from impressing soldiers or seamen without the consent of the local legislature. The supreme executive power was to be vested in a president or governor-general, appointed and paid by the Crown. He was to nominate all military officers, subject to the approval of the Grand Council, and was to have a veto on all the acts of the Grand Council. No money could be issued save by joint order of the governor- general and the council. This plan, said Franklin, “is not altogether to my mind, but it is as I could get it.” It should be observed, to the credit of its author, that this scheme, long afterward known as the “Albany Plan,” contemplated the formation of a self-sustaining federal government, and not of a mere league. As Frothingham well says, “It designed to confer on the representatives of the people the power of making laws acting directly on individuals, and appointing officers to execute them, and yet not to interfere with the execution of the laws operating on the same individuals by the local officers.” It would have erected “a public authority as obligatory in its sphere as the local governments were in their spheres.” In this respect it was much more complete than the scheme of confederation agreed on in Congress in 1777, and it afforded a valuable precedent for the more elaborate and perfect Federal Constitution of 1787. It was in its main features a noble scheme, and the great statesman who devised it was already looking forward to the immense growth of the American Union, though he had not yet foreseen the separation of the colonies from the mother-country. In less than a century, he said, the great country
- 57. Rejection of the plan behind the Alleghanies must become “a populous and powerful dominion;” and he recommended that two new colonies should at once be founded in the West,—the one on Lake Erie, the other in the valley of the Ohio,—with free chartered governments like those of Rhode Island and Connecticut. But public opinion was not yet ripe for the adoption of Franklin’s bold and comprehensive ideas. Of the royal governors who were anxious to see the colonies united on any terms, none opposed the plan except Delancey of New York, who wished to reserve to the governors a veto upon all elections of representatives to the Grand Council. To this it was rightly objected that such a veto power would virtually destroy the freedom of elections, and make the Grand Council an assembly of creatures of the governors. On the popular side the objections were many. The New England delegates, on the whole, were the least
- 58. Shirley recommend s a stamp act disinclined to union; yet Connecticut urged that the veto power of the governor-general might prove ruinous to the whole scheme; that the concentration of all the military forces in his hands would be fraught with dangers to liberty; and that even the power of taxation, lodged in the hands of an assembly so remote from local interests, was hardly compatible with the preservation of the ancient rights of Englishmen. After long debate, the assembly at Albany decided to adopt Franklin’s plan, and copies of it were sent to all the colonies for their consideration. But nowhere did it meet with approval. The mere fact that the royal governors were all in favour of it—though their advocacy was at present, no doubt, determined mainly by sound military reasons—was quite enough to create an insuperable prejudice against it on the part of the people. The Massachusetts legislature seems to have been the only one which gave it a respectful consideration, albeit a large town meeting in Boston denounced it as subversive of liberty. Pennsylvania rejected it without a word of discussion. None of the assemblies favoured it. On the other hand, when sent over to England to be inspected by the Lords of Trade, it only irritated and disgusted them. As they truly said, it was a scheme of union “complete in itself;” and ever since the days of the New England confederacy the Crown had looked with extreme jealousy upon all attempts at concerted action among the colonies which did not originate with itself. Besides this, the Lords of Trade were now considering a plan of their own for remodelling the governments of the colonies, establishing a standing army, enforcing the navigation acts, and levying taxes by authority of Parliament. Accordingly little heed was paid to Franklin’s ideas. Though the royal governors had approved the Albany plan, in default of any scheme of union more to their minds, they had no real sympathy with it.In 1756, Shirley wrote to the Lords of Trade, urging upon them the paramount necessity for a union of the American colonies, in order to withstand the French; while at the same time he disparaged Franklin’s scheme, as containing principles of government unfit even for a single colony like Rhode Island, and much more unfit for a great American confederacy. The union, he urged, should be effected
- 59. Writs of assistance by act of Parliament, and by the same authority a general fund should be raised to meet the expenses of the war,—an end which Shirley thought might be most speedily and quietly attained by means of a “stamp duty.” As Shirley had been for fifteen years governor of Massachusetts, and was now commander-in-chief of all the troops in America, his opinion had great weight with the Lords of Trade; and the same views being reiterated by Dinwiddie of Virginia, Sharpe of Maryland, Hardy of New York, and other governors, the notion that Parliament must tax the Americans became deeply rooted in the British official mind. Nothing was done, however, until the work of the French war had been accomplished. In 1761, it was decided to enforce the Navigation Act, and one of the revenue officers at Boston applied to the superior court for a “writ of assistance,” or general search-warrant, to enable him to enter private houses and search for smuggled goods, but without specifying either houses or goods. Such general warrants had been allowed by a statute of the bad