![Degenerate first order identification
problems in Banach spaces 1
Mohammed Al-Horani and Angelo Favini
Abstract We study a first order identification problem in a Banach space. We
discuss both the nondegenerate and (mainly) the degenerate case. As a first step,
suitable hypotheses on the involved closed linear operators are made in order to
obtain unique solvability after reduction to a nondegenerate case; the general case
is then handled with the help of new results on convolutions. Various applications
to partial differential equations motivate this abstract approach.
1 Introduction
In this article we are concerned with an identification problem for first order
linear systems extending the theory and methods discussed in [7] and [1]. See
also [2] and [9]. Related nonsingular results were obtained in [11] under differ-
ent additional conditions even in the regular case. There is a wide literature
on inverse problems motivated by applied sciences. We refer to [11] for an
extended list of references. Inverse problems for degenerate differential and
integrodifferential equations are a new branch of research. Very recent results
have been obtained in [7], [5] and [6] relative to identification problems for de-
generate integrodifferential equations. Here we treat similar equations without
the integral term and this allows us to lower the required regularity in time of
the data by one. The singular case for infinitely differentiable semigroups and
second order equations in time will be treated in some forthcoming papers.
The contents of the paper are as follows. In Section 2 we present the non-
singular case, precisely, we consider the problem
u0
(t) + Au(t) = f(t)z , 0 ≤ t ≤ τ ,
u(0) = u0 ,
Φ[u(t)] = g(t) , 0 ≤ t ≤ τ ,
1Work partially supported by the Italian Ministero dell’Istruzione, dell’Università e della
Ricerca (M.I.U.R.), PRIN no. 2004011204, Project Analisi Matematica nei Problemi Inversi
and by the University of Bologna, Funds for Selected Research Topics.
1
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![2 M. Al-Horani and A. Favini
where −A generates an analytic semigroup in X, X being a Banach space,
Φ ∈ X∗
, g ∈ C1
([0, τ], R), τ > 0 fixed, u0, z ∈ D(A) and the pair (u, f) ∈
C1+θ
([0, τ]; X) × Cθ
([0, τ]; R), θ ∈ (0, 1), is to be found. Here Cθ
([0, τ]; X)
denotes the space of all X-valued Hölder-continuous functions on [0, τ] with
exponent θ, and
C1+θ
([0, τ]; X) = {u ∈ C1
([0, τ]; X); u0
∈ Cθ
([0, τ]; X)}.
In Section 3 we consider the possibly degenerate problem
d
dt
((Mu)(t)) + Lu(t) = f(t)z , 0 ≤ t ≤ τ ,
(Mu)(0) = Mu0 ,
Φ[Mu(t)] = g(t) , 0 ≤ t ≤ τ ,
where L, M are two closed linear operators in X with D(L) ⊆ D(M), L
being invertible, Φ ∈ X∗
and g ∈ C1+θ
([0, τ]; R), for some θ ∈ (0, 1). In this
possibly degenerate problem, M may have no bounded inverse and the pair
(u, f) ∈ Cθ
([0, τ]; D(L))×Cθ
([0, τ]; R) is to be found. This problem was solved
(see [1]) when λ = 0 is a simple pole for the resolvent (λL + M)−1
. Here we
consider this problem under the assumption that M and L act in a reflexive
Banach space X with the resolvent estimate
kλM(λM + L)−1
kL(X) ≤ C, Re λ ≥ 0 ,
or the equivalent one
kL(λM + L)−1
kL(X) = k(λT + I)−1
kL(X) ≤ C, Re λ ≥ 0 ,
where T = ML−1
. Reflexivity of X allows to use the representation of X
as a direct sum of the null space N(T) and the closure of its range R(T), a
consequence of the ergodic theorem (see [13], pp. 216-217). Here, a basic role
is played by real interpolation space, see [12].
In Section 4 we give some examples from partial differential equations de-
scribing the range of applications of the previous abstract results.
2 The nonsingular case
Let X be a Banach space with norm k · kX (sometimes, k · k will be used for
the sake of brevity), τ > 0 fixed, u0, z ∈ D(A), where −A is the generator of
an analytic semigroup in X, Φ ∈ X∗
and g ∈ C1
([0, τ], R). We want to find a
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![Degenerate first order identification problems in Banach spaces 3
pair (u, f) ∈ C1+θ
([0, τ]; X) × Cθ
([0, τ]; R), θ ∈ (0, 1), such that
u0
(t) + Au(t) = f(t)z , 0 ≤ t ≤ τ , (2.1)
u(0) = u0 , (2.2)
Φ[u(t)] = g(t) , 0 ≤ t ≤ τ , (2.3)
under the compatibility relation
Φ[u0] = g(0) . (2.4)
Let us remark that the compatibility relation (2.4) follows from (2.2)-(2.3).
To solve our problem we first apply Φ to (2.1) and take equation (2.3) into
account; we obtain the following equation in the unknown f(t):
g0
(t) + Φ[Au(t)] = f(t)Φ[z] . (2.5)
Suppose the condition
Φ[z] 6= 0 (2.6)
to be satisfied. Then we can write (2.5) under the form:
f(t) =
1
Φ[z]
{g0
(t) + Φ[Au(t)]} , 0 ≤ t ≤ τ , (2.7)
and the solution u of (2.1)-(2.3) is assigned by the formula
u(t) = e−tA
u0 +
Z t
0
e−(t−s)A {g0
(s) + Φ[Au(s)]}
Φ[z]
z ds
=
Z t
0
e−(t−s)A Φ[Au(s)]
Φ[z]
z ds + e−tA
u0
+
1
Φ[z]
Z t
0
e−(t−s)A
g0
(s)z ds . (2.8)
Apply the operator A to (2.8) and obtain
Au(t) =
Z t
0
e−(t−s)A Φ[Au(s)]
Φ[z]
Az ds + e−tA
Au0
+
1
Φ[z]
Z t
0
e−(t−s)A
g0
(s)Az ds . (2.9)
Let Au(t) = v(t); then (2.7) and (2.9) can be written, respectively, as follows:
f(t) =
1
Φ[z]
{g0
(t) + Φ[v(t)]} , 0 ≤ t ≤ τ , (2.10)
v(t) =
Z t
0
e−(t−s)A Φ[v(s)]
Φ[z]
Az ds + e−tA
Au0
+
1
Φ[z]
Z t
0
e−(t−s)A
g0
(s)Az ds . (2.11)
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![4 M. Al-Horani and A. Favini
Let us introduce the operator S
Sw(t) =
Z t
0
e−(t−s)A Φ[w(s)]
Φ[z]
Az ds .
Then (2.11) can be written in the form
v − Sv = h (2.12)
where
h(t) = e−tA
Au0 +
1
Φ[z]
Z t
0
e−(t−s)A
g0
(s)Az ds .
It is easy to notice that h ∈ C([0, τ]; X).
To prove that (2.12) has a unique solution in C([0, τ]; X), it is sufficient to
show that Sn
is a contraction for some n ∈ N. For this, we note
kSv(t)k ≤
M kΦkX∗
|Φ(z)|
Z t
0
kv(s)k kAzk ds
kS2
v(t)k ≤
M kΦkX∗
|Φ(z)|
Z t
0
kTv(s)k kAzk ds
≤
µ
M kΦkX∗ kAzk
|Φ(z)|
¶2 Z t
0
µZ s
0
kv(σ)k dσ
¶
ds
≤
µ
M kΦkX∗ kAzk
|Φ(z)|
¶2 Z t
0
(t − σ)kv(σ)k dσ
≤
µ
M kΦkX∗ kAzk
|Φ(z)|
¶2
kvk∞
t2
2
,
where kvk∞ = kvkC([0,τ];X) .
Proceeding by induction, we can find the estimate
kSn
v(t)k ≤
µ
M kΦkX∗ kAzk
|Φ(z)|
¶n
tn
n!
kvk∞ ,
which implies that
kSn
vk∞ ≤
µ
M kΦkX∗ kAzk
|Φ(z)|
τ
¶n
1
n!
kvk∞ .
Consequently, Sn
is a contraction for sufficiently large n. At last notice that
f(t) z is then a continuous D(A)-valued function on [0, τ], so that (2.1), (2.2)
has in fact a unique strict solution. However, we want to discuss the maximal
regularity for the solution v = Au, and for this we need some additional
conditions. We now recall that if −A generates a bounded analytic semigroup
in X, then the real interpolation space (X, D(A))θ,∞ = DA(θ, ∞) coincides
with {x ∈ X; supt>0 t1−θ
kAe−tA
xk < ∞}, (see [3]).
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![Degenerate first order identification problems in Banach spaces 5
Consider formula (2.11) and notice that (see [10])
e−tA
Au0 ∈ Cθ
([0, τ]; X) if and only if Au0 ∈ DA(θ, ∞) .
Moreover, if g ∈ C1+θ
([0, τ]; R) and Az ∈ DA(θ, ∞), then
Z t
0
e−(t−s)A
g0
(s)Az ds ∈ Cθ
([0, τ]; X)
and
Z t
0
e−(t−s)A
Az Φ[v(s)] ds =
¡
e−tA
Az ∗ Φ[v]
¢
(t) ∈ Cθ
([0, τ]; X) .
See [7] and [6].
Therefore, if we assume
Au0, Az ∈ DA(θ, ∞) , (2.13)
then v(t) ∈ Cθ
([0, τ]; X), i.e., Au(t) ∈ Cθ
([0, τ]; X) which implies that f(t) ∈
Cθ
([0, τ]; R). Then there exists a unique solution (u, f) ∈ C1+θ
([0, τ]; X) ×
Cθ
([0, τ]; R).
We summarize our discussion in the following theorem.
THEOREM 2.1 Let −A be the generator of an analytic semigroup, Φ ∈
X∗
, u0, z ∈ DA(θ+1, ∞) and g ∈ C1+θ
([0, τ]; R). If Φ[z] 6= 0 and (2.4) holds,
then problem (2.1)-(2.3) admits a unique solution (u, f) ∈ [C1+θ
([0, τ]; X) ∩
Cθ
([0, τ]; D(A))] × Cθ
([0, τ]; R).
3 The singular case
Consider the possibly degenerate problem
Dt(Mu) + Lu = f(t)z , 0 ≤ t ≤ τ , (3.1)
(Mu)(0) = Mu0 , (3.2)
Φ[Mu(t)] = g(t) , 0 ≤ t ≤ τ , (3.3)
where L, M are two closed linear operators with D(L) ⊆ D(M), L being
invertible, Φ ∈ X∗
and g ∈ C1+θ
([0, τ]; R) for θ ∈ (0, 1). Here M may have
no bounded inverse and the pair (u, f) ∈ C([0, τ]; D(L)) × Cθ
([0, τ]; R), with
Mu ∈ C1+θ
([0, τ]; X), is to be determined so that the following compatibility
condition must hold:
Φ[Mu(0)] = Φ[Mu0] = g(0) . (3.4)
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![6 M. Al-Horani and A. Favini
Let us assume that the pair (M, L) satisfies the estimate
kλM(λM + L)−1
kL(X) ≤ C, Re λ ≥ 0 , (3.5)
or the equivalent one
kL(λM + L)−1
kL(X) = k(λT + I)−1
kL(X) ≤ C, Re λ ≥ 0 , (3.6)
where T = ML−1
.
Various concrete examples of this relation can be found in [8]. One may
note that λ = 0 is not necessarily a simple pole for (λ + T)−1
, T = ML−1
.
Let Lu = v and observe that T = ML−1
∈ L(X). Then (3.1)-(3.3) can be
written as
Dt(Tv) + v = f(t)z , 0 ≤ t ≤ τ , (3.7)
(Tv)(0) = Tv0 = ML−1
v0 , (3.8)
Φ[Tv(t)] = g(t) , 0 ≤ t ≤ τ , (3.9)
where v0 = Lu0.
Since X is a reflexive Banach space and (3.5) holds, we can represent X as
a direct sum (cfr. [8, p. 153], see also [13], pp. 216-217)
X = N(T) ⊕ R(T)
where N(T) is the null space of T and R(T) is the range of T. Let T̃ = TR(T )
:
R(T) → TR(T )
be the restriction of T to R(T). Clearly T̃ is a one to one map
from R(T) onto R(T) (T̃ is an abstract potential operator in R(T). Indeed,
in view of the assumptions, −T̃−1
generates an analytic semigroup on R(T),
(see [8, p. 154]).
Finally, let P be the corresponding projection onto N(T) along R(T).
We can now prove the following theorem:
THEOREM 3.1 Let L, M be two closed linear operators in the reflex-
ive Banach space X with D(L) ⊆ D(M), L being invertible, Φ ∈ X∗
and g ∈
C1+θ
([0, τ]; R). Suppose the condition (3.5) to hold with (3.4), too. Then prob-
lem (3.1)-(3.3) admits a unique solution (u, f) ∈ Cθ
([0, τ]; D(L))×Cθ
([0, τ]; R)
provided that
Φ[(I − P)z] 6= 0 , sup
t>0
tθ
k(tT̃ + 1)−1
yikX < +∞ , i = 1, 2
where y1 = (I − P)Lu0 and y2 = T̃−1
(I − P)z.
Proof. Since P is the projection onto N(T) along R(T), it is easy to check
that problem (3.7)-(3.9) is equivalent to the couple of problems
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![Degenerate first order identification problems in Banach spaces 7
DtT̃(I − P)v + (I − P)v = f(t)(I − P)z , 0 ≤ t ≤ τ , (3.10)
T̃(I − P)v(0) = T̃(I − P)v0 , (3.11)
Φ[T̃(I − P)v(t)] = g(t) , 0 ≤ t ≤ τ , (3.12)
and
Pv(t) = f(t)Pz . (3.13)
Let w = T̃(I −P)v, so that (I −P)v = T̃−1
w, and hence system (3.10)-(3.12)
becomes
w0
(t) + T̃−1
w = f(t)(I − P)z , 0 ≤ t ≤ τ , (3.14)
w(0) = w0 = T̃(I − P)v0 = Tv0 , (3.15)
Φ[w(t)] = g(t) , 0 ≤ t ≤ τ . (3.16)
Then, according to Theorem 2.1, there exists a unique solution (w, f) ∈
C1+θ
([0, τ]; R(T)) × Cθ
([0, τ]; R) with T̃−1
w ∈ Cθ
([0, τ]; R(T)) to problem
(3.14)-(3.16) provided that
Φ[(I − P)z] 6= 0 , (I − P)Lu0 , T̃−1
(I − P)z ∈ DT̃ −1 (θ, ∞) .
Therefore, (I −P)v ∈ Cθ
([0, τ]; R(T)), Pv ∈ Cθ
([0, τ]; N(T)) and hence there
exists a unique solution (u, f) ∈ Cθ
([0, τ]; D(L)) × Cθ
([0, τ]; R) with Mu ∈
C1+θ
([0, τ]; X) to problem (3.1)-(3.3) . ¤
Our next goal is to weaken the assumptions on the data in the Theorems
1 and 2. To this end we again suppose −A to be the generator of an analytic
semigroup in X of negative type, i.e., ke−tA
k ≤ ce−ωt
, t ≥ 0, where c, ω > 0,
g ∈ C1+θ
([0, τ]; R), but we take u0 ∈ DA(θ + 1; X), z ∈ DA(θ0, ∞), where
0 < θ < θ0 < 1. Our goal is to find a pair (u, f) ∈ C1
([0, τ]; X) × C([0, τ]; R),
Au ∈ Cθ
([0, τ]; X) such that equations (2.1)-(2.3) hold under the compatibil-
ity relation (2.4).
THEOREM 3.2 Let −A be a generator of an analytic semigroup in X
of positive type, 0 < θ < θ0 < 1, g ∈ C1+θ
([0, τ]; R), u0 ∈ DA(θ + 1, ∞),
z ∈ DA(θ0, ∞). If, in addition, (2.4), (2.6) hold, then problem (2.1)-(2.3) has
a unique solution (u, f) ∈ Cθ
([0, τ], D(A)) × Cθ
([0, τ]; R).
Proof. Recall (see [10, p. 145]) that if u0 ∈ D(A), f ∈ C([0, τ]; R), z ∈
DA(θ0, ∞), then problem (2.1)-(2.2) has a unique strict solution. Moreover, if
u0 ∈ DA(θ + 1; X), then the solution u to (2.1)-(2.2) has the maximal regu-
larity u0
, Au ∈ C([0, τ]; X) ∩ B([0, τ]; DA(θ0, ∞)), where B([0, τ]; Y ) denotes
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![8 M. Al-Horani and A. Favini
the space of all bounded functions from [0, τ] into the Banach space Y . In
addition Au ∈ Cθ
([0, τ]; X).
In order to prove our statement, we need to study suitably the properties of
the function u and to use carefully some properties of the convolution operator
and real interpolation spaces.
One readily sees that u satisfies
Au(t) =
Z t
0
Φ[Au(s)]
Φ[z]
Ae−(t−s)A
z ds + e−tA
Au0
+
1
Φ[z]
Z t
0
A e−(t−s)A
z g0
(s) ds
so that v(t) = Au(t) must satisfy
v(t) =
Z t
0
Ae−(t−s)A
z
Φ[v(s)]
Φ[z]
ds + e−tA
Au0
+
1
Φ[z]
Z t
0
A e−(t−s)A
z g0
(s) ds .
Let us introduce the operator S : C([0, τ]; X) → C([0, τ]; X) by
(Sw)(t) =
Z t
0
Ae−(t−s)A
z
Φ[w(s)]
Φ[z]
ds .
Since z ∈ DA(θ0, ∞), i.e.,
kAe−tA
zk ≤
c
t1−θ0
, t > 0 ,
we deduce
kSw(t)k ≤ c
Z t
0
kΦkX∗ kzkθ0, ∞
kw(s)k
(t − s)1−θ0
ds ,
kS2
w(t)k ≤ [ckΦkX∗ kzkθ0, ∞]
Z t
0
kSw(s)k
(t − s)1−θ0
ds
≤ [ckΦkX∗ kzkθ0, ∞]2
Z t
0
ds
(t − s)1−θ0
Z s
0
kw(σ)k
(s − σ)1−θ0
dσ
= [ckΦkX∗ kzkθ0, ∞]2
Z t
0
µZ t
σ
ds
(t − s)1−θ0 (s − σ)1−θ0
¶
kw(σ)k dσ
= c2
1
·Z 1
0
dη
(1 − η)1−θ0 η1−θ0
¸
(t − σ)1−2(1−θ0)
kw(σ)k dσ ,
where c1 = ckΦkX∗ kzkθ0, ∞, k · kDA(θ0,∞) denoting the norm in DA(θ0, ∞).
Recall that
B(p, q) =
Z 1
0
(1 − η)p−1
ηq−1
dη =
Γ(p) Γ(q)
Γ(p + q)
.
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![Degenerate first order identification problems in Banach spaces 9
Then
kS3
w(t)k ≤ c3
1
Z 1
0
dη
(1 − η)1−θ0 η1−θ0
Z 1
0
dη
(1 − η)1−θ0 η2(1−θ0)−1
×
Z 1
0
(t − σ)2−3(1−θ0)
kw(σ)k dσ
≤ c3
1 B(θ0, θ0) B(θ0, 2θ0)
Z 1
0
(t − σ)2−3(1−θ0)
kw(σ)k dσ
≤ c3
1
Γ(θ0)3
Γ(3θ0)
t3θ0
3θ0
kwkC([0,t];X) .
By induction, we easily verify that
kSn
w(t)k ≤ cn
1
Γ(θ0)n
Γ(nθ0)
tnθ0
nθ0
kwkC([0,t];X) .
Since n
p
Γ(nθ0) → ∞ as n → ∞, we conclude that the operator S has spectral
radius equal to 0. On the other hand, since z ∈ DA(θ0, ∞), θ0 > θ, and
g0
∈ Cθ
([0, τ]; R), we deduce by [6] (Lemma 3.3) that the convolution
Z t
0
g0
(s)Ae−(t−s)A
z ds
belongs to Cθ
([0, τ]; X).
Moreover, since Au0 ∈ DA(θ, ∞), e−tA
Au0 ∈ Cθ
([0, τ]; X). It follows that
equation (2.12), i.e.,
v − Sv = h ,
with
h(t) = e−tA
Au0 +
1
Φ[z]
Z t
0
Ae−(t−s)A
z g0
(s) ds
has a unique solution v ∈ C([0, τ]; X). In order to obtain more regularity for
v, we use Lemma 3.3 in [6] (see also [7]) again. To this end, we introduce the
following Lp
-spaces related to any positive constant δ:
Lp
δ((0, τ); X) =
©
u : (0, τ) → X : e−tδ
u ∈ Lp
((0, τ); X)
ª
,
endowed with the norms kukδ,0,p = ke−tδ
ukLp((0,τ);X). Moreover,
kgkδ,θ,∞ = ke−tδ
gkCθ([0,τ];X) .
Lemma 3.3 in [6] establishes that, in fact, if z ∈ DA(θ0, ∞)), 0 < θ < θ0 < 1,
then
°
°
°
Z t
0
Ae−(t−s)A
z Φ[v(s)] ds
°
°
°
δ,θ,∞
≤ c δ−θ0+θ+1/p
kΦ[v(.)]kδ,0,p
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![10 M. Al-Horani and A. Favini
provided that (θ0 − θ)−1
< p. Now,
Z t
0
|Φ[v(t)]|p
e−δpt
dt ≤ kΦkp
X∗ kvkp
Lp
δ ((0,τ);X)
≤ τ kΦkp
X∗ kvkp
δ,θ,∞ .
Choose δ suitably large and recall that h ∈ Cθ
([0, τ]; X). Then the norm of
S as an operator from Cθ
([0, τ]; X) (with norm k · kδ,θ,∞) into itself is less
than 1, so that we can deduce that the solution v = Au has the regularity
Cθ
([0, τ]; X), as desired. ¤
As a consequence, Theorem 3.1 has the following improvement.
THEOREM 3.3 Let L, M be two closed linear operators in the reflexive
Banach space X with D(L) ⊆ D(M), L being invertible, Φ ∈ X∗
and g ∈
C1+θ
([0, τ]; R). Suppose (3.4), (3.5) to hold.
If 0 < θ < θ0 < 1 and Φ[(I−P)z] 6= 0 , sup
t>0
tθ0
k(tT +1)−1
(I−P)zkX < +∞,
sup
t>0
tθ
k(tT + 1)−1
(I − P)Lu0kX < +∞, then problem (3.1)-(3.3) admits a
unique solution (u, f) ∈ Cθ
([0, τ]; D(L))×Cθ
([0, τ]; R) with Mu ∈ C1+θ
([0, τ];
X).
4 Applications
In this section we show that our abstract results can be applied to some con-
crete identification problems. For further examples for which the theory works
we refer to [8].
Problem 1. Consider the following identification problem related to a bounded
region Ω in Rn
with a smooth boundary ∂Ω
Dtu(x, t) =
n
X
i,j=1
Dxi
(aij(x)Dxj
u(x, t)) + f(t)v(x) , (x, t) ∈ Ω × [0, τ] ,
u(x, t) = 0 , ∀ (x, t) ∈ ∂Ω × [0, τ] ,
u(x, 0) = u0(x) , x ∈ Ω ,
Φ[u(x, t)] =
Z
Ω
η(x)u(x, t) dx = g(t) , ∀ t ∈ [0, τ] ,
where the coefficients aij enjoy the properties
aij ∈ C(Ω) , aij = aji , i, j = 1, 2, ..., n
n
X
i,j=1
aij(x) ξi ξj ≥ c0|ξ|2
∀ x ∈ Ω , ∀ ξ ∈ Rn
,
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![Degenerate first order identification problems in Banach spaces 11
c0 being a positive constant. Moreover, g ∈ C1
([0, τ]; R). We take
Au = −
n
X
i,j=1
Dxi (aijDxj u) , D(A) = W2,p
(Ω) ∩ W1,p
0 (Ω) ,
where 1 < p < +∞ is assumed. Concerning η, we suppose η ∈ Lq
(Ω), where
1/p + 1/q = 1. As it is well known, −A generates an analytic semigroup in
Lp
(Ω) and thus we can apply Theorem 3.2 provided that u0 ∈ DA(θ + 1; ∞),
i.e., Au0 ∈ DA(θ, ∞), v ∈ DA(θ0; ∞), 0 < θ < θ0 < 1. On the other hand,
the interpolation spaces DA(θ, ∞) are well characterized. Then our problem
admits a unique solution
(u, f) ∈ Cθ
([0, τ]; W2,p
(Ω) ∩ W1,p
0 (Ω)) × Cθ
([0, τ]; R),
if g ∈ C1+θ
([0, τ]; R), g(0) =
R
Ω
η(x) u0(x) dx and
R
Ω
η(x) v(x) dx 6= 0.
Problem 2. Let Ω be a bounded region in Rn
with a smooth boundary ∂Ω.
Let us consider the identification problem
Dtu(x, t) =
n
X
i,j=1
Dxi (aij(x)Dxj u(x, t)) + f(t)v(x) , (x, t) ∈ Ω × [0, τ] ,
u(x, t) = 0 , (x, t) ∈ ∂Ω × [0, τ] ,
u(x, 0) = u0(x) , x ∈ Ω ,
Φ[u(x, t)] = u(x, t) = g(t) , t ∈ [0, τ] ,
where x ∈ Ω is fixed, and the pair (f, u) is the unknown.
Here we take
X = C0(Ω) =
©
u ∈ C(Ω), u(x) = 0 ∀ x ∈ ∂Ω
ª
,
endowed with the sup norm kukX = kuk∞.
If the coefficients aij are assumed as in Problem 1, and
Au = −
n
X
i,j=1
Dxi
(aij(x)Dxj
u(x)) , D(A) =
©
u ∈ C0(Ω) ; Au ∈ C0(Ω)
ª
,
then −A generates an analytic semigroup in X. The interpolation spaces
DA(θ; ∞) have no simple characterization, in view of the boundary condi-
tions imposed to Au. Hence we notice that Theorem 3.2 applies provided that
u0 ∈ D(A2
) and v0 ∈ D(A), 0 < θ < 1, g ∈ C1+θ
([0, τ]; R), u0(x) = g(0) and
v(x) 6= 0.
Notice that we could develop a corresponding result to Theorem 3.2 related
to operators A with a nondense domain, but this is not so simple and the
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![12 M. Al-Horani and A. Favini
problem will be handled elsewhere.
Problem 3. Let us consider the following identification problem on a bounded
region Ω in R, n ≥ 1, with a smooth boundary ∂Ω:
Dt[m(x)u] = ∆u + f(t)w(x), (x, t) ∈ Ω × [0, τ] , (4.1)
u = 0 on ∂Ω × [0, τ] , (4.2)
(mu)(x, 0) = m(x)u0(x) , x ∈ Ω , (4.3)
Z
Ω
η(x) (mu)(x, t) dx = g(t) , ∀t ∈ [0, τ] , (4.4)
where m ∈ L∞
(Ω), ∆ : H1
0 (Ω) :→ H−1
(Ω) is the Laplacian, u0 ∈ H1
0 (Ω),
w ∈ H−1
(Ω), η ∈ H1
0 (Ω), g ∈ C1+θ
([0, τ]; R), 0 < θ < 1, and the pair (u, f) ∈
Cθ
([0, τ]; H1
0 (Ω))×Cθ
([0, τ]; R) is the unknown. Of course, the integral in (4.4)
stands for the duality between H−1
(Ω) and H1
0 (Ω). Theorem 3.3 applies with
X = H−1
(Ω), see [8, p. 75]. We deduce that if g(0) =
R
Ω
η(x) m(x)u0(x) dx,
w(x) = m(x)ζ(x) for some ζ ∈ H1
0 (Ω),
R
Ω
η(x) m(x)ζ(x) dx 6= 0 and (∆u0)(x)
= m(x)ζ1(x) for some ζ1 ∈ H1
0 (Ω), then problem (4.1)-(4.4) has a unique
solution (u, f) ∈ Cθ
([0, τ]; H1
0 (Ω))×Cθ
([0, τ]; R), mu ∈ C1+θ
([0, τ]; H−1
(Ω)).
Problem 4. Consider the degenerate parabolic equation
Dtv = ∆[a(x)v] + f(t)w(x) , (x, t) ∈ Ω × [0, τ] , (4.5)
together with the initial-boundary conditions
a(x)v(x, t) = 0 , (x, t) ∈ ∂Ω × [0, τ] , (4.6)
v(x, 0) = v0(x) , x ∈ Ω , (4.7)
and the additional information
Z
Ω
η(x)v(x, t) dx = g(t) , t ∈ [0, τ] . (4.8)
Here Ω is a bounded region in Rn
, n ≥ 1, with a smooth boundary ∂Ω, a(x) ≥
0 on Ω and a(x) > 0 almost everywhere in Ω is a given function in L∞
(Ω),
w ∈ H−1
(Ω), v0 ∈ H1
0 (Ω), η ∈ H1
0 (Ω), g is a real valued-function on [0, τ], at
least continuous, and the pair (v, f) is the unknown. Of course, we shall see
that functions w, v0 and g need much more regularity. Call a(x)v = u. Then,
if m(x) = a(x)−1
and u0(x) = a(x)v0(x) we obtain a system like (4.1)-(4.4).
Let M be the multiplication operator by m from H1
0 (Ω) into H−1
(Ω) and let
L = −∆ be endowed with Dirichlet condition, that is, L : H1
0 (Ω) → H−1
(Ω),
as previously. Take X = H−1
(Ω). Then it is seen in [8, p. 81] that (3.5) holds
if
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![Degenerate first order identification problems in Banach spaces 13
i) a−1
∈ L1
(Ω), when n = 1,
ii) a−1
∈ Lr
(Ω) with some r > 1, when n = 2,
iii) a−1
∈ L
n
2 (Ω), when n ≥ 3.
In order to apply Theorem 3.3 we suppose u0(x) = a(x)v0(x) ∈ H1
0 (Ω). As-
sumption (3.4) reads
Z
Ω
η(x)v0(x) dx =
Z
Ω
η(x)
u0(x)
a(x)
dx = g(0) .
Take g ∈ C1+θ
([0, τ]; R), 0 < θ < 1. Since R(T) = R((1/a)∆−1
), let aw =
ζ ∈ H1
0 (Ω), a∆u0 = a∆(av0) = ζ1 ∈ H1
0 (Ω),
R
Ω
η(x)ζ(x)
a(x) dx 6= 0.
Then we conclude that there exists a unique pair (v, f) satisfying (4.5)-(4.8)
with regularity
∆(av) ∈ Cθ
([0, τ]; H−1
(Ω)) , v ∈ C1+θ
([0, τ]; H−1
(Ω)) .
In many applications a(x) is comparable with some power of the distance
of x to the boundary ∂Ω and hence the assumptions depend heavily from
the geometrical properties of the domain Ω. For example, if Ω = (−1, 1),
a(x) = (1 − x2
)α
or a(x) = (1 − x)α
(1 + x)β
, 0 < α, β < 1 are allowed.
More generally, in Rn
, one can handle a(x) = (1−kxk2
)α
for some α > 0 with
Ω = {x ∈ Rn
: kxk < r}, r > 0. Precisely, if n = 2, then 0 < α < 1, if n ≥ 3
then 0 < α < 2/n.
Problem 5. Let us consider another degenerate parabolic equation, precisely
Dtv = x(1 − x)D2
xv + f(t)w(x), (x, t) ∈ (0, 1) × (0, τ), (4.9)
with the initial condition
v(x, 0) = v0(x), x ∈ (0, 1), (4.10)
but with a Wentzell boundary condition (basic in probability theory and in
applied sciences)
lim
x→0
x(1 − x)D2
xv(x, t) = 0, t ∈ (0, 1).
We add the additional information:
Φ[v(·, t)] = v(x̄, t) = g(t), t ∈ [0, τ], (4.11)
where x̄ ∈ (0, 1) is fixed. Here we take X = H1
(0, 1), with the norm
kuk2
X := kuk2
L2(0,1) + ku0
k2
L2(0,1) + |u(0)|2
+ |u(1)|2
.
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![14 M. Al-Horani and A. Favini
Introduce operator (A, D(A)) defined by
D(A) :=
©
u ∈ H1
(0, 1); u00
∈ L1
loc(0, 1) and x(1 − x)u00
∈ H1
0 (0, 1)
ª
,
Au = −x(1 − x)u00
, u ∈ D(A).
Then −A generates an analytic semigroup in H1
(0, 1), see [8, pp. 249-250],
[4]. So, we can apply Theorem 3.2; therefore, if 0 < θ < θ0 < 1, g ∈
C1+θ
([0, τ]; R), v0 ∈ DA(θ + 1, ∞), w ∈ DA(θ0, ∞) (in particular, v0 ∈
D(A2
), w ∈ D(A)), g(0) = v0(x̄), w(x̄) 6= 0, then there exists a unique
pair (v, f) ∈ Cθ
([0, τ]; D(A)) × Cθ
([0, τ]; R) satisfying (4.9)–(4.11) and Dtv ∈
Cθ
([0, τ]; H1
(0, 1)). Of course, general functionals Φ in the dual space H(0, 1)∗
could be treated.
References
[1] M.H. Al-Horani: An identification problem for some degenerate differ-
ential equations, Le Matematiche, 57, 217–227, 2002.
[2] A. Asanov and E.R. Atamanov: Nonclassical and inverse problems for
pseudoparabolic equations, 1st ed., VSP, Utrecht, 1997.
[3] G. Da Prato: Abstract differential equations, maximal regularity, and
linearization, Proceedings Symp. Pure Math., 45, 359–370, 1986.
[4] A. Favini, J.A. Goldstein and S. Romanelli: An analytic semigroup as-
sociated to a degenerate evolution equation, Stochastic processes and
Functional Analysis , M. Dekker, New York, 88–100, 1997.
[5] A. Favini and A. Lorenzi: Identification problems for singular integro-
differential equations of parabolic type II, Nonlinear Analysis T.M.A.,
56, 879–904, 2004.
[6] A. Favini and A. Lorenzi: Singular integro-differential equations of
parabolic type and inverse problems, Math. Models and Methods in
Applied Sciences, 13, 1745–1766, 2003.
[7] A. Favini and A. Lorenzi: Identification problems in singular integro-
differential equations of parabolic type I, Dynamics of continuous, dis-
crete, and impulsive systems, series A: Mathematical Analysis, 12, 303–
328, 2005.
[8] A. Favini and A. Yagi: Degenerate differential equations in Banach
spaces, 1st ed., Dekker, New York, 1999.
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![Degenerate first order identification problems in Banach spaces 15
[9] A. Lorenzi: Introduction to identification problem via Functional Anal-
ysis, 1st ed., VSP, Utrecht, 2001.
[10] A. Lunardi: Analytic semigroups and optimal regularity in parabolic
problems, 1st ed., Birkhäuser, Basel, 1995.
[11] A.I. Prilepko, D.G. Orlovsky and I.A. Vasin: Methods for solving inverse
problems in Mathematical Physics, 1st ed., M. Dekker, New York, 2000.
[12] H. Triebel: Interpolation theory, function spaces, differential operators,
North-Holland, Amsterdam, 1978.
[13] K. Yosida: Functional Analysis, 6th ed., Springer-Verlag, Berlin, 1980.
Mohammed Al-Horani Angelo Favini
Department of Mathematics Department of Mathematics
University of Jordan University of Bologna
Amman Bologna
Jordan Italy
horani@ju.edu.jo favini@dm.unibo.it
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![A nonisothermal dynamical Ginzburg-
Landau model of superconductivity.
Existence and uniqueness theorems
Valeria Berti and Mauro Fabrizio
Abstract A time-dependent Ginzburg-Landau model describing superconduc-
tivity with thermal effects into account is studied. For this problem, the absolute
temperature is a state variable for the superconductor. Therefore, we modify the
classical time-dependent Ginzburg-Landau equations by including the tempera-
ture dependence. Finally, the existence and the uniqueness of this nonisothermal
Ginzburg-Landau system is proved.