reign of Charles II., and a statute of William III., in general terms, had granted to revenue officers in America like powers to those they possessed in England. But James Otis showed that the issue of such writs was contrary to the whole spirit of the British constitution. To issue such universal warrants allowing the menials of the custom house, on mere suspicion, and perhaps from motives of personal enmity, to invade the home of any citizen, without being held responsible for any rudeness they might commit there,—such, he said, was “a kind of power, the exercise of which cost one king of England his head and another his throne;” and he plainly declared that even an act of Parliament which should sanction so gross an infringement of the immemorial rights of Englishmen would be treated as null and void. Chief Justice Hutchinson granted the writs of assistance, and as an interpreter of the law he was doubtless right in so doing; but Otis’s argument suggested the question whether Americans were bound to obey laws which they had no share in making, and his passionate eloquence made so great an impression upon the people that this scene in the court room has been since
- 60. The chief justice of New York remembered—and not unjustly—as the opening scene of the American Revolution. In the same year the arbitrary temper of the government was exhibited in New York. Down to this time the chief justice of the colony had held office only during good behaviour, and had been liable to dismissal at the hands of the colonial assembly. The chief justice was now made removable only by the Crown, a measure which struck directly at the independent administration of justice in the colony. The assembly tried to protect itself by refusing to assign a fixed salary to the chief justice, whereupon the king ordered that the salary should be paid out of the quit-rents for the public lands. At the same time instructions were sent to all the royal governors to grant no judicial commissions for any other period than “during the king’s
- 61. Otis’s “Vindication ” pleasure;” and to show that this was meant in earnest, the governor of New Jersey was next year peremptorily dismissed for commissioning a judge “during good behaviour.” In 1762, a question distinctly involving the right of the people to control the expenditure of their own money came up in Massachusetts. Governor Bernard, without authority from the assembly, had sent a couple of ships to the northward, to protect the fisheries against French privateers, and an expense of some £400 had been thus incurred. The assembly was now ordered to pay this sum, but it refused to do so. It would be of little consequence to the people,” said Otis, in the debate on the question, “whether they were subject to George or Louis, the king of Great Britain or the French king, if both were arbitrary, as both would be, if both could levy taxes without Parliament.” A cry of “Treason!” from one of the less clear- headed members greeted this bold statement; and Otis, being afterward taken to task for his language, published a “Vindication,” in which he maintained that the rights of a colonial assembly, as regarded the expenditure of public money, were as sacred as the rights of the House of Commons.
- 62. Expenses of the French war In April, 1763, just three years after the accession of George III., George Grenville became Prime Minister of England, while at the same time Charles Townshend was First Lord of Trade. Townshend had paid considerable attention to American affairs, and was supposed to know more about them than any other man in England. But his studies had led him to the conclusion that the colonies ought to be deprived of their self-government, and that a standing army ought to be maintained in America by means of taxes arbitrarily assessed upon the people by Parliament. Grenville was far from approving of such extreme measures as these, but he thought that a tax ought to be imposed upon the colonies, in order to help defray the expenses of the French war. Yet in point of fact, as Franklin truly said, the colonies had “raised, paid, and clothed nearly twenty-five thousand men during the last war,—a number equal to those sent from Great Britain, and far beyond their proportion. They went deeply into debt
- 63. Grenville’s Resolves Reply of the in doing this; and all their estates and taxes are mortgaged for many years to come for discharging that debt.” That the colonies had contributed more than an equitable share toward the expenses of the war, that their contributions had even been in excess of their ability, had been freely acknowledged by Parliament, which, on several occasions between 1756 and 1763, had voted large sums to be paid over to the colonies, in partial compensation for their excessive outlay. Parliament was therefore clearly estopped from making the defrayal of the war debt the occasion for imposing upon the colonies a tax of a new and strange character, and under circumstances which made the payment of such a tax seem equivalent to a surrender of their rights as free English communities. In March, 1764, Grenville introduced in the House of Commons a series of Declaratory Resolves, announcing the intention of the government to raise a revenue in America by requiring various commercial and legal documents, newspapers, etc., to bear stamps, varying in price from threepence to ten pounds. A year was to elapse, however, before these resolutions should take effect in a formal enactment. It marks the inferiority of the mother-country to the colonies in political development, at that time, that the only solicitude as yet entertained by the British official mind, with regard to this measure, seems to have been concerned with the question how far the Americans would be willing to part with their money. With the Americans it was as far as possible from being a question of pounds, shillings, and pence; but this was by no means correctly understood in England. The good Shirley, although he had lived so long in Massachusetts, had thought that a revenue might be most easily and quietly raised by means of a stamp duty. Of all kinds of direct tax, none, perhaps, is less annoying. But the position taken by the Americans had little to do with mere convenience; it rested from the outset upon the deepest foundations of political justice, and from this foothold neither threatening nor coaxing could stir it. The first deliberate action with reference to the proposed Stamp Act was taken in the Boston town
- 64. colonies The Stamp Act meeting in May, 1764. In this memorable town meeting Samuel Adams drew up a series of resolutions, which contained the first formal and public denial of the right of Parliament to tax the colonies without their consent; and while these resolutions were adopted by the Massachusetts assembly, a circular letter was at the same time sent to all the other colonies, setting forth the need for concerted and harmonious action in respect of so grave a matter. In response, the assemblies of Connecticut, New York, Pennsylvania, Virginia, and South Carolina joined with Massachusetts in remonstrating against the proposed Stamp Act. All these memorials were remarkable for clearness of argument and simple dignity of language. They all took their stand on the principle that, as free-born Englishmen, they could not rightfully be taxed by the House of Commons unless they were represented in that body. But the proviso was added, that if a letter from the secretary of state, coming in the king’s name, should be presented to the colonial assemblies, asking them to contribute something from their general resources to the needs of the British Empire, they would cheerfully, as heretofore, grant liberal sums of money, in token of their loyalty and of their interest in all that concerned the welfare of the mighty empire to which they belonged. These able and temperate memorials were sent to England; and in order to reinforce them by personal tact and address, Franklin went over to London as agent for the colony of Pennsylvania. The alternative proposed by the colonies was virtually the same as the system of requisitions already in use, and the inefficiency of which, in securing a revenue, had been abundantly proved by the French war. Parliament therefore rejected it, and early in 1765 the Stamp Act was passed. It is worthy of remark that the idea that the Americans would resist its execution did not at once occur to Franklin. Acquiescence seemed to him, for the present, the only safe policy. In writing to his friend
- 65. Charles Thomson, he said that he could no more have hindered the passing of the Stamp Act than he could have hindered the sun’s setting. “That,” he says, “we could not do. But since it is down, my friend, and it may be long before it rises again, let us make as good a night of it as we can. We may still light candles. Frugality and industry will go a great way towards indemnifying us.” But Thomson, in his answer, with truer foresight, observed, “I much fear, instead of the candles you mentioned being lighted, you will hear of the works of darkness!” The news of the passage of the Stamp Act was greeted in America with a burst of indignation. In New York, the act was reprinted with a death’s-head upon it in place of the royal arms, and it was hawked about the streets under the title of “The Folly of England and the Ruin of America.” In Boston, the church-bells were tolled, and the flags on the shipping put at half- mast.
- 66. The Parson’s Cause SPEAKER’S CHAIR, HOUSE OF BURGESSES But formal defiance came first from Virginia. A year and a half before, a famous lawsuit, known as the “Parsons’ Cause,” had brought into public notice a young man who was destined to take high rank among modern orators. The lawsuit which made Patrick Henry’s reputation was one of the straws which showed how the stream of tendency in America was then strongly setting toward independence. Tobacco had not yet ceased to be a legal currency in Virginia, and by virtue of an old statute each clergyman of the Established Church was entitled to sixteen thousand pounds of tobacco as his yearly salary. In 1755 and 1758, under the severe pressure of the French war, the assembly had passed relief acts, allowing all public dues, including the salaries of the clergy, to be paid either in kind or in money, at a fixed rate of twopence for a pound of tobacco. The policy of these acts was thoroughly unsound, as they involved a partial repudiation of debts; but the extreme distress of the community was pleaded in excuse,
- 67. and every one, clergy as well as laymen, at first acquiesced in them. But in 1759 tobacco was worth sixpence per pound, and the clergy became dissatisfied. Their complaints reached the ears of Sherlock, the Bishop of London, and the act of 1758 was summarily vetoed by the king in council. The clergy brought suits to recover the unpaid portions of their salaries; in the test case of Rev. James Maury, the court decided the point of the law in their favour, on the ground of the royal veto, and nothing remained but to settle before a jury the amount of the damages. On this occasion, Henry appeared for the first time in court, and after a few timid and awkward sentences burst forth with an eloquent speech, in which he asserted the indefeasible right of Virginia to make laws for herself, and declared that in annulling a salutary ordinance at the request of a favoured class in the community “a king, from being the father of his people, degenerates into a tyrant, and forfeits all right to obedience.” Cries of “Treason!” were heard in the court room, but the jury immediately returned a verdict of one penny in damages, and Henry became the popular idol of Virginia. The clergy tried in vain to have him indicted for treason, alleging that his crime was hardly less heinous than that which had brought old Lord Lovat to the block. But the people of Louisa county replied, in 1765, by choosing him to represent them in the colonial assembly.