1 Introduction
There are some materials which exhibit a sharp rise in conductivity at tem-
peratures of the order of 5o
K and currents started in these metals persist for
a long time. This is the essence of superconductivity which was discovered by
Kamerlingh Onnes in 1911 (cf.[1], [2], [5], [6], [7], [15], [16], [17]). He observed
that the electrical resistance of various metals such as mercury, lead and tin
disappeared completely in a small temperature range at a critical tempera-
ture Tc which is characteristic of the material. The complete disappearance
of resistance is most sensitively demonstrated by experiments with persistent
currents in superconducting rings.
In 1914 Kamerlingh Onnes discovered that the resistance of a superconduc-
tor could be restored to its value in the normal state by the application of
a large magnetic field. About ten years later, Tuyn and Kamerlingh Onnes
performed experiments on cylindrical specimens, with the axis along the di-
rection of the applied field, and showed that the resistance increases rapidly in
a very small field interval. The value Hc of H at which the jump in resistance
occurs is termed threshold field. This value Hc is zero at T = Tc and increases
as the T is lowered below Tc.
In the first part of the paper we recall the London model of superconduc-
tivity, the traditional Ginzburg-Landau theory and the dynamical extension
presented by Gor’kov and Éliashberg [11]. These models are able to describe
the phase transition which occurs in a metal or alloy superconductor, when
17
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![18 V. Berti and M. Fabrizio
the temperature is constant, but under the critical value Tc. In these hy-
potheses the material will pass from the normal to the superconductor state
if the magnetic field is lowered under the threshold field Hc. In this paper we
present a generalization of the Ginzburg-Landau theory which considers vari-
able both the magnetic field and the temperature. Also this model describes
the phenomenon of superconductivity as a second-order phase transition. The
two phases are represented in the plane H − T by two regions divided by a
parabola.
The second part of the paper is devoted to the proof of existence and unique-
ness of the solutions of the nonisothermal Ginzburg-Landau equations. In a
previous paper ([3]) we have shown the well posedness of the problem ob-
tained by neglecting the magnetic field. In this paper, the existence and the
uniqueness of the solutions of the nonisothermal Ginzburg-Landau equations
are proved after formulating the problem by means of the classical state vari-
ables (ψ, A, φ) together with the temperature u = T/Tc. The existence of
the weak solutions in a bounded time interval is established by applying the
Galerkin’s technique. Then, by means of energy estimates we obtain the ex-
istence of global solutions in time. Finally, we prove further regularity and
uniqueness of the solutions.
2 Superconductivity and London theory
Until 1933 the magnetic properties of a superconductor were tacitly assumed
to be a consequence of the property of infinite conductivity. Meissner and
Ochsenfeld checked experimentally this assumption and found that such is
not the case. They observed the behavior of a cylinder in an applied uniform
magnetic field. When the temperature is above the critical value Tc, the sample
is in the normal state and the internal magnetic field is equal to the external
magnetic field. If the cylinder is cooled through Tc, the magnetic field inside
the sample is expelled, showing that a superconducting material exhibits a
perfect diamagnetism (Meissner effect).
The phenomenological theory of the brothers Heinz and Fritz London, de-
veloped in 1935 soon after the discovery of the Meissner effect, is based on the
diamagnetic approach in that it gives a unique relation between current and
magnetic field. At the same time it is closely related to the infinite conductiv-
ity approach in that the allowed current distributions represent a particular
class of solutions for electron motion in the absence of scattering.
In the London theory the electrons of a superconducting material are di-
vided in normal (as the electrons in a normal material scatter and suffer
resistance to their motion) and superconducting (they cross the metal with-
out suffering any resistance). Below the critical temperature Tc, the current
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![20 V. Berti and M. Fabrizio
Take the curl of (2.1) and compare with (2.4) and the induction law; we
have
α∇×J = −
B
µ0
− σαḂ.
Similarly, take the time derivative of (2.1) and compare with (2.2) to have
αJ̇ =
1
µ0
E + ασĖ.
These equations have the advantage that only the total current density J
occurs rather than Jn and Js. This is appreciated if the separation of the
total current J into the two components Jn, Js looks somewhat artificial.
Equations (2.3) and (2.4) can be used to evaluate the magnetic induction
(field) in a superconductor. If, for simplicity, B is allowed to depend on a
single Cartesian coordinate, x say, then by (2.3) the only bounded solution as
x ≥ 0 is
B(x) = B(0) exp(−x/
√
α).
This result shows that, roughly, B penetrates in the half-space x ≥ 0 of a
distance
√
α =
p
m/(µ0nsq2). That is why the quantity
λL =
m
µ0nsq2
is called London penetration depth. This implies that a magnetic field is ex-
ponentially screened from the interior of a sample with penetration depth λL,
i.e., the Meissner effect.
3 Ginzburg-Landau theory
The Ginzburg-Landau theory [10] deals with the transition of a material from
a normal state to a superconducting state. If a magnetic field occurs then the
transition involves a latent heat which means that the transition is of the first
order. If, instead, the magnetic field is zero the transition is associated with a
jump of the specific heat and no latent heat (second-order transition). Landau
[14] argued that a second-order transition induces a sudden change in the sym-
metry of the material and suggested that the symmetry can be measured by a
complex-valued parameter ψ, called order parameter. The physical meaning of
ψ is specified by saying that |ψ|2
is the number density, ns, of superconducting
electrons. Hence ψ = 0 means that the material is in the normal state, T > Tc,
while |ψ| = 1 corresponds to the state of a perfect superconductor (T = 0).
There must exist a relation between ψ and the absolute temperature T and
this occurs through the free energy e. Incidentally, at first Gorter and Casimir
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![A nonisothermal dynamical Ginzburg-Landau model 21
[12] elaborated a thermodynamic potential with a real-valued order parame-
ter. Later, Ginzburg and Landau argued that the order parameter should be
complex-valued so as to make the theory gauge-invariant.
With a zero magnetic field, at constant pressure and around the critical
temperature Tc the free energy e0 is written as
e0 = −a(T)|ψ|2
+
1
2
b(T)|ψ|4
; (3.1)
higher-order terms in |ψ|2
are neglected which means that the model is valid
around the critical temperature Tc for small values of |ψ|. If a magnetic field
occurs then the free energy of the material is given by
Z
Ω
e(ψ, T, H)dv =
Z
Ω
[e0(ψ, T) +
1
2
µH2
+
1
2m
| − i~∇ψ − qAψ|2
]dv (3.2)
where ~ is Planck’s constant and A is the vector potential associated to H,
i.e., µH = ∇ × A. The free energy (3.2) turns out to be gauge-invariant.
Assume that the free energy is stationary (extremum) at equilibrium. Regard
T as fixed, which means that quasi-static processes are considered whereby
Js = ∇×H. The corresponding Euler-Lagrange equations, for the unknowns
ψ and A, are
1
2m
(i~∇ + qA)2
ψ − aψ + b|ψ|2
ψ = 0, (3.3)
Js = −i
~q
2m
(ψ̄∇ψ − ψ∇ψ̄) −
q2
m
|ψ|2
A. (3.4)
Examine the consequences of (3.3)–(3.4). The boundary condition takes the
form
(−i~∇ψ − qAψ) · n
¯
¯
∂Ω
= 0.
By (3.4) this implies that
Js · n
¯
¯
∂Ω
= 0.
Also, letting ψ = |ψ| exp(iθ), we obtain from (3.4) that
Js = −
~q
m
|ψ|2
∇θ −
q2
m
|ψ|2
A = −Λ−1
(
~
q
∇θ + A). (3.5)
Hence the London equation
∇×(ΛJs) = −B (3.6)
follows.
To make the theory apparently gauge-invariant, we express the free energy
in terms of Js rather than of A. As shown in [7], §3.1, we have
|i~∇ψ + qAψ|2
= ~2
(∇|ψ|)2
+ |ψ|2
(~∇θ + qA)2
= ~2
(∇|ψ|)2
+ ΛJ2
s.
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![22 V. Berti and M. Fabrizio
Hence we can write the free energy (3.2) as a functional of f = |ψ| and T, H
in the form
Z
Ω
e(f, T, H)dv =
Z
Ω
h
− a(T)f2
+
1
2
b(T)f4
+
1
2
µH2
+
~2
2m
(∇f)2
−
1
2
ΛJ2
s
i
dv, (3.7)
the sign before ΛJ2
s arising from the Legendre transformation between A and
H. The term ~2
(∇f)2
/2m represents the energy density associated with the
interaction between the superconducting phase and the normal phase.
As is the case in Ginzburg-Landau theory, we restrict attention to time-
independent processes where Js = J = ∇×H. Hence the functional (3.7) is
stationary with respect to f and H, with H×n fixed at the boundary ∂Ω, if
the Euler-Lagrange equations
−
~2
2m
∆f +
m
2q2f3
J2
s − af + bf3
= 0 (3.8)
µH = −∇×ΛJs (3.9)
hold together with the boundary condition
∇f · n
¯
¯
∂Ω
= 0.
Equation (3.9) coincides with (3.6) and hence with the Ginzburg-Landau equa-
tion (3.5) or (3.4). Also, equation (3.8) reduces to equation (3.3) when the
phase θ of ψ is chosen to be zero as is the case for the system (3.8), (3.9).
Since the vector potential A is a nonmeasurable quantity, equation (3.9)
may seem more convenient than (3.4) as long as the relation ∇×ΛJs = −B
may be preferable to (3.5).
4 Quasi-steady model
Starting from the BCS theory of superconductivity, Schmid ([18]) and Gor’kov
& Éliashberg ([11]) have elaborated a generalization to the dynamic case of
the Ginzburg-Landau theory within the approximation that the temperature
T is near the transition value Tc. They consider the variables ψ, A and the
electrical potential φ which, together with the vector potential A, is subject
to the equations
∇×A = B , E = −Ȧ + ∇φ. (4.1)
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![A nonisothermal dynamical Ginzburg-Landau model 23
By adhering to [9] we complete the quasi-steady model of superconductivity
through the equations
γ(ψ̇ − i
q
~
φψ) = −
1
2m
(i~∇ + qA)2
+ aψ − b|ψ|2
ψ,
σ(Ȧ − ∇φ) = −∇×∇×A + Js, (4.2)
Js = −
i~q
2m
(ψ̄∇ψ − ψ∇ψ̄) −
q2
m
|ψ|2
A,
where γ is an appropriate relaxation coefficient.
The system of equations must be invariant under a gauge transformation
(ψ, A, φ) ←→ (ψei(q/~)χ
, A + ∇χ, φ + χ̇)
where the gauge χ is an arbitrary smooth function of (x, t). Among the possible
gauges we mention the London gauge
∇ · A = 0 , A · n|∂Ω = 0 ,
the Lorentz gauge
∇ · A = −φ
and the zero-electrical potential gauge φ = 0. Reference [13] investigates these
gauges and shows that the condition φ = 0 is incompatible with the London
gauge ∇ · A = 0.
The system (4.2) is associated with the initial conditions
ψ(x, 0) = ψ0(x) , A(x, 0) = A0(x). (4.3)
Equation (4.2)2 follows from the Maxwell equation
∇×H = Js + Jn + εĖ
by disregarding the derivative Ė and letting Jn = σE. That is why the problem
(4.2) is called quasi-steady.
Moreover, by letting ψ = f exp(iθ), from (4.2)1, we deduce the evolution
equation for the variable f. In terms of the observable variables f, ps, H, E,
the system (4.2) can be written in the form
γ ˙
f =
~2
2m
∇2
f −
q2
2m
p2
sf + af − bf3
(4.4)
∇×ps = −µH (4.5)
∇×H = Λ−1
(f)ps + σE (4.6)
∇×E = −µḢ (4.7)
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![24 V. Berti and M. Fabrizio
along with the boundary conditions
∇f · n|∂Ω = 0, H × n|∂Ω = g , ps · n|∂Ω = 0 (4.8)
and the initial conditions
f(x, 0) = f0(x) , H(x, 0) = H0(x). (4.9)
Observe that by (4.5) and (4.7) we have
ṗs =
m
q
v̇s = E − ∇φs.
This result can be viewed as the Euler equation for a nonviscous electronic
liquid (see [15]), where the scalar function φs represents the thermodynamic
potential per electron. The previous relation allows the quasi-steady problem
(4.4)–(4.7) to be written as
γ ˙
f =
~2
2m
∆f −
q2
2m
p2
sf + af − bf3
,
1
µ
∇×∇×ps = −Λ−1
(f)ps − σṗs − σ∇φs.
Moreover (4.6) provides
∇ · (Λ−1
(f)ps) = ∇ · Js = −∇ · (σE) = −σρ. (4.10)
In the theory of Gor’kov and Éliashberg [11], which is based on the system
(4.2), the function φs is assumed to depend on f and on the total electron
density ρ in the form
φs = Λ(f)ρ. (4.11)
The comparison of (4.10) and (4.11) gives
∇ · (Λ−1
(f)ps) = −σΛ−1
(f)φs. (4.12)
5 Phase transition in superconductivity
with thermal effects
We present a generalization of the model which describes the phase transition
in superconductivity without neglecting thermal effects. The main assumption
is that the phase transition is of second order and that the effects due to the
variation of the temperature are like the ones shown by varying the magnetic
field. In this sense the temperature T can be considered as the dual variable
of the magnetic field H.
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![A nonisothermal dynamical Ginzburg-Landau model 25
In order to justify the model here examined, we consider the expression of
Gauss free energy in terms of the variables (ψ, T, A)
E(ψ, u, A)
=
Z
Ω
·
−a(T)|ψ|2
+
1
2
b(T)|ψ|4
+
1
2µ
|∇ × A|2
+
1
2m
|i~∇ψ + qAψ|2
¸
dv.
Following [8] and [19] we consider the linear approximation of a(T) in a neigh-
borhood of the critical temperature, namely
a(T) = −a0
µ
T
Tc
− 1
¶
= −a0(u − 1),
where u = T
Tc
> 0. Finally, we suppose constant the coefficient b(T). By means
of the temperature u, the critical value uc is now given by uc = 1, while the
domain of definition is R+
.
Under these hypotheses the free energy takes the following form
E(ψ, u, A)
=
Z
Ω
h
a0(u − 1)|ψ|2
+
b0
2
|ψ|4
+
1
2µ
|∇ × A|2
+
1
2m
|i~∇ψ + qAψ|2
i
dv
(5.1)
which as a function of f, u, H can be written as
E(f, u, H) =
Z
Ω
h
a0(u − 1)f2
+
b0
2
f4
+
µ
2
H2
+
~2
2m
|∇f|2
+
1
2
Λ(f)|∇ × H|2
i
dv.
When we use the representation (5.1) as free energy with a0 = b0 = 1, then
the first Gor’kov Éliashberg equation takes the dimensionless form
γ ˙
f =
1
κ2
4f − (f2
− 1 + u)f − f|A −
1
κ
∇θ|2
(5.2)
4φ + γf2
(θ̇ − κφ) = 0 (5.3)
where κ > 0 is the Ginzburg-Landau parameter. From (5.1) or (5.2) it is
possible to retrieve the phase diagram, which separates the normal from su-
perconductor zone. This relation is represented by a parabola in the H − T
plane (see [1]), which can be approximated considering the points for which
the coefficient of f is zero. Namely, the points such that
−1 + u +
¯
¯
¯A −
1
κ
∇θ
¯
¯
¯
2
= 0 .
The temperature effect will be supposed negligible on the first Maxwell
equation, which we write in the London gauge (∇ · A = 0)
Ȧ − ∇φ + ∇ × ∇ × A + f2
(A −
1
κ
∇θ) = 0. (5.4)
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![A nonisothermal dynamical Ginzburg-Landau model 27
0, the system (6.1)–(6.5) reduces to
γ(ψt − iκφψ) −
1
κ2
4ψ +
2i
κ
( b
A + Aex) · ∇ψ + | b
A + Aex|2
ψ
+ψ
¡
|ψ|2
− 1 + b
u + e
u
¢
= 0 , (6.6)
b
At − ∇φ + ∇ × ∇ × b
A −
i
2κ
¡
ψ∇ψ̄ − ψ̄∇ψ
¢
+ |ψ|2
( b
A + Aex) = 0 , (6.7)
αb
ut − k4b
u −
1
2
¡
ψψ̄t + ψ̄ψt
¢
= 0 , (6.8)
∇ψ · n|∂Ω = 0 , b
A · n|∂Ω = 0 , (∇ × b
A) × n|∂Ω = 0 ,
∇φ · n|∂Ω = 0 , b
u|∂Ω = 0 , (6.9)
ψ(x, 0) = ψ0(x) , b
A(x, 0) = b
A0(x) , b
u(x, 0) = b
u0(x) . (6.10)
Let us denote by Lp
(Ω), p > 0 and Hs
(Ω), s ∈ R, the usual Lebesgue and
Sobolev spaces, endowed with the standard norms k·kp and k·kHs . In partic-
ular, we denote by k · k the norm in L2
(Ω). Given a time interval [a, b] and a
Banach space X, we denote by C(a, b, X) [Lp
(a, b, X)] the space of continuous
[Lp
] functions from [a, b] into X, with the usual norms
kfkC(a,b,X) = sup
t∈[a,b]
kf(t)kX ,
h
kfkp
Lp(a,b,X) =
Z b
a
kf(t)kp
X
i
.
Finally let us introduce the following functional spaces
D(Ω) =
©
A : A ∈ H1
(Ω), ∇ · A = 0, A · n|∂Ω = 0
ª
,
H1
m(Ω) =
½
φ : φ ∈ H1
(Ω),
Z
Ω
φdv = 0
¾
.