- 68. Patrick Henry’s resolutions PATRICK HENRY MAKING HIS TARQUIN AND CÆSAR SPEECH Hardly had Henry taken his seat in the assembly when the news of the Stamp Act arrived. In a committee of the whole house, he drew up a series of resolutions, declaring that the colonists were entitled to all the liberties and privileges of natural-born subjects, and that “the taxation of the people by themselves, or by persons chosen by themselves to represent them, ... is the distinguishing characteristic of British freedom, without which the ancient constitution cannot exist.” It was further declared that any attempt to vest the power of taxation in any other body than the colonial assembly was a menace to British no less than to American freedom; that the people of Virginia were not bound to obey any law enacted in disregard of
- 69. The Stamp Act Congress these fundamental principles; and that any one who should maintain the contrary should be regarded as a public enemy. It was in the lively debate which ensued upon these resolutions, that Henry uttered those memorable words commending the example of Tarquin and Cæsar and Charles I. to the attention of George III. Before the vote had been taken upon all the resolutions, Governor Fauquier dissolved the assembly; but the resolutions were printed in the newspapers, and hailed with approval all over the country. See Transcription Meanwhile, the Massachusetts legislature, at the suggestion of Otis, had issued a circular letter to all the colonies, calling for a general congress, in order to
- 70. concert measures of resistance to the Stamp Act. The first cordial response came from South Carolina, at the instance of Christopher Gadsden, a wealthy merchant of Charleston and a scholar learned in Oriental languages, a man of rare sagacity and most liberal spirit. On the 7th of October, the proposed congress assembled at New York, comprising delegates from Massachusetts, South Carolina, Pennsylvania, Rhode Island, Connecticut, Delaware, Maryland, New Jersey, and New York, in all nine colonies, which are here mentioned in the order of the dates at which they chose their delegates. In Virginia, the governor succeeded in preventing the meeting of the legislature, so that this great colony did not send delegates; and, for various reasons, New Hampshire, North Carolina, and Georgia were likewise unrepresented at the congress. But the sentiment of all the thirteen colonies was none the less unanimous, and those which did not attend lost no time in declaring their full concurrence with what was done at New York. At this memorable meeting, held under the very guns of the British fleet and hard by the headquarters of General Gage, the commander-in-chief of the regular forces in America, a series of resolutions were adopted, echoing the spirit of Patrick Henry’s resolves, though couched in language somewhat more conciliatory, and memorials were addressed to the king and to both Houses of Parliament. Of all the delegates present, Gadsden took the broadest ground, in behalf both of liberty and of united action among the colonies. He objected to sending petitions to Parliament, lest thereby its paramount authority should implicitly and unwittingly be acknowledged. “A confirmation of our essential and common rights as Englishmen,” said he, “may be pleaded from charters safely enough; but any further dependence on them may be fatal. We should stand upon the broad common ground of those natural rights that we all feel and know as men and as descendants of Englishmen. I wish the charters may not ensnare us at last, by drawing different colonies to act differently in this great cause. Whenever that is the case, all will be over with the whole. There ought to be no New England man, no New Yorker, known on the continent; but all of us Americans.” So thought and said this broad-minded South Carolinian.