DEFINITION 6.1 A triplet (ψ, b
A, b
u) such that ψ ∈ L2
(0, τ, H1
(Ω)) ∩
H1
(0, τ, L2
(Ω)), b
A ∈ L2
(0, τ, D(Ω))∩H1
(0, τ, H1
(Ω)0
), b
u ∈ L2
(0, τ, H1
0 (Ω))∩
H1
(0, τ, H−1
(Ω)), satisfying (6.10), is a weak solution of the problem (6.6)–
(6.10) with φ ∈ L2
(0, τ, H1
m(Ω)), Aex ∈ D(Ω) if
Z
Ω
h
γ(ψt − iκφψ)χ +
1
κ2
∇ψ · ∇χ −
2i
κ
ψ( b
A + Aex) · ∇χ + | b
A + Aex|2
ψχ
+ψχ(|ψ|2
− 1 + b
u + e
u)
i
dv = 0 , (6.11)
Z
Ω
h
b
At · b + φ∇ · b + ∇ × b
A · ∇ × b −
i
2κ
¡
ψ∇ψ̄ − ψ̄∇ψ
¢
· b
+|ψ|2
( b
A + Aex) · b
i
dv = 0 , (6.12)
Z
Ω
·
αb
utv + k∇b
u · ∇v −
1
2
¡
ψψ̄t + ψ̄ψt
¢
v
¸
dv = 0 , (6.13)
for each χ ∈ H1
(Ω), b ∈ H1
(Ω), v ∈ H1
0 (Ω) and for a.e. t ∈ [0, τ].
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![30 V. Berti and M. Fabrizio
the inequality
dF
dt
+
γ
2
kψm
t k2
+
1
2κ2
k4ψm
k2
+
1
2
k∇φm
k2
+
k
2
k∇b
um
k2
+ k b
Am
t k2
+k∇ × b
Am
k2
≤ cF5
(6.18)
can be proved. See [4] for details. An integration in (0, t) leads to
F ≤ (F(0)−4
− ct)−1/4
t < τ0, (6.19)
where τ0 depends on the norms kψ0mkH1 , kA0mkH1 , ku0mk. The previous in-
equalities allow to pass to the limit as m → ∞ and prove the existence of a
solution (ψ, A, u) of the problem (6.6)–(6.10) satisfying ψ ∈ C(0, τ0, H1
(Ω)),
b
A ∈ C(0, τ0, H1
(Ω)) and b
u ∈ C(0, τ0, L2
(Ω)).
The local solutions, defined in the time interval (0, τ0) by Theorem 6.1, can
be extended to the whole interval (0, +∞). Indeed we construct a Lyapunov
functional for the system
γft −
1
κ2
4f + (f2
− 1 + u)f − f|A −
1
κ
∇θ|2
= 0 , (6.20)
At − ∇φ + ∇ × ∇ × A + f2
(A −
1
κ
∇θ) = 0 , (6.21)
4φ + γf2
(θt − κφ) = 0 , (6.22)
αut − fft − k4u = 0 , (6.23)
by multiplying the equations respectively by ft, At − κ−1
∇θt, −φ + κ−1
θt, b
u
and integrating in Ω. We obtain
kftk2
+
1
2
d
dt
·
1
κ2
k∇fk2
+
1
2
kf2
− 1k2
¸
+
Z
Ω
fft
·
|A −
1
κ
∇θ|2
+ u
¸
dv = 0,
kAtk2
+
1
2
d
dt
µ
k∇ × Ak2
− 2
Z
∂Ω
A × Hex · nda
¶
+
Z
Ω
·
f2
(A −
1
κ
∇θ) ·
µ
At −
1
κ
∇θt
¶
+
1
κ
∇φ · ∇θt
¸
dv = 0,
k∇φk2
+ γκk
1
κ
fθt − fφk2
−
1
κ
Z
Ω
∇φ · ∇θtdv = 0,
α
2
d
dt
kb
uk2
+ kk∇b
uk2
−
Z
Ω
b
ufftdv = 0 .
Adding the previous equations, we get
dG
dt
+ kftk2
+ kAtk2
+ k∇φk2
+ γκkf(
1
κ
θt − φ)k2
+ kk∇uk2
= 0 , (6.24)
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![A nonisothermal dynamical Ginzburg-Landau model 31
where the functional G is defined as
G =
1
2
µ
1
κ2
k∇fk2
+
1
2
kf2
− 1k2
+ kf(A −
1
κ
∇θ)k2
+ k∇ × Ak2
−2
Z
∂Ω
A × Hex · nda + νkHexk2
H−1/2(∂Ω) + αkuk2
+
Z
Ω
e
uf2
dv
¶
and the constant ν is sufficiently large in order to make G positive.
The relation (6.24) yields
G(t) ≤ G(0) , ∀t ≥ 0,
which guarantees that the local solutions defined in (0, τ0) can be extended
in (0, ∞). As a consequence of last inequality, we can prove some a priori
estimates of the solutions. In particular, if the initial data are chosen such
that the energy is finite, we have
kfk2
H1 + kAk2
H1 + kf∇θk2
+ kuk2
≤ C . (6.25)
Moreover, by integrating the relation (6.24) in (0, t) we obtain the further
estimate
Z t
0
[kftk2
+ kAtk2
+ k∇φk2
+ kfθtk2
+ k∇uk2
]ds ≤ C . (6.26)
The inequalities (6.25) and (6.26) lead to an estimate for the variable ψ
kψk2
H1 +
Z t
0
kψtk2
ds ≤ C . (6.27)
It can be proved ([3]) that if f0(x) ≤ 1 almost everywhere in Ω, then
f(x, t) ≤ 1 , (6.28)
for all t > 0. Accordingly, the relations (6.1), (6.25), (6.26) and (6.27) yield
Z t
0
k4ψk2
ds ≤ C . (6.29)
THEOREM 6.2 The solution (ψ, A, u) of the system (6.1)–(6.5), with ini-
tial data (ψ0, A0, u0) ∈ H1
(Ω) × D(Ω) × L2
(Ω) is unique.
Proof. Let (ψ1, A1, u1), (ψ2, A2, u2) be two solutions of the problem (6.6)–
(6.10) with the same initial data (ψ0, A0, u0) and sources Aex, e
u. By denoting
by ψ = ψ1 − ψ2, A = A1 − A2, φ = φ1 − φ2 and u = u1 − u2, from the
equations (6.6)–(6.8) and the inequalities (6.25)–(6.29) we deduce ([4])
1
2
d
dt
·
γkψk2
+
1
κ2
k∇ψk2
+ kAk2
+ k∇ × Ak2
+ αkuk2
¸
≤ ϕ1(t)kψk2
H1 + ϕ2(t)kAk2
H1 + Ckuk2
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![32 V. Berti and M. Fabrizio
where ϕ1, ϕ2 are L1
-functions of time. Therefore, an application of Gronwall’s
inequality proves ψ = 0, A = 0, u = 0.
References
[1] J. Bardeen: Theory of superconductivity. In Handbuch der Physik XV
(Edited by S. Flügge), (Springer, 1956), 274–369.
[2] J. G. Bednorz and K. A. Müller: Earlier and Recent Aspects of Super-
conductivity, (Springer, 1990).
[3] V. Berti and M. Fabrizio: A nonisothermal Ginzburg-Landau model in
superconductivity: existence, uniqueness and asymptotic behaviour, sub-
mitted.
[4] V. Berti and M. Fabrizio: Existence and uniqueness for a nonisothermal
dynamical Ginzburg-Landau model of superconductivity, submitted.
[5] R. de Bruyn Oubouter: Superconductivity: Discoveries during the early
years of low temperature research at Leiden 1908-1914, IEEE Transac-
tions on Magnetics Mag-23 (1987), 355–370.
[6] B. S. Chandrasekhar: Early experiments and phenomenological theories,
Superconductivity (I), edited by R.D. Parks, Marcel Dekker, 1969.
[7] S. J. Chapman, S. D. Howison, and J. R. Ockendon: Macroscopic models
for superconductivity, SIAM Rev. 34 (1992), 529–560.
[8] E. Coskun, Z. Cakir and P. Takac: Nucleation of vortices with a tempera-
ture and time-dependent Ginzburg-Landau model of superconductivity,
Euro. J. Appl. Math. 14 (2003), 111–127.
[9] M. Fabrizio, G. Gentili and B. Lazzari: A nonlocal phenomenological
theory in superconductivity, Math. Models Methods Appl. Sci. 7 (1997),
345–362.
[10] V. L. Ginzburg and L. D. Landau: On the theory of superconductivity,
Zh. Eksp. Teor. Fiz. 20 (1950), 1064–1082.
[11] L. P. Gor’kov and G.M. Éliashberg: Generalization of the Ginzburg-
Landau equations for nonstationary problems in the case of alloys with
paramagnetic impurities, Soviet Phys. JETP 27 (1968), 328–334.
[12] C. J. Gorter and H. Casimir: Zur thermodynamik des supaleitenden
Copyright © 2006 Taylor & Francis Group, LLC](https://image.slidesharecdn.com/7446-250706184920-73e56669/85/Differential-equations-inverse-and-direct-problems-1st-Edition-Angelo-Favini-46-320.jpg)
![A nonisothermal dynamical Ginzburg-Landau model 33
zustantles, Phys. Zs. 35 (1934), 963–966.
[13] H.G. Kaper and P. Takac: An equivalence for the Ginzburg-Landau
equations of superconductivity, ZAMP 48 (1997), 665–675
[14] L. D. Landau: Unbranched model of intermediate state, Phys. Z. Sowjet.
11 (1937), 129–138.
[15] F. London: Superfluids I, Wiley, 1950.
[16] H. London: An experimental examination of electrostatic behaviour of
superconductors, Proc. Roy. Soc. A 155 (1936), 102–108.
[17] A. B. Pippard: An experimental and theoretical study of relations be-
tween magnetic fields and current in superconductors, Proc. Roy. Soc.
London A 216 (1953), 547–568.
[18] A. Schmid: A time dependent Ginzburg-Landau equation and its appli-
cation to the problem of resistivity in the mixed state, Phys. Kondeus.
Mater. 5 (1966), 302–317.
[19] M. Tinkham: Introduction to superconductivity, McGraw-Hill, 1975.
Valeria Berti Mauro Fabrizio
Department of Mathematics Department of Mathematics
University of Bologna University of Bologna
P.zza Porta S.Donato 5 P.zza Porta S.Donato 5
40126 Bologna 40126 Bologna
Italy Italy
berti@dm.unibo.it fabrizio@dm.unibo.it
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![Some global in time results
for integrodifferential parabolic
inverse problems
Fabrizio Colombo, Davide Guidetti
and Vincenzo Vespri
Abstract We discuss a global in time existence and uniqueness result for an
inverse problem arising in the theory of heat conduction for materials with mem-
ory. The novelty lies in the fact this is a global in time well posed problem in the
sense of Hadamard, for semilinear parabolic inverse problems of integrodifferen-
tial type.
1 Introduction
In this paper we discuss some strategies we can use in the study of parabolic
integrodifferential inverse problems. The choice of the strategy depends on
what type of nonlinearities are involved. We consider the heat equation for
materials with memory since it is one of the most important physical examples
to which our methods apply. Other models, for instance in the theory of
population dynamics, can also be considered within our framework. We recall,
for the sake of completeness, the heat equation for materials with memory.
Let Ω be an open and bounded set in R3
and T be a positive real number.
The evolution equation for the temperature u is given, for (t, x) ∈ [0, T] × Ω,
by
Dtu(t, x) = k∆u(t, x) +
Z t
0
h(t − s)∆u(s, x) ds + F(u(t, x)), (1.1)
where k is the diffusivity coefficient, h accounts for the memory effects and F is
the heat source. In the inverse problem we consider, besides the temperature
u, also h as a further unknown, and to determine it we add an additional
measurement on u represented in integral form by
Z
Ω
φ(x)u(t, x) dx = G(t), ∀t ∈ [0, T], (1.2)
35
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![36 F. Colombo, D. Guidetti and V. Vespri
where φ and G are given functions representing the type of device used to mea-
sure u (on a suitable part of the body Ω) and the result of the measurement,
respectively. We associate with (1.1)–(1.2) the initial-boundary conditions, for
example of Neumann type:
(
u(0, x) = u0(x), x ∈ Ω,
Dνu(t, x) = 0, (t, x) ∈ [0, T] × ∂Ω,
(1.3)
ν denoting the outward normal unit vector.
So one of the problems we are going to investigate is the following.
PROBLEM 1.1 (The Inverse Problem with two types of nonlinearities):
determine the temperature u : [0, T] × Ω −→ R and the convolution kernel
h : [0, T] × Ω −→ R satisfying (1.1)–(1.3).
In the case when F is independent of u, but depends only on x and on t,
we assume that the heat source is placed in a given position, but its time
dependence is unknown, so we can suppose that
F(t, x) = f(t)g(x),
where f has to be determined and g is a given datum. Then we also assume
that the diffusion coefficient k is unknown. The second inverse problem we
will study is as follows.
PROBLEM 1.2 (An inverse problem with a nonlinearity of convolution
type): determine the temperature u : [0, T] × Ω −→ R, the diffusion coefficient
k and the functions h : [0, T] −→ R, f : [0, T] −→ R satisfying the system
Dtu(t, x) = k∆u(t, x) +
R t
0
h(t − s)∆u(s, x) ds + f(t)g(x),
u(0, x) = u0(x), x ∈ Ω,
∂u
∂ν
(t, x) = 0, (t, x) ∈ [0, T] × ∂Ω,
(1.4)
with the additional conditions
Z
Ω
u(t, x)µj(dx) = Gj(t), ∀t ∈ [0, T], j = 1, 2, (1.5)
where g, u0, G1, G2 are given data and µ1 and µ2 are finite Borel measures
in C(Ω).
REMARK 1.1 The additional conditions considered for Problem 1.2 (cf.