- 71. Declaration of the Massachuse tts assembly Resistance to the Stamp Act in Boston While these things were going on at New York, the Massachusetts assembly, under the lead of Samuel Adams, who had just taken his seat in it, drew up a very able state paper, in which it was declared, among other things, that “the Stamp Act wholly cancels the very conditions upon which our ancestors, with much toil and blood and at their sole expense, settled this country and enlarged his majesty’s dominions. It tends to destroy that mutual confidence and affection, as well as that equality, which ought ever to subsist among all his majesty’s subjects in this wide and extended empire; and what is the worst of all evils, if his majesty’s American subjects are not to be governed according to the known and stated rules of the constitution, their minds may in time become disaffected.” This moderate and dignified statement was applauded by many in England and by others derided as the “raving of a parcel of wild enthusiasts,” but from the position here taken Massachusetts never afterward receded. But it was not only in these formal and decorous proceedings that the spirit of resistance was exhibited. The first announcement of the Stamp Act had called into existence a group of secret societies of workingmen known as “Sons of Liberty,” in allusion to a famous phrase in one of Colonel Barré’s speeches. These societies were solemnly pledged to resist the execution of the obnoxious law. On the 14th of August, the quiet town of Boston witnessed some extraordinary proceedings. At daybreak, the effigy of the stamp officer, Oliver, was seen hanging from a great elm-tree, while near it was suspended a boot, to represent the late prime minister, Lord Bute; and from the top of the boot-leg there issued a grotesque head, garnished with horns, to represent the devil. At nightfall the Sons of Liberty cut down these figures, and bore them on a bier through the streets until they reached King Street, where they demolished the frame of a house which was supposed to be erecting for a stamp office. Thence, carrying the beams of this frame to Fort Hill, where Oliver lived, they made a bonfire of them in front of his
- 72. and in New York house, and in the bonfire they burned up the effigies. Twelve days after, a mob sacked the splendid house of Chief Justice Hutchinson, threw his plate into the street, and destroyed the valuable library which he had been thirty years in collecting, and which contained many manuscripts, the loss of which was quite irreparable. As usual with mobs, the vengeance fell in the wrong place, for Hutchinson had done his best to prevent the passage of the Stamp Act. In most of the colonies, the stamp officers were compelled to resign their posts. Boxes of stamps arriving by ship were burned or thrown into the sea. Leading merchants agreed to import no more goods from England, and wealthy citizens set the example of dressing in homespun garments. Lawyers agreed to overlook the absence of the stamp on legal documents, while editors derisively issued their newspapers with a death’s-head in the place where the stamp was required to be put.In New York, the presence of the troops for a moment encouraged the lieutenant- governor, Colden, to take a bold stand in behalf of the law. He talked of firing upon the people, but was warned that if he did so he would be speedily hanged on a lamp-post, like Captain Porteous of Edinburgh. A torchlight procession, carrying images of Colden and of the devil, broke into the governor’s coach-house, and, seizing his best chariot, paraded it about town with the images upon it, and finally burned up chariot and images on the Bowling Green, in full sight of Colden and the garrison, who looked on from the Battery, speechless with rage, but afraid to interfere. Gage did not dare to have the troops used, for fear of bringing on a civil war; and the next day the discomfited Colden was obliged to surrender all the stamps to the common council of New York, by whom they were at once locked up in the City Hall. Nothing more was needed to prove the impossibility of carrying the Stamp Act into effect. An act which could be thus rudely defied under the very eyes of the commander-in-chief plainly could never be enforced without a war. But nobody wanted a war, and the matter
- 73. Debate in the House of Commons began to be reconsidered in England. In July, the Grenville ministry had gone out of office, and the Marquis of Rockingham was now prime minister, while Conway, who had been one of the most energetic opponents of the Stamp Act, was secretary of state for the colonies. The new ministry would perhaps have been glad to let the question of taxing America remain in abeyance, but that was no longer possible. The debate on the proposed repeal of the Stamp Act was one of the keenest that has ever been heard in the House of Commons. Grenville and his friends, now in opposition, maintained in all sincerity that no demand could ever be more just, or more honourably intended, than that which had lately been made upon the Americans. Of the honest conviction of Grenville and his supporters that they were entirely in the right, and that the Americans were governed by purely sordid and vulgar motives in resisting the Stamp Act, there cannot be the slightest doubt. To refute this gross misconception of the American position, Pitt hastened from a sick-bed to the House of Commons, and delivered those speeches in which he avowed that he rejoiced in the resistance of the Americans, and declared that, had they submitted tamely to the measures of Grenville, they would have shown themselves only fit to be slaves. He pointed out distinctly that the Americans were