(1.5)) is more general than the one considered for Problem 1.1 (cf. (1.2)). This
is due to the fact that in Problem 1.2 we will choose the space of continuous
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Differential equations inverse and direct problems 1st Edition Angelo Favini
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- 5. Differential Equations Inverse and Direct Problems Copyright © 2006 Taylor & Francis Group, LLC
- 6. M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology S. Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University Anil Nerode Cornell University Freddy van Oystaeyen University of Antwerp, Belgium Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universität Siegen Mark Teply University of Wisconsin, Milwaukee PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EDITORIAL BOARD EXECUTIVE EDITORS Earl J. Taft Rutgers University Piscataway, New Jersey Zuhair Nashed University of Central Florida Orlando, Florida Copyright © 2006 Taylor & Francis Group, LLC
- 7. LECTURE NOTES IN PURE AND APPLIED MATHEMATICS Recent Titles G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. Ali Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra Á. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis R. Salvi, The Navier-Stokes Equations F. U. Coelho and H. A. Merklen, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Carnielli et al., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators and Related Problems G. R. Goldstein et al., Evolution Equations A. Giambruno et al., Polynomial Identities and Combinatorial Methods A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups J. Bergen et al., Hopf Algebras A. C. Krinik and R. J. Swift, Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao S. Caenepeel and F. van Oystaeyen, HopfAlgebras in Noncommutative Geometry and Physics J. Cagnol and J.-P. Zolésio, Control and Boundary Analysis S. T. Chapman, Arithmetical Properties of Commutative Rings and Monoids O. Imanuvilov, et al., Control Theory of Partial Differential Equations Corrado De Concini, et al., Noncommutative Algebra and Geometry A. Corso, et al., Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications – VII L. Sabinin, et al., Non-Associative Algebra and Its Application K. M. Furati, et al., Mathematical Models and Methods for Real World Systems A. Giambruno, et al., Groups, Rings and Group Rings P. Goeters and O. Jenda, Abelian Groups, Rings, Modules, and Homological Algebra J. Cannon and B. Shivamoggi, Mathematical and Physical Theory of Turbulence A. Favini and A. Lorenzi, Differential Equations: Inverse and Direct Problems Copyright © 2006 Taylor & Francis Group, LLC
- 8. Edited by Angelo Favini Università degli Studi di Bologna Italy Alfredo Lorenzi Università degli Studi di Milano Italy Differential Equations Inverse and Direct Problems Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business Copyright © 2006 Taylor & Francis Group, LLC
- 9. Published in 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-58488-604-8 (Softcover) International Standard Book Number-13: 978-1-58488-604-4 (Softcover) Library of Congress Card Number 2006008692 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Favini, A. (Angelo), 1946- Differential equations : inverse and direct problems / Angelo Favini, Alfredo Lorenzi. p. cm. -- (Lecture notes in pure and applied mathematics ; v. 251) Includes bibliographical references. ISBN 1-58488-604-8 1. Differential equations. 2. Inverse problems (Differential equations) 3. Banach spaces. I. Lorenzi, Alfredo. II. Title. III. Series. QA371.F28 2006 515'.35--dc22 2006008692 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Taylor & Francis Group is the Academic Division of Informa plc. C6048_Discl.fm Page 1 Thursday, March 16, 2006 11:29 AM Copyright © 2006 Taylor & Francis Group, LLC
- 10. Preface The meeting on Differential Equations: Inverse and Direct Problems was held in Cortona, June 21-25, 2004. The topics discussed by well-known specialists in the various disciplinary fields during the Meeting included, among others: differential and integrodifferential equations in Banach spaces, linear and non- linear theory of semigroups, direct and inverse problems for regular and singu- lar elliptic and parabolic differential and/or integrodifferential equations, blow up of solutions, elliptic equations with Wentzell boundary conditions, models in superconductivity, phase transition models, theory of attractors, Ginzburg- Landau and Schrödinger equations and, more generally, applications to partial differential and integrodifferential equations from Mathematical Physics. The reports by the lecturers highlighted very recent, interesting and original research results in the quoted fields contributing to make the Meeting very attractive and stimulating also to younger participants. After a lot of discussions related to the reports, some of the senior lecturers were asked by the organizers to provide a paper on their contribution or some developments of them. The present volume is the result of all this. In this connection we want to emphasize that almost all the contributions are original and are not expositive papers of results published elsewhere. Moreover, a few of the contributions started from the discussions in Cortona and were completed in the very end of 2005. So, we can say that the main purpose of the editors of this volume has con- sisted in stimulating the preparation of new research results. As a consequence, the editors want to thank in a particular way the authors that have accepted this suggestion. Of course, we warmly thank the Italian Istituto Nazionale di Alta Matematica that made the Meeting in Cortona possible and also the Universitá degli Studi di Milano for additional support. Finally, the editors thank the staff of Taylor & Francis for their help and useful suggestions they supplied during the preparation of this volume. Angelo Favini and Alfredo Lorenzi Bologna and Milan, December 2005 vii Copyright © 2006 Taylor & Francis Group, LLC
- 11. Contents M. Al-Horani and A. Favini: Degenerate first order identification problems in Banach spaces 1 V. Berti and M. Fabrizio: A nonisothermal dynamical Ginzburg-Landau model of supercon- ductivity. Existence and uniqueness theorems 17 F. Colombo, D. Guidetti and V. Vespri: Some global in time results for integrodifferential parabolic inverse problems 35 A. Favini, G. Ruiz Goldstein, J. A. Goldstein, and S. Romanelli: Fourth order ordinary differential operators with general Wentzell boundary conditions 59 A. Favini, R. Labbas, S. Maingot, H. Tanabe and A. Yagi: Study of elliptic differential equations in UMD spaces 73 A. Favini, A. Lorenzi and H. Tanabe: Degenerate integrodifferential equations of parabolic type 91 A. Favini, A. Lorenzi and A. Yagi: Exponential attractors for semiconductor equations 111 S. Gatti and M. Grasselli: Convergence to stationary states of solutions to the semilinear equa- tion of viscoelasticity 131 S. Gatti and A. Miranville: Asymptotic behavior of a phase field system with dynamic boundary conditions 149 M. Geissert, B. Grec, M. Hieber and E. Radkevich: The model-problem associated to the Stefan problem with surface tension: an approach via Fourier-Laplace multipliers 171 G. Ruiz Goldstein, J. A. Goldstein and I. Kombe: The power potential and nonexistence of positive solutions 183 A. Lorenzi and H. Tanabe: Inverse and direct problems for nonautonomous degenerate inte- grodifferential equations of parabolic type with Dirichlet boundary con- ditions 197 ix Copyright © 2006 Taylor & Francis Group, LLC
- 12. x F. Luterotti, G. Schimperna and U. Stefanelli: Existence results for a phase transition model based on microscopic movements 245 N. Okazawa: Smoothing effects and strong L2 -wellposedness in the complex Ginzburg-Landau equation 265 Copyright © 2006 Taylor & Francis Group, LLC
- 13. Contributors Mohammed Al-Horani Department of Mathematics, University of Jordan, Amman, Jordan horani@ju.edu.jo Valeria Berti Department of Mathematics, University of Bologna, Piazza di Porta S.Donato 5, 40126 Bologna, Italy berti@dm.unibo.it Fabrizio Colombo Department of Mathematics, Polytechnic of Milan, Via Bonardi 9, 20133 Milan, Italy fabcol@mate.polimi.it Mauro Fabrizio Department of Mathematics, University of Bologna, Piazza di Porta S.Donato 5, 40126 Bologna, Italy fabrizio@dm.unibo.it Angelo Favini Department of Mathematics, University of Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy favini@dm.unibo.it Stefania Gatti Department of Mathematics, University of Ferrara, Via Machiavelli 35, Ferrara, Italy stefania.gatti@.unife.it Matthias Geissert Department of Mathematics, Technische Universität Darmstadt, Darmstadt, Germany geissert@mathematik.tu-darmstadt.de Gisle Ruiz Goldstein Department of Mathematical Sciences University of Memphis, Memphis Tennessee 38152 ggoldste@memphis.edu Jerome A. Goldstein Department of Mathematical Sciences University of Memphis, Memphis Tennessee 38152 jgoldste@memphis.edu Maurizio Grasselli Department of Mathematics, Polytechnic of Milan Via Bonardi 9, 20133 Milan, Italy maugra@mate.polimi.it Bérénice Grec Department of Mathematics, Technische Universität Darmstadt, Darmstadt, Germany berenice.grec@web.de xi Copyright © 2006 Taylor & Francis Group, LLC
- 14. xii Davide Guidetti Department of Mathematics, University of Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy guidetti@dm.unibo.it Matthias Hieber Department of Mathematics, Technische Universität Darmstadt, Darmstadt, Germany hieber@mathematik.tu-darmstadt.de Ismail Kombe Mathematics Department, Oklahoma City University 2501 North Blackwelder, Oklahoma City OK 73106-1493, U.S.A. ikombe@okcu.edu Rabah Labbas Laboratoire de Mathématiques, Faculté des Sciences et Techniques, Université du Havre, B.P 540, 76058 Le Havre Cedex, France rabah.labbas@univ-lehavre.fr Alfredo Lorenzi Department of Mathematics, Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italy lorenzi@mat.unimi.it Fabio Luterotti Department of Mathematics, University of Brescia Via Branze 38, 25123 Brescia, Italy luterott@ing.unibs.it Stéphane Maingot Laboratoire de Mathématiques, Faculté des Sciences et Techniques, Université du Havre, B.P 540, 76058 Le Havre Cedex, France rabah.labbas@univ-lehavre.fr Alain Miranville Laboratoire de Mathématiques et Applications, UMR CNRS 6086 - SP2MI, Boulevard Marie et Pierre Curie - Tlport 2 F-86962 Chasseneuil Futuroscope Cedex, France miranv@math.univ-poitiers.fr Noboru Okazawa Department of Mathematics, Science University of Tokyo Wakamiya-cho 26, Shinjuku-ku Tokyo 162-8601, Japan okazawa@ma.kagu.sut.ac.jp Evgeniy Radkevich Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia evrad@land.ru Silvia Romanelli Department of Mathematics, University of Bari Via E. Orabona 4, 70125 Bari, Italy romans@dm.uniba.it Giulio Schimperna Department of Mathematics, University of Pavia Via Ferrata 1, 27100 Pavia, Italy giulio@dimat.unipv.it Ulisse Stefanelli IMATI, Università degli Studi di Pavia Via Ferrata 1, 27100 Pavia, Italy ulisse@imati.cnr.it Copyright © 2006 Taylor & Francis Group, LLC
- 15. xiii Hiroki Tanabe Hirai Sanso 12-13, Takarazuka, 665-0817, Japan h7tanabe@jttk.zaq.ne.jp Vincenzo Vespri Department of Mathematics, Università degli Studi di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy vespri@math.unifi.it Atsushi Yagi Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan yagi@ap.eng.osaka-u.ac.jp Copyright © 2006 Taylor & Francis Group, LLC
- 16. Degenerate first order identification problems in Banach spaces 1 Mohammed Al-Horani and Angelo Favini Abstract We study a first order identification problem in a Banach space. We discuss both the nondegenerate and (mainly) the degenerate case. As a first step, suitable hypotheses on the involved closed linear operators are made in order to obtain unique solvability after reduction to a nondegenerate case; the general case is then handled with the help of new results on convolutions. Various applications to partial differential equations motivate this abstract approach. 1 Introduction In this article we are concerned with an identification problem for first order linear systems extending the theory and methods discussed in [7] and [1]. See also [2] and [9]. Related nonsingular results were obtained in [11] under differ- ent additional conditions even in the regular case. There is a wide literature on inverse problems motivated by applied sciences. We refer to [11] for an extended list of references. Inverse problems for degenerate differential and integrodifferential equations are a new branch of research. Very recent results have been obtained in [7], [5] and [6] relative to identification problems for de- generate integrodifferential equations. Here we treat similar equations without the integral term and this allows us to lower the required regularity in time of the data by one. The singular case for infinitely differentiable semigroups and second order equations in time will be treated in some forthcoming papers. The contents of the paper are as follows. In Section 2 we present the non- singular case, precisely, we consider the problem u0 (t) + Au(t) = f(t)z , 0 ≤ t ≤ τ , u(0) = u0 , Φ[u(t)] = g(t) , 0 ≤ t ≤ τ , 1Work partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (M.I.U.R.), PRIN no. 2004011204, Project Analisi Matematica nei Problemi Inversi and by the University of Bologna, Funds for Selected Research Topics. 1 Copyright © 2006 Taylor & Francis Group, LLC
- 17. 2 M. Al-Horani and A. Favini where −A generates an analytic semigroup in X, X being a Banach space, Φ ∈ X∗ , g ∈ C1 ([0, τ], R), τ > 0 fixed, u0, z ∈ D(A) and the pair (u, f) ∈ C1+θ ([0, τ]; X) × Cθ ([0, τ]; R), θ ∈ (0, 1), is to be found. Here Cθ ([0, τ]; X) denotes the space of all X-valued Hölder-continuous functions on [0, τ] with exponent θ, and C1+θ ([0, τ]; X) = {u ∈ C1 ([0, τ]; X); u0 ∈ Cθ ([0, τ]; X)}. In Section 3 we consider the possibly degenerate problem d dt ((Mu)(t)) + Lu(t) = f(t)z , 0 ≤ t ≤ τ , (Mu)(0) = Mu0 , Φ[Mu(t)] = g(t) , 0 ≤ t ≤ τ , where L, M are two closed linear operators in X with D(L) ⊆ D(M), L being invertible, Φ ∈ X∗ and g ∈ C1+θ ([0, τ]; R), for some θ ∈ (0, 1). In this possibly degenerate problem, M may have no bounded inverse and the pair (u, f) ∈ Cθ ([0, τ]; D(L))×Cθ ([0, τ]; R) is to be found. This problem was solved (see [1]) when λ = 0 is a simple pole for the resolvent (λL + M)−1 . Here we consider this problem under the assumption that M and L act in a reflexive Banach space X with the resolvent estimate kλM(λM + L)−1 kL(X) ≤ C, Re λ ≥ 0 , or the equivalent one kL(λM + L)−1 kL(X) = k(λT + I)−1 kL(X) ≤ C, Re λ ≥ 0 , where T = ML−1 . Reflexivity of X allows to use the representation of X as a direct sum of the null space N(T) and the closure of its range R(T), a consequence of the ergodic theorem (see [13], pp. 216-217). Here, a basic role is played by real interpolation space, see [12]. In Section 4 we give some examples from partial differential equations de- scribing the range of applications of the previous abstract results. 2 The nonsingular case Let X be a Banach space with norm k · kX (sometimes, k · k will be used for the sake of brevity), τ > 0 fixed, u0, z ∈ D(A), where −A is the generator of an analytic semigroup in X, Φ ∈ X∗ and g ∈ C1 ([0, τ], R). We want to find a Copyright © 2006 Taylor & Francis Group, LLC
- 18. Degenerate first order identification problems in Banach spaces 3 pair (u, f) ∈ C1+θ ([0, τ]; X) × Cθ ([0, τ]; R), θ ∈ (0, 1), such that u0 (t) + Au(t) = f(t)z , 0 ≤ t ≤ τ , (2.1) u(0) = u0 , (2.2) Φ[u(t)] = g(t) , 0 ≤ t ≤ τ , (2.3) under the compatibility relation Φ[u0] = g(0) . (2.4) Let us remark that the compatibility relation (2.4) follows from (2.2)-(2.3). To solve our problem we first apply Φ to (2.1) and take equation (2.3) into account; we obtain the following equation in the unknown f(t): g0 (t) + Φ[Au(t)] = f(t)Φ[z] . (2.5) Suppose the condition Φ[z] 6= 0 (2.6) to be satisfied. Then we can write (2.5) under the form: f(t) = 1 Φ[z] {g0 (t) + Φ[Au(t)]} , 0 ≤ t ≤ τ , (2.7) and the solution u of (2.1)-(2.3) is assigned by the formula u(t) = e−tA u0 + Z t 0 e−(t−s)A {g0 (s) + Φ[Au(s)]} Φ[z] z ds = Z t 0 e−(t−s)A Φ[Au(s)] Φ[z] z ds + e−tA u0 + 1 Φ[z] Z t 0 e−(t−s)A g0 (s)z ds . (2.8) Apply the operator A to (2.8) and obtain Au(t) = Z t 0 e−(t−s)A Φ[Au(s)] Φ[z] Az ds + e−tA Au0 + 1 Φ[z] Z t 0 e−(t−s)A g0 (s)Az ds . (2.9) Let Au(t) = v(t); then (2.7) and (2.9) can be written, respectively, as follows: f(t) = 1 Φ[z] {g0 (t) + Φ[v(t)]} , 0 ≤ t ≤ τ , (2.10) v(t) = Z t 0 e−(t−s)A Φ[v(s)] Φ[z] Az ds + e−tA Au0 + 1 Φ[z] Z t 0 e−(t−s)A g0 (s)Az ds . (2.11) Copyright © 2006 Taylor & Francis Group, LLC
- 19. 4 M. Al-Horani and A. Favini Let us introduce the operator S Sw(t) = Z t 0 e−(t−s)A Φ[w(s)] Φ[z] Az ds . Then (2.11) can be written in the form v − Sv = h (2.12) where h(t) = e−tA Au0 + 1 Φ[z] Z t 0 e−(t−s)A g0 (s)Az ds . It is easy to notice that h ∈ C([0, τ]; X). To prove that (2.12) has a unique solution in C([0, τ]; X), it is sufficient to show that Sn is a contraction for some n ∈ N. For this, we note kSv(t)k ≤ M kΦkX∗ |Φ(z)| Z t 0 kv(s)k kAzk ds kS2 v(t)k ≤ M kΦkX∗ |Φ(z)| Z t 0 kTv(s)k kAzk ds ≤ µ M kΦkX∗ kAzk |Φ(z)| ¶2 Z t 0 µZ s 0 kv(σ)k dσ ¶ ds ≤ µ M kΦkX∗ kAzk |Φ(z)| ¶2 Z t 0 (t − σ)kv(σ)k dσ ≤ µ M kΦkX∗ kAzk |Φ(z)| ¶2 kvk∞ t2 2 , where kvk∞ = kvkC([0,τ];X) . Proceeding by induction, we can find the estimate kSn v(t)k ≤ µ M kΦkX∗ kAzk |Φ(z)| ¶n tn n! kvk∞ , which implies that kSn vk∞ ≤ µ M kΦkX∗ kAzk |Φ(z)| τ ¶n 1 n! kvk∞ . Consequently, Sn is a contraction for sufficiently large n. At last notice that f(t) z is then a continuous D(A)-valued function on [0, τ], so that (2.1), (2.2) has in fact a unique strict solution. However, we want to discuss the maximal regularity for the solution v = Au, and for this we need some additional conditions. We now recall that if −A generates a bounded analytic semigroup in X, then the real interpolation space (X, D(A))θ,∞ = DA(θ, ∞) coincides with {x ∈ X; supt>0 t1−θ kAe−tA xk < ∞}, (see [3]). Copyright © 2006 Taylor & Francis Group, LLC
- 20. Degenerate first order identification problems in Banach spaces 5 Consider formula (2.11) and notice that (see [10]) e−tA Au0 ∈ Cθ ([0, τ]; X) if and only if Au0 ∈ DA(θ, ∞) . Moreover, if g ∈ C1+θ ([0, τ]; R) and Az ∈ DA(θ, ∞), then Z t 0 e−(t−s)A g0 (s)Az ds ∈ Cθ ([0, τ]; X) and Z t 0 e−(t−s)A Az Φ[v(s)] ds = ¡ e−tA Az ∗ Φ[v] ¢ (t) ∈ Cθ ([0, τ]; X) . See [7] and [6]. Therefore, if we assume Au0, Az ∈ DA(θ, ∞) , (2.13) then v(t) ∈ Cθ ([0, τ]; X), i.e., Au(t) ∈ Cθ ([0, τ]; X) which implies that f(t) ∈ Cθ ([0, τ]; R). Then there exists a unique solution (u, f) ∈ C1+θ ([0, τ]; X) × Cθ ([0, τ]; R). We summarize our discussion in the following theorem. THEOREM 2.1 Let −A be the generator of an analytic semigroup, Φ ∈ X∗ , u0, z ∈ DA(θ+1, ∞) and g ∈ C1+θ ([0, τ]; R). If Φ[z] 6= 0 and (2.4) holds, then problem (2.1)-(2.3) admits a unique solution (u, f) ∈ [C1+θ ([0, τ]; X) ∩ Cθ ([0, τ]; D(A))] × Cθ ([0, τ]; R). 3 The singular case Consider the possibly degenerate problem Dt(Mu) + Lu = f(t)z , 0 ≤ t ≤ τ , (3.1) (Mu)(0) = Mu0 , (3.2) Φ[Mu(t)] = g(t) , 0 ≤ t ≤ τ , (3.3) where L, M are two closed linear operators with D(L) ⊆ D(M), L being invertible, Φ ∈ X∗ and g ∈ C1+θ ([0, τ]; R) for θ ∈ (0, 1). Here M may have no bounded inverse and the pair (u, f) ∈ C([0, τ]; D(L)) × Cθ ([0, τ]; R), with Mu ∈ C1+θ ([0, τ]; X), is to be determined so that the following compatibility condition must hold: Φ[Mu(0)] = Φ[Mu0] = g(0) . (3.4) Copyright © 2006 Taylor & Francis Group, LLC
- 21. 6 M. Al-Horani and A. Favini Let us assume that the pair (M, L) satisfies the estimate kλM(λM + L)−1 kL(X) ≤ C, Re λ ≥ 0 , (3.5) or the equivalent one kL(λM + L)−1 kL(X) = k(λT + I)−1 kL(X) ≤ C, Re λ ≥ 0 , (3.6) where T = ML−1 . Various concrete examples of this relation can be found in [8]. One may note that λ = 0 is not necessarily a simple pole for (λ + T)−1 , T = ML−1 . Let Lu = v and observe that T = ML−1 ∈ L(X). Then (3.1)-(3.3) can be written as Dt(Tv) + v = f(t)z , 0 ≤ t ≤ τ , (3.7) (Tv)(0) = Tv0 = ML−1 v0 , (3.8) Φ[Tv(t)] = g(t) , 0 ≤ t ≤ τ , (3.9) where v0 = Lu0. Since X is a reflexive Banach space and (3.5) holds, we can represent X as a direct sum (cfr. [8, p. 153], see also [13], pp. 216-217) X = N(T) ⊕ R(T) where N(T) is the null space of T and R(T) is the range of T. Let T̃ = TR(T ) : R(T) → TR(T ) be the restriction of T to R(T). Clearly T̃ is a one to one map from R(T) onto R(T) (T̃ is an abstract potential operator in R(T). Indeed, in view of the assumptions, −T̃−1 generates an analytic semigroup on R(T), (see [8, p. 154]). Finally, let P be the corresponding projection onto N(T) along R(T). We can now prove the following theorem: THEOREM 3.1 Let L, M be two closed linear operators in the reflex- ive Banach space X with D(L) ⊆ D(M), L being invertible, Φ ∈ X∗ and g ∈ C1+θ ([0, τ]; R). Suppose the condition (3.5) to hold with (3.4), too. Then prob- lem (3.1)-(3.3) admits a unique solution (u, f) ∈ Cθ ([0, τ]; D(L))×Cθ ([0, τ]; R) provided that Φ[(I − P)z] 6= 0 , sup t>0 tθ k(tT̃ + 1)−1 yikX < +∞ , i = 1, 2 where y1 = (I − P)Lu0 and y2 = T̃−1 (I − P)z. Proof. Since P is the projection onto N(T) along R(T), it is easy to check that problem (3.7)-(3.9) is equivalent to the couple of problems Copyright © 2006 Taylor & Francis Group, LLC
- 22. Degenerate first order identification problems in Banach spaces 7 DtT̃(I − P)v + (I − P)v = f(t)(I − P)z , 0 ≤ t ≤ τ , (3.10) T̃(I − P)v(0) = T̃(I − P)v0 , (3.11) Φ[T̃(I − P)v(t)] = g(t) , 0 ≤ t ≤ τ , (3.12) and Pv(t) = f(t)Pz . (3.13) Let w = T̃(I −P)v, so that (I −P)v = T̃−1 w, and hence system (3.10)-(3.12) becomes w0 (t) + T̃−1 w = f(t)(I − P)z , 0 ≤ t ≤ τ , (3.14) w(0) = w0 = T̃(I − P)v0 = Tv0 , (3.15) Φ[w(t)] = g(t) , 0 ≤ t ≤ τ . (3.16) Then, according to Theorem 2.1, there exists a unique solution (w, f) ∈ C1+θ ([0, τ]; R(T)) × Cθ ([0, τ]; R) with T̃−1 w ∈ Cθ ([0, τ]; R(T)) to problem (3.14)-(3.16) provided that Φ[(I − P)z] 6= 0 , (I − P)Lu0 , T̃−1 (I − P)z ∈ DT̃ −1 (θ, ∞) . Therefore, (I −P)v ∈ Cθ ([0, τ]; R(T)), Pv ∈ Cθ ([0, τ]; N(T)) and hence there exists a unique solution (u, f) ∈ Cθ ([0, τ]; D(L)) × Cθ ([0, τ]; R) with Mu ∈ C1+θ ([0, τ]; X) to problem (3.1)-(3.3) . ¤ Our next goal is to weaken the assumptions on the data in the Theorems 1 and 2. To this end we again suppose −A to be the generator of an analytic semigroup in X of negative type, i.e., ke−tA k ≤ ce−ωt , t ≥ 0, where c, ω > 0, g ∈ C1+θ ([0, τ]; R), but we take u0 ∈ DA(θ + 1; X), z ∈ DA(θ0, ∞), where 0 < θ < θ0 < 1. Our goal is to find a pair (u, f) ∈ C1 ([0, τ]; X) × C([0, τ]; R), Au ∈ Cθ ([0, τ]; X) such that equations (2.1)-(2.3) hold under the compatibil- ity relation (2.4). THEOREM 3.2 Let −A be a generator of an analytic semigroup in X of positive type, 0 < θ < θ0 < 1, g ∈ C1+θ ([0, τ]; R), u0 ∈ DA(θ + 1, ∞), z ∈ DA(θ0, ∞). If, in addition, (2.4), (2.6) hold, then problem (2.1)-(2.3) has a unique solution (u, f) ∈ Cθ ([0, τ], D(A)) × Cθ ([0, τ]; R). Proof. Recall (see [10, p. 145]) that if u0 ∈ D(A), f ∈ C([0, τ]; R), z ∈ DA(θ0, ∞), then problem (2.1)-(2.2) has a unique strict solution. Moreover, if u0 ∈ DA(θ + 1; X), then the solution u to (2.1)-(2.2) has the maximal regu- larity u0 , Au ∈ C([0, τ]; X) ∩ B([0, τ]; DA(θ0, ∞)), where B([0, τ]; Y ) denotes Copyright © 2006 Taylor & Francis Group, LLC
- 23. 8 M. Al-Horani and A. Favini the space of all bounded functions from [0, τ] into the Banach space Y . In addition Au ∈ Cθ ([0, τ]; X). In order to prove our statement, we need to study suitably the properties of the function u and to use carefully some properties of the convolution operator and real interpolation spaces. One readily sees that u satisfies Au(t) = Z t 0 Φ[Au(s)] Φ[z] Ae−(t−s)A z ds + e−tA Au0 + 1 Φ[z] Z t 0 A e−(t−s)A z g0 (s) ds so that v(t) = Au(t) must satisfy v(t) = Z t 0 Ae−(t−s)A z Φ[v(s)] Φ[z] ds + e−tA Au0 + 1 Φ[z] Z t 0 A e−(t−s)A z g0 (s) ds . Let us introduce the operator S : C([0, τ]; X) → C([0, τ]; X) by (Sw)(t) = Z t 0 Ae−(t−s)A z Φ[w(s)] Φ[z] ds . Since z ∈ DA(θ0, ∞), i.e., kAe−tA zk ≤ c t1−θ0 , t > 0 , we deduce kSw(t)k ≤ c Z t 0 kΦkX∗ kzkθ0, ∞ kw(s)k (t − s)1−θ0 ds , kS2 w(t)k ≤ [ckΦkX∗ kzkθ0, ∞] Z t 0 kSw(s)k (t − s)1−θ0 ds ≤ [ckΦkX∗ kzkθ0, ∞]2 Z t 0 ds (t − s)1−θ0 Z s 0 kw(σ)k (s − σ)1−θ0 dσ = [ckΦkX∗ kzkθ0, ∞]2 Z t 0 µZ t σ ds (t − s)1−θ0 (s − σ)1−θ0 ¶ kw(σ)k dσ = c2 1 ·Z 1 0 dη (1 − η)1−θ0 η1−θ0 ¸ (t − σ)1−2(1−θ0) kw(σ)k dσ , where c1 = ckΦkX∗ kzkθ0, ∞, k · kDA(θ0,∞) denoting the norm in DA(θ0, ∞). Recall that B(p, q) = Z 1 0 (1 − η)p−1 ηq−1 dη = Γ(p) Γ(q) Γ(p + q) . Copyright © 2006 Taylor & Francis Group, LLC
- 24. Degenerate first order identification problems in Banach spaces 9 Then kS3 w(t)k ≤ c3 1 Z 1 0 dη (1 − η)1−θ0 η1−θ0 Z 1 0 dη (1 − η)1−θ0 η2(1−θ0)−1 × Z 1 0 (t − σ)2−3(1−θ0) kw(σ)k dσ ≤ c3 1 B(θ0, θ0) B(θ0, 2θ0) Z 1 0 (t − σ)2−3(1−θ0) kw(σ)k dσ ≤ c3 1 Γ(θ0)3 Γ(3θ0) t3θ0 3θ0 kwkC([0,t];X) . By induction, we easily verify that kSn w(t)k ≤ cn 1 Γ(θ0)n Γ(nθ0) tnθ0 nθ0 kwkC([0,t];X) . Since n p Γ(nθ0) → ∞ as n → ∞, we conclude that the operator S has spectral radius equal to 0. On the other hand, since z ∈ DA(θ0, ∞), θ0 > θ, and g0 ∈ Cθ ([0, τ]; R), we deduce by [6] (Lemma 3.3) that the convolution Z t 0 g0 (s)Ae−(t−s)A z ds belongs to Cθ ([0, τ]; X). Moreover, since Au0 ∈ DA(θ, ∞), e−tA Au0 ∈ Cθ ([0, τ]; X). It follows that equation (2.12), i.e., v − Sv = h , with h(t) = e−tA Au0 + 1 Φ[z] Z t 0 Ae−(t−s)A z g0 (s) ds has a unique solution v ∈ C([0, τ]; X). In order to obtain more regularity for v, we use Lemma 3.3 in [6] (see also [7]) again. To this end, we introduce the following Lp -spaces related to any positive constant δ: Lp δ((0, τ); X) = © u : (0, τ) → X : e−tδ u ∈ Lp ((0, τ); X) ª , endowed with the norms kukδ,0,p = ke−tδ ukLp((0,τ);X). Moreover, kgkδ,θ,∞ = ke−tδ gkCθ([0,τ];X) . Lemma 3.3 in [6] establishes that, in fact, if z ∈ DA(θ0, ∞)), 0 < θ < θ0 < 1, then ° ° ° Z t 0 Ae−(t−s)A z Φ[v(s)] ds ° ° ° δ,θ,∞ ≤ c δ−θ0+θ+1/p kΦ[v(.)]kδ,0,p Copyright © 2006 Taylor & Francis Group, LLC
- 25. 10 M. Al-Horani and A. Favini provided that (θ0 − θ)−1 < p. Now, Z t 0 |Φ[v(t)]|p e−δpt dt ≤ kΦkp X∗ kvkp Lp δ ((0,τ);X) ≤ τ kΦkp X∗ kvkp δ,θ,∞ . Choose δ suitably large and recall that h ∈ Cθ ([0, τ]; X). Then the norm of S as an operator from Cθ ([0, τ]; X) (with norm k · kδ,θ,∞) into itself is less than 1, so that we can deduce that the solution v = Au has the regularity Cθ ([0, τ]; X), as desired. ¤ As a consequence, Theorem 3.1 has the following improvement. THEOREM 3.3 Let L, M be two closed linear operators in the reflexive Banach space X with D(L) ⊆ D(M), L being invertible, Φ ∈ X∗ and g ∈ C1+θ ([0, τ]; R). Suppose (3.4), (3.5) to hold. If 0 < θ < θ0 < 1 and Φ[(I−P)z] 6= 0 , sup t>0 tθ0 k(tT +1)−1 (I−P)zkX < +∞, sup t>0 tθ k(tT + 1)−1 (I − P)Lu0kX < +∞, then problem (3.1)-(3.3) admits a unique solution (u, f) ∈ Cθ ([0, τ]; D(L))×Cθ ([0, τ]; R) with Mu ∈ C1+θ ([0, τ]; X). 4 Applications In this section we show that our abstract results can be applied to some con- crete identification problems. For further examples for which the theory works we refer to [8]. Problem 1. Consider the following identification problem related to a bounded region Ω in Rn with a smooth boundary ∂Ω Dtu(x, t) = n X i,j=1 Dxi (aij(x)Dxj u(x, t)) + f(t)v(x) , (x, t) ∈ Ω × [0, τ] , u(x, t) = 0 , ∀ (x, t) ∈ ∂Ω × [0, τ] , u(x, 0) = u0(x) , x ∈ Ω , Φ[u(x, t)] = Z Ω η(x)u(x, t) dx = g(t) , ∀ t ∈ [0, τ] , where the coefficients aij enjoy the properties aij ∈ C(Ω) , aij = aji , i, j = 1, 2, ..., n n X i,j=1 aij(x) ξi ξj ≥ c0|ξ|2 ∀ x ∈ Ω , ∀ ξ ∈ Rn , Copyright © 2006 Taylor & Francis Group, LLC
- 26. Degenerate first order identification problems in Banach spaces 11 c0 being a positive constant. Moreover, g ∈ C1 ([0, τ]; R). We take Au = − n X i,j=1 Dxi (aijDxj u) , D(A) = W2,p (Ω) ∩ W1,p 0 (Ω) , where 1 < p < +∞ is assumed. Concerning η, we suppose η ∈ Lq (Ω), where 1/p + 1/q = 1. As it is well known, −A generates an analytic semigroup in Lp (Ω) and thus we can apply Theorem 3.2 provided that u0 ∈ DA(θ + 1; ∞), i.e., Au0 ∈ DA(θ, ∞), v ∈ DA(θ0; ∞), 0 < θ < θ0 < 1. On the other hand, the interpolation spaces DA(θ, ∞) are well characterized. Then our problem admits a unique solution (u, f) ∈ Cθ ([0, τ]; W2,p (Ω) ∩ W1,p 0 (Ω)) × Cθ ([0, τ]; R), if g ∈ C1+θ ([0, τ]; R), g(0) = R Ω η(x) u0(x) dx and R Ω η(x) v(x) dx 6= 0. Problem 2. Let Ω be a bounded region in Rn with a smooth boundary ∂Ω. Let us consider the identification problem Dtu(x, t) = n X i,j=1 Dxi (aij(x)Dxj u(x, t)) + f(t)v(x) , (x, t) ∈ Ω × [0, τ] , u(x, t) = 0 , (x, t) ∈ ∂Ω × [0, τ] , u(x, 0) = u0(x) , x ∈ Ω , Φ[u(x, t)] = u(x, t) = g(t) , t ∈ [0, τ] , where x ∈ Ω is fixed, and the pair (f, u) is the unknown. Here we take X = C0(Ω) = © u ∈ C(Ω), u(x) = 0 ∀ x ∈ ∂Ω ª , endowed with the sup norm kukX = kuk∞. If the coefficients aij are assumed as in Problem 1, and Au = − n X i,j=1 Dxi (aij(x)Dxj u(x)) , D(A) = © u ∈ C0(Ω) ; Au ∈ C0(Ω) ª , then −A generates an analytic semigroup in X. The interpolation spaces DA(θ; ∞) have no simple characterization, in view of the boundary condi- tions imposed to Au. Hence we notice that Theorem 3.2 applies provided that u0 ∈ D(A2 ) and v0 ∈ D(A), 0 < θ < 1, g ∈ C1+θ ([0, τ]; R), u0(x) = g(0) and v(x) 6= 0. Notice that we could develop a corresponding result to Theorem 3.2 related to operators A with a nondense domain, but this is not so simple and the Copyright © 2006 Taylor & Francis Group, LLC