upholding those eternal principles of political justice which should be to all Englishmen most dear, and that a victory over the colonies would be of ill-omen for English liberty, whether in the Old World or in the New. Beware, he said, how you persist in this ill-considered policy. “In such a cause your success would be hazardous. America, if she fell, would fall like the strong man with his arms around the pillars of the Constitution.” There could be no sounder political philosophy than was contained in these burning sentences of Pitt. From all the history of the European world since the later days of the Roman Republic, there is no more important lesson to be learned than this,—that it is impossible for a free people to govern a dependent people despotically without endangering its own freedom. Pitt therefore urged that the Stamp Act should instantly be repealed, and that the reason for the repeal should be explicitly stated to be because the act “was founded on an
- 74. Repeal of the Stamp Act erroneous principle.” At the same time he recommended the passage of a Declaratory Act, in which the sovereign authority of Parliament over the colonies should be strongly asserted with respect to everything except direct taxation. Similar views were set forth in the House of Lords, with great learning and ability, by Lord Camden; but he was vehemently opposed by Lord Mansfield, and when the question came to a decision, the only peers who supported Camden were Lords Shelburne, Cornwallis, Paulet, and Torrington. The result finally reached was the unconditional repeal of the Stamp Act, and the simultaneous passage of a Declaratory Act, in which the views of Pitt and Camden were ignored and Parliament asserted its right to make laws binding on the colonies “in all cases whatsoever.” By the people of London the repeal was received with enthusiastic delight, and Pitt and Conway, as they appeared on the street, were loudly cheered, while Grenville was greeted with a storm of hisses. In America the effect of the news was electric. There were bonfires in every town, while addresses of thanks to the king were voted in all the legislatures. Little heed was paid to the Declaratory Act, which was regarded merely as an artifice for saving the pride of the British government. There was a unanimous outburst of loyalty all over the country, and never did the people seem less in a mood for rebellion than at that moment. The quarrel had now been made up. On the question of principle, the British had the last word. The government had got out of its dilemma remarkably well, and the plain and obvious course for British statesmanship was not to allow another such direct issue to come up between the colonies and the mother-country. To force on another such issue while the memory of this one was fresh in everybody’s mind was sheer madness. To raise the question wantonly, as Charles Townshend did in the course of the very next year, was one of those blunders that are worse than crimes.
- 75. The Duke of Grafton’s ministry FUNERAL PROCESSION OF THE STAMP ACT In July, 1766,—less than six months after the repeal of the Stamp Act,—the Rockingham ministry fell, and the formation of a new ministry was entrusted to Pitt, the man who best appreciated the value of the American colonies. But the state of Pitt’s health was not such as to warrant his taking upon himself the arduous duties of prime minister. He took the great seal, and, accepting the earldom of Chatham, passed into the House of Lords. The Duke of Grafton became prime minister, under Pitt’s guidance; Conway and Lord Shelburne were secretaries of state, and Camden became Lord Chancellor,—all three of them warm friends of America, and adopting the extreme American view of the constitutional questions lately at issue; and along with these was Charles Townshend, the evil spirit of the administration, as chancellor of the exchequer. From such a ministry, it might at first sight seem strange that a fresh quarrel with America should have proceeded. But Chatham’s illness soon overpowered him,
- 76. so that he was kept at home suffering excruciating pain, and could neither guide nor even pay due attention to the proceedings of his colleagues. Of the rest of the ministry, only Conway and Townshend were in the House of Commons, where the real direction of affairs rested; and when Lord Chatham was out of the way, as the Duke of Grafton counted for nothing, the strongest man in the cabinet was unquestionably Townshend. Now when an act for raising an American revenue was proposed by Townshend, a prejudice against it was sure to be excited at once, simply because every American knew well what Townshend’s views were. It would have been difficult for such a man even to assume a conciliatory attitude without having his motives suspected; and if the question with Great Britain had been simply that of raising a revenue on statesmanlike principles, it would have been well to entrust the business to some one like Lord Shelburne, in whom the Americans had confidence. In 1767, Townshend ventured to do what in any English ministry of the present day would be impossible. In flat opposition to the policy of Chatham and the rest of his colleagues, trusting in the favour of the king and in his own ability to coax or browbeat the House of Commons, he brought in a series of new measures for taxing America. “I expect to be dismissed for my pains,” he said in the House, with flippant defiance; and indeed he came very near it. As soon as he heard what was going on, Chatham mustered up strength enough to go to London and insist upon Townshend’s dismissal. But Lord North was the only person that
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