- 27. 12 M. Al-Horani and A. Favini problem will be handled elsewhere. Problem 3. Let us consider the following identification problem on a bounded region Ω in R, n ≥ 1, with a smooth boundary ∂Ω: Dt[m(x)u] = ∆u + f(t)w(x), (x, t) ∈ Ω × [0, τ] , (4.1) u = 0 on ∂Ω × [0, τ] , (4.2) (mu)(x, 0) = m(x)u0(x) , x ∈ Ω , (4.3) Z Ω η(x) (mu)(x, t) dx = g(t) , ∀t ∈ [0, τ] , (4.4) where m ∈ L∞ (Ω), ∆ : H1 0 (Ω) :→ H−1 (Ω) is the Laplacian, u0 ∈ H1 0 (Ω), w ∈ H−1 (Ω), η ∈ H1 0 (Ω), g ∈ C1+θ ([0, τ]; R), 0 < θ < 1, and the pair (u, f) ∈ Cθ ([0, τ]; H1 0 (Ω))×Cθ ([0, τ]; R) is the unknown. Of course, the integral in (4.4) stands for the duality between H−1 (Ω) and H1 0 (Ω). Theorem 3.3 applies with X = H−1 (Ω), see [8, p. 75]. We deduce that if g(0) = R Ω η(x) m(x)u0(x) dx, w(x) = m(x)ζ(x) for some ζ ∈ H1 0 (Ω), R Ω η(x) m(x)ζ(x) dx 6= 0 and (∆u0)(x) = m(x)ζ1(x) for some ζ1 ∈ H1 0 (Ω), then problem (4.1)-(4.4) has a unique solution (u, f) ∈ Cθ ([0, τ]; H1 0 (Ω))×Cθ ([0, τ]; R), mu ∈ C1+θ ([0, τ]; H−1 (Ω)). Problem 4. Consider the degenerate parabolic equation Dtv = ∆[a(x)v] + f(t)w(x) , (x, t) ∈ Ω × [0, τ] , (4.5) together with the initial-boundary conditions a(x)v(x, t) = 0 , (x, t) ∈ ∂Ω × [0, τ] , (4.6) v(x, 0) = v0(x) , x ∈ Ω , (4.7) and the additional information Z Ω η(x)v(x, t) dx = g(t) , t ∈ [0, τ] . (4.8) Here Ω is a bounded region in Rn , n ≥ 1, with a smooth boundary ∂Ω, a(x) ≥ 0 on Ω and a(x) > 0 almost everywhere in Ω is a given function in L∞ (Ω), w ∈ H−1 (Ω), v0 ∈ H1 0 (Ω), η ∈ H1 0 (Ω), g is a real valued-function on [0, τ], at least continuous, and the pair (v, f) is the unknown. Of course, we shall see that functions w, v0 and g need much more regularity. Call a(x)v = u. Then, if m(x) = a(x)−1 and u0(x) = a(x)v0(x) we obtain a system like (4.1)-(4.4). Let M be the multiplication operator by m from H1 0 (Ω) into H−1 (Ω) and let L = −∆ be endowed with Dirichlet condition, that is, L : H1 0 (Ω) → H−1 (Ω), as previously. Take X = H−1 (Ω). Then it is seen in [8, p. 81] that (3.5) holds if Copyright © 2006 Taylor & Francis Group, LLC
- 28. Degenerate first order identification problems in Banach spaces 13 i) a−1 ∈ L1 (Ω), when n = 1, ii) a−1 ∈ Lr (Ω) with some r > 1, when n = 2, iii) a−1 ∈ L n 2 (Ω), when n ≥ 3. In order to apply Theorem 3.3 we suppose u0(x) = a(x)v0(x) ∈ H1 0 (Ω). As- sumption (3.4) reads Z Ω η(x)v0(x) dx = Z Ω η(x) u0(x) a(x) dx = g(0) . Take g ∈ C1+θ ([0, τ]; R), 0 < θ < 1. Since R(T) = R((1/a)∆−1 ), let aw = ζ ∈ H1 0 (Ω), a∆u0 = a∆(av0) = ζ1 ∈ H1 0 (Ω), R Ω η(x)ζ(x) a(x) dx 6= 0. Then we conclude that there exists a unique pair (v, f) satisfying (4.5)-(4.8) with regularity ∆(av) ∈ Cθ ([0, τ]; H−1 (Ω)) , v ∈ C1+θ ([0, τ]; H−1 (Ω)) . In many applications a(x) is comparable with some power of the distance of x to the boundary ∂Ω and hence the assumptions depend heavily from the geometrical properties of the domain Ω. For example, if Ω = (−1, 1), a(x) = (1 − x2 )α or a(x) = (1 − x)α (1 + x)β , 0 < α, β < 1 are allowed. More generally, in Rn , one can handle a(x) = (1−kxk2 )α for some α > 0 with Ω = {x ∈ Rn : kxk < r}, r > 0. Precisely, if n = 2, then 0 < α < 1, if n ≥ 3 then 0 < α < 2/n. Problem 5. Let us consider another degenerate parabolic equation, precisely Dtv = x(1 − x)D2 xv + f(t)w(x), (x, t) ∈ (0, 1) × (0, τ), (4.9) with the initial condition v(x, 0) = v0(x), x ∈ (0, 1), (4.10) but with a Wentzell boundary condition (basic in probability theory and in applied sciences) lim x→0 x(1 − x)D2 xv(x, t) = 0, t ∈ (0, 1). We add the additional information: Φ[v(·, t)] = v(x̄, t) = g(t), t ∈ [0, τ], (4.11) where x̄ ∈ (0, 1) is fixed. Here we take X = H1 (0, 1), with the norm kuk2 X := kuk2 L2(0,1) + ku0 k2 L2(0,1) + |u(0)|2 + |u(1)|2 . Copyright © 2006 Taylor & Francis Group, LLC
- 29. 14 M. Al-Horani and A. Favini Introduce operator (A, D(A)) defined by D(A) := © u ∈ H1 (0, 1); u00 ∈ L1 loc(0, 1) and x(1 − x)u00 ∈ H1 0 (0, 1) ª , Au = −x(1 − x)u00 , u ∈ D(A). Then −A generates an analytic semigroup in H1 (0, 1), see [8, pp. 249-250], [4]. So, we can apply Theorem 3.2; therefore, if 0 < θ < θ0 < 1, g ∈ C1+θ ([0, τ]; R), v0 ∈ DA(θ + 1, ∞), w ∈ DA(θ0, ∞) (in particular, v0 ∈ D(A2 ), w ∈ D(A)), g(0) = v0(x̄), w(x̄) 6= 0, then there exists a unique pair (v, f) ∈ Cθ ([0, τ]; D(A)) × Cθ ([0, τ]; R) satisfying (4.9)–(4.11) and Dtv ∈ Cθ ([0, τ]; H1 (0, 1)). Of course, general functionals Φ in the dual space H(0, 1)∗ could be treated. References [1] M.H. Al-Horani: An identification problem for some degenerate differ- ential equations, Le Matematiche, 57, 217–227, 2002. [2] A. Asanov and E.R. Atamanov: Nonclassical and inverse problems for pseudoparabolic equations, 1st ed., VSP, Utrecht, 1997. [3] G. Da Prato: Abstract differential equations, maximal regularity, and linearization, Proceedings Symp. Pure Math., 45, 359–370, 1986. [4] A. Favini, J.A. Goldstein and S. Romanelli: An analytic semigroup as- sociated to a degenerate evolution equation, Stochastic processes and Functional Analysis , M. Dekker, New York, 88–100, 1997. [5] A. Favini and A. Lorenzi: Identification problems for singular integro- differential equations of parabolic type II, Nonlinear Analysis T.M.A., 56, 879–904, 2004. [6] A. Favini and A. Lorenzi: Singular integro-differential equations of parabolic type and inverse problems, Math. Models and Methods in Applied Sciences, 13, 1745–1766, 2003. [7] A. Favini and A. Lorenzi: Identification problems in singular integro- differential equations of parabolic type I, Dynamics of continuous, dis- crete, and impulsive systems, series A: Mathematical Analysis, 12, 303– 328, 2005. [8] A. Favini and A. Yagi: Degenerate differential equations in Banach spaces, 1st ed., Dekker, New York, 1999. Copyright © 2006 Taylor & Francis Group, LLC
- 30. Degenerate first order identification problems in Banach spaces 15 [9] A. Lorenzi: Introduction to identification problem via Functional Anal- ysis, 1st ed., VSP, Utrecht, 2001. [10] A. Lunardi: Analytic semigroups and optimal regularity in parabolic problems, 1st ed., Birkhäuser, Basel, 1995. [11] A.I. Prilepko, D.G. Orlovsky and I.A. Vasin: Methods for solving inverse problems in Mathematical Physics, 1st ed., M. Dekker, New York, 2000. [12] H. Triebel: Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, 1978. [13] K. Yosida: Functional Analysis, 6th ed., Springer-Verlag, Berlin, 1980. Mohammed Al-Horani Angelo Favini Department of Mathematics Department of Mathematics University of Jordan University of Bologna Amman Bologna Jordan Italy horani@ju.edu.jo favini@dm.unibo.it Copyright © 2006 Taylor & Francis Group, LLC
- 31. A nonisothermal dynamical Ginzburg- Landau model of superconductivity. Existence and uniqueness theorems Valeria Berti and Mauro Fabrizio Abstract A time-dependent Ginzburg-Landau model describing superconduc- tivity with thermal effects into account is studied. For this problem, the absolute temperature is a state variable for the superconductor. Therefore, we modify the classical time-dependent Ginzburg-Landau equations by including the tempera- ture dependence. Finally, the existence and the uniqueness of this nonisothermal Ginzburg-Landau system is proved. 1 Introduction There are some materials which exhibit a sharp rise in conductivity at tem- peratures of the order of 5o K and currents started in these metals persist for a long time. This is the essence of superconductivity which was discovered by Kamerlingh Onnes in 1911 (cf.[1], [2], [5], [6], [7], [15], [16], [17]). He observed that the electrical resistance of various metals such as mercury, lead and tin disappeared completely in a small temperature range at a critical tempera- ture Tc which is characteristic of the material. The complete disappearance of resistance is most sensitively demonstrated by experiments with persistent currents in superconducting rings. In 1914 Kamerlingh Onnes discovered that the resistance of a superconduc- tor could be restored to its value in the normal state by the application of a large magnetic field. About ten years later, Tuyn and Kamerlingh Onnes performed experiments on cylindrical specimens, with the axis along the di- rection of the applied field, and showed that the resistance increases rapidly in a very small field interval. The value Hc of H at which the jump in resistance occurs is termed threshold field. This value Hc is zero at T = Tc and increases as the T is lowered below Tc. In the first part of the paper we recall the London model of superconduc- tivity, the traditional Ginzburg-Landau theory and the dynamical extension presented by Gor’kov and Éliashberg [11]. These models are able to describe the phase transition which occurs in a metal or alloy superconductor, when 17 Copyright © 2006 Taylor & Francis Group, LLC
- 32. 18 V. Berti and M. Fabrizio the temperature is constant, but under the critical value Tc. In these hy- potheses the material will pass from the normal to the superconductor state if the magnetic field is lowered under the threshold field Hc. In this paper we present a generalization of the Ginzburg-Landau theory which considers vari- able both the magnetic field and the temperature. Also this model describes the phenomenon of superconductivity as a second-order phase transition. The two phases are represented in the plane H − T by two regions divided by a parabola. The second part of the paper is devoted to the proof of existence and unique- ness of the solutions of the nonisothermal Ginzburg-Landau equations. In a previous paper ([3]) we have shown the well posedness of the problem ob- tained by neglecting the magnetic field. In this paper, the existence and the uniqueness of the solutions of the nonisothermal Ginzburg-Landau equations are proved after formulating the problem by means of the classical state vari- ables (ψ, A, φ) together with the temperature u = T/Tc. The existence of the weak solutions in a bounded time interval is established by applying the Galerkin’s technique. Then, by means of energy estimates we obtain the ex- istence of global solutions in time. Finally, we prove further regularity and uniqueness of the solutions. 2 Superconductivity and London theory Until 1933 the magnetic properties of a superconductor were tacitly assumed to be a consequence of the property of infinite conductivity. Meissner and Ochsenfeld checked experimentally this assumption and found that such is not the case. They observed the behavior of a cylinder in an applied uniform magnetic field. When the temperature is above the critical value Tc, the sample is in the normal state and the internal magnetic field is equal to the external magnetic field. If the cylinder is cooled through Tc, the magnetic field inside the sample is expelled, showing that a superconducting material exhibits a perfect diamagnetism (Meissner effect). The phenomenological theory of the brothers Heinz and Fritz London, de- veloped in 1935 soon after the discovery of the Meissner effect, is based on the diamagnetic approach in that it gives a unique relation between current and magnetic field. At the same time it is closely related to the infinite conductiv- ity approach in that the allowed current distributions represent a particular class of solutions for electron motion in the absence of scattering. In the London theory the electrons of a superconducting material are di- vided in normal (as the electrons in a normal material scatter and suffer resistance to their motion) and superconducting (they cross the metal with- out suffering any resistance). Below the critical temperature Tc, the current Copyright © 2006 Taylor & Francis Group, LLC
- 33. A nonisothermal dynamical Ginzburg-Landau model 19 consists of superconducting electrons and normal electrons. Above the critical temperature only normal electrons occur. Accordingly we write the current density as the sum of a normal and superconducting part, i.e., J = Jn + Js. The normal density current Jn is required to satisfy Ohm’s law, namely Jn = σE, (2.1) σ being the conductivity of normal electrons. In the London theory, the behav- ior of Js is derived through a corpuscular scheme. Since the superconducting electrons suffer no resistance, their motion in the electric field E is governed by mv̇s = −qE where m, −q, vs are the mass, the charge and the velocity of the supercon- ducting electrons. Let ns be the density of superconducting electrons so that Js = −nsqvs. Multiplication by −nsq/m and the assumption that ns is con- stant yield J̇s = nsq2 m E. (2.2) Assume further that the superconductor is diamagnetic and that time vari- ations are slow enough that the displacement current is negligible. Maxwell’s equations become Ḃ = −∇×E, ∇×B = µ0Js. Comparison gives Ḃ = − m nsq2 ∇×J̇s whence Ḃ = −α∇×(∇×Ḃ) where α = m/(µ0nsq2 ). The usual identity ∇×(∇×) = ∇(∇·) − ∆ and the divergence-free condition of B give ∆Ḃ = 1 α Ḃ. Appropriate initial values of B and Js and an integration in time allow B and Js to satisfy the equations ∆B = 1 α B, (2.3) B = −µ0α∇×Js. (2.4) Equations (2.1) through (2.4) are the basic relations of the London theory. Copyright © 2006 Taylor & Francis Group, LLC
- 34. 20 V. Berti and M. Fabrizio Take the curl of (2.1) and compare with (2.4) and the induction law; we have α∇×J = − B µ0 − σαḂ. Similarly, take the time derivative of (2.1) and compare with (2.2) to have αJ̇ = 1 µ0 E + ασĖ. These equations have the advantage that only the total current density J occurs rather than Jn and Js. This is appreciated if the separation of the total current J into the two components Jn, Js looks somewhat artificial. Equations (2.3) and (2.4) can be used to evaluate the magnetic induction (field) in a superconductor. If, for simplicity, B is allowed to depend on a single Cartesian coordinate, x say, then by (2.3) the only bounded solution as x ≥ 0 is B(x) = B(0) exp(−x/ √ α). This result shows that, roughly, B penetrates in the half-space x ≥ 0 of a distance √ α = p m/(µ0nsq2). That is why the quantity λL = m µ0nsq2 is called London penetration depth. This implies that a magnetic field is ex- ponentially screened from the interior of a sample with penetration depth λL, i.e., the Meissner effect. 3 Ginzburg-Landau theory The Ginzburg-Landau theory [10] deals with the transition of a material from a normal state to a superconducting state. If a magnetic field occurs then the transition involves a latent heat which means that the transition is of the first order. If, instead, the magnetic field is zero the transition is associated with a jump of the specific heat and no latent heat (second-order transition). Landau [14] argued that a second-order transition induces a sudden change in the sym- metry of the material and suggested that the symmetry can be measured by a complex-valued parameter ψ, called order parameter. The physical meaning of ψ is specified by saying that |ψ|2 is the number density, ns, of superconducting electrons. Hence ψ = 0 means that the material is in the normal state, T > Tc, while |ψ| = 1 corresponds to the state of a perfect superconductor (T = 0). There must exist a relation between ψ and the absolute temperature T and this occurs through the free energy e. Incidentally, at first Gorter and Casimir Copyright © 2006 Taylor & Francis Group, LLC
- 35. A nonisothermal dynamical Ginzburg-Landau model 21 [12] elaborated a thermodynamic potential with a real-valued order parame- ter. Later, Ginzburg and Landau argued that the order parameter should be complex-valued so as to make the theory gauge-invariant. With a zero magnetic field, at constant pressure and around the critical temperature Tc the free energy e0 is written as e0 = −a(T)|ψ|2 + 1 2 b(T)|ψ|4 ; (3.1) higher-order terms in |ψ|2 are neglected which means that the model is valid around the critical temperature Tc for small values of |ψ|. If a magnetic field occurs then the free energy of the material is given by Z Ω e(ψ, T, H)dv = Z Ω [e0(ψ, T) + 1 2 µH2 + 1 2m | − i~∇ψ − qAψ|2 ]dv (3.2) where ~ is Planck’s constant and A is the vector potential associated to H, i.e., µH = ∇ × A. The free energy (3.2) turns out to be gauge-invariant. Assume that the free energy is stationary (extremum) at equilibrium. Regard T as fixed, which means that quasi-static processes are considered whereby Js = ∇×H. The corresponding Euler-Lagrange equations, for the unknowns ψ and A, are 1 2m (i~∇ + qA)2 ψ − aψ + b|ψ|2 ψ = 0, (3.3) Js = −i ~q 2m (ψ̄∇ψ − ψ∇ψ̄) − q2 m |ψ|2 A. (3.4) Examine the consequences of (3.3)–(3.4). The boundary condition takes the form (−i~∇ψ − qAψ) · n ¯ ¯ ∂Ω = 0. By (3.4) this implies that Js · n ¯ ¯ ∂Ω = 0. Also, letting ψ = |ψ| exp(iθ), we obtain from (3.4) that Js = − ~q m |ψ|2 ∇θ − q2 m |ψ|2 A = −Λ−1 ( ~ q ∇θ + A). (3.5) Hence the London equation ∇×(ΛJs) = −B (3.6) follows. To make the theory apparently gauge-invariant, we express the free energy in terms of Js rather than of A. As shown in [7], §3.1, we have |i~∇ψ + qAψ|2 = ~2 (∇|ψ|)2 + |ψ|2 (~∇θ + qA)2 = ~2 (∇|ψ|)2 + ΛJ2 s. Copyright © 2006 Taylor & Francis Group, LLC
- 36. 22 V. Berti and M. Fabrizio Hence we can write the free energy (3.2) as a functional of f = |ψ| and T, H in the form Z Ω e(f, T, H)dv = Z Ω h − a(T)f2 + 1 2 b(T)f4 + 1 2 µH2 + ~2 2m (∇f)2 − 1 2 ΛJ2 s i dv, (3.7) the sign before ΛJ2 s arising from the Legendre transformation between A and H. The term ~2 (∇f)2 /2m represents the energy density associated with the interaction between the superconducting phase and the normal phase. As is the case in Ginzburg-Landau theory, we restrict attention to time- independent processes where Js = J = ∇×H. Hence the functional (3.7) is stationary with respect to f and H, with H×n fixed at the boundary ∂Ω, if the Euler-Lagrange equations − ~2 2m ∆f + m 2q2f3 J2 s − af + bf3 = 0 (3.8) µH = −∇×ΛJs (3.9) hold together with the boundary condition ∇f · n ¯ ¯ ∂Ω = 0. Equation (3.9) coincides with (3.6) and hence with the Ginzburg-Landau equa- tion (3.5) or (3.4). Also, equation (3.8) reduces to equation (3.3) when the phase θ of ψ is chosen to be zero as is the case for the system (3.8), (3.9). Since the vector potential A is a nonmeasurable quantity, equation (3.9) may seem more convenient than (3.4) as long as the relation ∇×ΛJs = −B may be preferable to (3.5). 4 Quasi-steady model Starting from the BCS theory of superconductivity, Schmid ([18]) and Gor’kov & Éliashberg ([11]) have elaborated a generalization to the dynamic case of the Ginzburg-Landau theory within the approximation that the temperature T is near the transition value Tc. They consider the variables ψ, A and the electrical potential φ which, together with the vector potential A, is subject to the equations ∇×A = B , E = −Ȧ + ∇φ. (4.1) Copyright © 2006 Taylor & Francis Group, LLC
- 37. A nonisothermal dynamical Ginzburg-Landau model 23 By adhering to [9] we complete the quasi-steady model of superconductivity through the equations γ(ψ̇ − i q ~ φψ) = − 1 2m (i~∇ + qA)2 + aψ − b|ψ|2 ψ, σ(Ȧ − ∇φ) = −∇×∇×A + Js, (4.2) Js = − i~q 2m (ψ̄∇ψ − ψ∇ψ̄) − q2 m |ψ|2 A, where γ is an appropriate relaxation coefficient. The system of equations must be invariant under a gauge transformation (ψ, A, φ) ←→ (ψei(q/~)χ , A + ∇χ, φ + χ̇) where the gauge χ is an arbitrary smooth function of (x, t). Among the possible gauges we mention the London gauge ∇ · A = 0 , A · n|∂Ω = 0 , the Lorentz gauge ∇ · A = −φ and the zero-electrical potential gauge φ = 0. Reference [13] investigates these gauges and shows that the condition φ = 0 is incompatible with the London gauge ∇ · A = 0. The system (4.2) is associated with the initial conditions ψ(x, 0) = ψ0(x) , A(x, 0) = A0(x). (4.3) Equation (4.2)2 follows from the Maxwell equation ∇×H = Js + Jn + εĖ by disregarding the derivative Ė and letting Jn = σE. That is why the problem (4.2) is called quasi-steady. Moreover, by letting ψ = f exp(iθ), from (4.2)1, we deduce the evolution equation for the variable f. In terms of the observable variables f, ps, H, E, the system (4.2) can be written in the form γ ˙ f = ~2 2m ∇2 f − q2 2m p2 sf + af − bf3 (4.4) ∇×ps = −µH (4.5) ∇×H = Λ−1 (f)ps + σE (4.6) ∇×E = −µḢ (4.7) Copyright © 2006 Taylor & Francis Group, LLC
- 38. 24 V. Berti and M. Fabrizio along with the boundary conditions ∇f · n|∂Ω = 0, H × n|∂Ω = g , ps · n|∂Ω = 0 (4.8) and the initial conditions f(x, 0) = f0(x) , H(x, 0) = H0(x). (4.9) Observe that by (4.5) and (4.7) we have ṗs = m q v̇s = E − ∇φs. This result can be viewed as the Euler equation for a nonviscous electronic liquid (see [15]), where the scalar function φs represents the thermodynamic potential per electron. The previous relation allows the quasi-steady problem (4.4)–(4.7) to be written as γ ˙ f = ~2 2m ∆f − q2 2m p2 sf + af − bf3 , 1 µ ∇×∇×ps = −Λ−1 (f)ps − σṗs − σ∇φs. Moreover (4.6) provides ∇ · (Λ−1 (f)ps) = ∇ · Js = −∇ · (σE) = −σρ. (4.10) In the theory of Gor’kov and Éliashberg [11], which is based on the system (4.2), the function φs is assumed to depend on f and on the total electron density ρ in the form φs = Λ(f)ρ. (4.11) The comparison of (4.10) and (4.11) gives ∇ · (Λ−1 (f)ps) = −σΛ−1 (f)φs. (4.12) 5 Phase transition in superconductivity with thermal effects We present a generalization of the model which describes the phase transition in superconductivity without neglecting thermal effects. The main assumption is that the phase transition is of second order and that the effects due to the variation of the temperature are like the ones shown by varying the magnetic field. In this sense the temperature T can be considered as the dual variable of the magnetic field H. Copyright © 2006 Taylor & Francis Group, LLC
- 39. A nonisothermal dynamical Ginzburg-Landau model 25 In order to justify the model here examined, we consider the expression of Gauss free energy in terms of the variables (ψ, T, A) E(ψ, u, A) = Z Ω · −a(T)|ψ|2 + 1 2 b(T)|ψ|4 + 1 2µ |∇ × A|2 + 1 2m |i~∇ψ + qAψ|2 ¸ dv. Following [8] and [19] we consider the linear approximation of a(T) in a neigh- borhood of the critical temperature, namely a(T) = −a0 µ T Tc − 1 ¶ = −a0(u − 1), where u = T Tc > 0. Finally, we suppose constant the coefficient b(T). By means of the temperature u, the critical value uc is now given by uc = 1, while the domain of definition is R+ . Under these hypotheses the free energy takes the following form E(ψ, u, A) = Z Ω h a0(u − 1)|ψ|2 + b0 2 |ψ|4 + 1 2µ |∇ × A|2 + 1 2m |i~∇ψ + qAψ|2 i dv (5.1) which as a function of f, u, H can be written as E(f, u, H) = Z Ω h a0(u − 1)f2 + b0 2 f4 + µ 2 H2 + ~2 2m |∇f|2 + 1 2 Λ(f)|∇ × H|2 i dv. When we use the representation (5.1) as free energy with a0 = b0 = 1, then the first Gor’kov Éliashberg equation takes the dimensionless form γ ˙ f = 1 κ2 4f − (f2 − 1 + u)f − f|A − 1 κ ∇θ|2 (5.2) 4φ + γf2 (θ̇ − κφ) = 0 (5.3) where κ > 0 is the Ginzburg-Landau parameter. From (5.1) or (5.2) it is possible to retrieve the phase diagram, which separates the normal from su- perconductor zone. This relation is represented by a parabola in the H − T plane (see [1]), which can be approximated considering the points for which the coefficient of f is zero. Namely, the points such that −1 + u + ¯ ¯ ¯A − 1 κ ∇θ ¯ ¯ ¯ 2 = 0 . The temperature effect will be supposed negligible on the first Maxwell equation, which we write in the London gauge (∇ · A = 0) Ȧ − ∇φ + ∇ × ∇ × A + f2 (A − 1 κ ∇θ) = 0. (5.4) Copyright © 2006 Taylor & Francis Group, LLC
- 40. 26 V. Berti and M. Fabrizio Finally, we need to consider the heat equation, which must be related to the equation (5.2) in order to have a thermodynamic compatibility. Hence let us consider the first law of thermodynamics or heat equation αuut − ufft = k∇ · u∇u (5.5) where α and k are two positive scalar constants. From (5.5), under the hy- pothesis of small perturbations for |∇u|2 , we obtain the entropy equation αut − fft = k4u (5.6) 6 Existence and uniqueness of the solutions In this section we prove the existence and the uniqueness of the solutions of the nonisothermal time dependent Ginzburg-Landau equations. To this purpose we write the system (4.2) in dimensionless form and the equation (5.6) by means of the complex variable ψ. Therefore we obtain γ(ψt − iκφψ) − 1 κ2 4ψ + 2i κ A · ∇ψ + |A|2 ψ + ψ ¡ |ψ|2 − 1 + u ¢ = 0 , (6.1) At − ∇φ + ∇ × ∇ × A − i 2κ ¡ ψ∇ψ̄ − ψ̄∇ψ ¢ + |ψ|2 A = 0 , (6.2) αut − k4u − 1 2 ¡ ψψ̄t + ψ̄ψt ¢ = 0 . (6.3) The problem is completed by the boundary conditions ∇ψ·n|∂Ω = 0, (∇×A)×n|∂Ω = Hex×n, ∇φ·n|∂Ω = 0, u|∂Ω = e u, (6.4) where Hex is the external magnetic field, and the initial data ψ(x, 0) = ψ0(x) , A(x, 0) = A0(x) , u(x, 0) = u0(x) . (6.5) In order to deal with homogeneous boundary conditions we introduce the new variables b u = u − e u and b A = A − Aex, where Aex is related to the external magnetic field by ∇ × Aex = Hex and satisfies ∇ · Aex = 0 , Aex · n|∂Ω = 0 . By assuming e u constant and Hex independent of time and such that ∇×Hex = Copyright © 2006 Taylor & Francis Group, LLC
- 41. A nonisothermal dynamical Ginzburg-Landau model 27 0, the system (6.1)–(6.5) reduces to γ(ψt − iκφψ) − 1 κ2 4ψ + 2i κ ( b A + Aex) · ∇ψ + | b A + Aex|2 ψ +ψ ¡ |ψ|2 − 1 + b u + e u ¢ = 0 , (6.6) b At − ∇φ + ∇ × ∇ × b A − i 2κ ¡ ψ∇ψ̄ − ψ̄∇ψ ¢ + |ψ|2 ( b A + Aex) = 0 , (6.7) αb ut − k4b u − 1 2 ¡ ψψ̄t + ψ̄ψt ¢ = 0 , (6.8) ∇ψ · n|∂Ω = 0 , b A · n|∂Ω = 0 , (∇ × b A) × n|∂Ω = 0 , ∇φ · n|∂Ω = 0 , b u|∂Ω = 0 , (6.9) ψ(x, 0) = ψ0(x) , b A(x, 0) = b A0(x) , b u(x, 0) = b u0(x) . (6.10) Let us denote by Lp (Ω), p > 0 and Hs (Ω), s ∈ R, the usual Lebesgue and Sobolev spaces, endowed with the standard norms k·kp and k·kHs . In partic- ular, we denote by k · k the norm in L2 (Ω). Given a time interval [a, b] and a Banach space X, we denote by C(a, b, X) [Lp (a, b, X)] the space of continuous [Lp ] functions from [a, b] into X, with the usual norms kfkC(a,b,X) = sup t∈[a,b] kf(t)kX , h kfkp Lp(a,b,X) = Z b a kf(t)kp X i . Finally let us introduce the following functional spaces D(Ω) = © A : A ∈ H1 (Ω), ∇ · A = 0, A · n|∂Ω = 0 ª , H1 m(Ω) = ½ φ : φ ∈ H1 (Ω), Z Ω φdv = 0 ¾ . DEFINITION 6.1 A triplet (ψ, b A, b u) such that ψ ∈ L2 (0, τ, H1 (Ω)) ∩ H1 (0, τ, L2 (Ω)), b A ∈ L2 (0, τ, D(Ω))∩H1 (0, τ, H1 (Ω)0 ), b u ∈ L2 (0, τ, H1 0 (Ω))∩ H1 (0, τ, H−1 (Ω)), satisfying (6.10), is a weak solution of the problem (6.6)– (6.10) with φ ∈ L2 (0, τ, H1 m(Ω)), Aex ∈ D(Ω) if Z Ω h γ(ψt − iκφψ)χ + 1 κ2 ∇ψ · ∇χ − 2i κ ψ( b A + Aex) · ∇χ + | b A + Aex|2 ψχ +ψχ(|ψ|2 − 1 + b u + e u) i dv = 0 , (6.11) Z Ω h b At · b + φ∇ · b + ∇ × b A · ∇ × b − i 2κ ¡ ψ∇ψ̄ − ψ̄∇ψ ¢ · b +|ψ|2 ( b A + Aex) · b i dv = 0 , (6.12) Z Ω · αb utv + k∇b u · ∇v − 1 2 ¡ ψψ̄t + ψ̄ψt ¢ v ¸ dv = 0 , (6.13) for each χ ∈ H1 (Ω), b ∈ H1 (Ω), v ∈ H1 0 (Ω) and for a.e. t ∈ [0, τ]. Copyright © 2006 Taylor & Francis Group, LLC
- 42. 28 V. Berti and M. Fabrizio Notice that, since any b ∈ H1 (Ω) can be decomposed as b = a + ∇ϕ, with a ∈ D(Ω) and ϕ ∈ H2 (Ω), the equation (6.12) can be replaced by Z Ω h b At · a + ∇ × b A · ∇ × a − i 2κ ¡ ψ∇ψ̄ − ψ̄∇ψ ¢ · a +|ψ|2 ( b A + Aex) · a i dv = 0 , Z Ω · φ4ϕ − i 2κ ¡ ψ∇ψ̄ − ψ̄∇ψ ¢ · ∇ϕ + |ψ|2 ( b A + Aex) · ∇ϕ ¸ dv = 0 . The following theorem proves the existence of the local solutions of the problem (6.6)–(6.10). THEOREM 6.1 Let ψ0 ∈ H1 (Ω), b A0 ∈ D(Ω), b u0 ∈ L2 (Ω). Then there exist τ0 > 0 and a solution (ψ, b A, b u) of the problem (6.6)–(6.10) in the time in- terval (0, τ0). Moreover ψ ∈ L2 (0, τ0, H2 (Ω)) ∩ C(0, τ0, H1 (Ω)), b A ∈ L2 (0, τ0, H2 (Ω)) ∩ C(0, τ0, H1 (Ω)), b u ∈ C(0, τ0, L2 (Ω)). Proof. The proof is based on the Faedo-Galerkin method. Let χj, aj and vj, j ∈ N be solutions of the boundary value problems −4χj = λjχj ∇χj · n|∂Ω = 0 ∇ × ∇ × aj = µjaj ∇ · aj = 0 aj · n|∂Ω = 0 (∇ × aj) × n|∂Ω = 0 ( −4vj = ξjvj vj|∂Ω = 0 where the eigenvalues λj, µj, ξj satisfy the inequalities 0 = λ1 < λ2 < ..., 0 < µ1 < µ2 < ..., 0 < ξ1 < ξ2 < ... and the eigenfunctions {χj}j∈N, {aj}j∈N and {vj}j∈N constitute orthonormal bases of L2 (Ω). Moreover χj ∈ H1 m(Ω) for each j ≥ 2. We denote by ψm (x, t) = m X j=1 αjm(t)χj(x) , b Am (x, t) = m X j=1 βjm(t)aj(x) , φm (x, t) = m X j=1 γjm(t)χj(x) , b um (x, t) = m X j=1 δjm(t)vj(x) , Copyright © 2006 Taylor & Francis Group, LLC
- 43. A nonisothermal dynamical Ginzburg-Landau model 29 which satisfy, for each j = 1, ..., m, the equations Z Ω h γ(ψm t − iκφm ψm )χj + 1 κ2 ∇ψm · ∇χj − 2i κ ψm ( b Am + Aex) · ∇χj +| b Am + Aex|2 ψm χj + ψm χj(|ψm |2 − 1 + b um + e u) i dv = 0 , (6.14) Z Ω h b Am t · aj + ∇ × b Am · ∇ × aj − i 2κ ¡ ψm ∇ψ̄m − ψ̄m ∇ψm ¢ · aj +|ψm |2 ( b Am + Aex) · aj i dv = 0 , (6.15) Z Ω · αb um t vj − 1 2 ¡ ψm ψ̄m t + ψ̄m ψm t ¢ vj + k∇b um · ∇vj ¸ dv = 0 , (6.16) Z Ω h φm 4χj − i 2κ ¡ ψ̄m ∇ψm − ψm ∇ψ̄m ¢ · ∇χj +|ψm |2 ( b Am + Aex) · ∇χj i dv = 0 . (6.17) The function φm is supposed to verify the condition Z Ω φm dv = 0 , for all t ∈ R. Moreover, since χj ∈ H1 m(Ω), j ≥ 2, from the previous equation we deduce γ1m = 0, for each m ∈ N, so that φm (x, t) = m X j=2 γjm(t)χj(x) . Let (ψ0m, b A0m, b u0m) be a sequence which converges to (ψ0, b A0, b u0) with re- spect to the norm of H1 (Ω) × H1 (Ω) × L2 (Ω) and denote by ψm (x, 0) = ψ0m(x) , b Am (x, 0) = A0m(x) , b um (x, 0) = u0m(x) . Then the equations (6.14)–(6.16) constitute a system of ordinary differential equations for the unknowns αjm, βjm and δjm with initial conditions αjm(0) = Z Ω ψ0mχjdv , βjm(0) = Z Ω A0m · ajdv , δjm(0) = Z Ω u0mvjdv . Notice that (6.17) allows to express γjm, j ≥ 2, as a function of αjm, βjm and δjm. Therefore the standard theory of ordinary differential equations ensures the existence and uniqueness of the local solutions. By letting F = γkψm k2 H1 + k b Am k2 + k∇ × b Am k2 + k i κ ∇ψm + ψm ( b Am + Aex)k2 + 1 2 k|ψm |2 − 1k2 + αkb um k2 + 1, Copyright © 2006 Taylor & Francis Group, LLC
- 44. 30 V. Berti and M. Fabrizio the inequality dF dt + γ 2 kψm t k2 + 1 2κ2 k4ψm k2 + 1 2 k∇φm k2 + k 2 k∇b um k2 + k b Am t k2 +k∇ × b Am k2 ≤ cF5 (6.18) can be proved. See [4] for details. An integration in (0, t) leads to F ≤ (F(0)−4 − ct)−1/4 t < τ0, (6.19) where τ0 depends on the norms kψ0mkH1 , kA0mkH1 , ku0mk. The previous in- equalities allow to pass to the limit as m → ∞ and prove the existence of a solution (ψ, A, u) of the problem (6.6)–(6.10) satisfying ψ ∈ C(0, τ0, H1 (Ω)), b A ∈ C(0, τ0, H1 (Ω)) and b u ∈ C(0, τ0, L2 (Ω)). The local solutions, defined in the time interval (0, τ0) by Theorem 6.1, can be extended to the whole interval (0, +∞). Indeed we construct a Lyapunov functional for the system γft − 1 κ2 4f + (f2 − 1 + u)f − f|A − 1 κ ∇θ|2 = 0 , (6.20) At − ∇φ + ∇ × ∇ × A + f2 (A − 1 κ ∇θ) = 0 , (6.21) 4φ + γf2 (θt − κφ) = 0 , (6.22) αut − fft − k4u = 0 , (6.23) by multiplying the equations respectively by ft, At − κ−1 ∇θt, −φ + κ−1 θt, b u and integrating in Ω. We obtain kftk2 + 1 2 d dt · 1 κ2 k∇fk2 + 1 2 kf2 − 1k2 ¸ + Z Ω fft · |A − 1 κ ∇θ|2 + u ¸ dv = 0, kAtk2 + 1 2 d dt µ k∇ × Ak2 − 2 Z ∂Ω A × Hex · nda ¶ + Z Ω · f2 (A − 1 κ ∇θ) · µ At − 1 κ ∇θt ¶ + 1 κ ∇φ · ∇θt ¸ dv = 0, k∇φk2 + γκk 1 κ fθt − fφk2 − 1 κ Z Ω ∇φ · ∇θtdv = 0, α 2 d dt kb uk2 + kk∇b uk2 − Z Ω b ufftdv = 0 . Adding the previous equations, we get dG dt + kftk2 + kAtk2 + k∇φk2 + γκkf( 1 κ θt − φ)k2 + kk∇uk2 = 0 , (6.24) Copyright © 2006 Taylor & Francis Group, LLC
- 45. A nonisothermal dynamical Ginzburg-Landau model 31 where the functional G is defined as G = 1 2 µ 1 κ2 k∇fk2 + 1 2 kf2 − 1k2 + kf(A − 1 κ ∇θ)k2 + k∇ × Ak2 −2 Z ∂Ω A × Hex · nda + νkHexk2 H−1/2(∂Ω) + αkuk2 + Z Ω e uf2 dv ¶ and the constant ν is sufficiently large in order to make G positive. The relation (6.24) yields G(t) ≤ G(0) , ∀t ≥ 0, which guarantees that the local solutions defined in (0, τ0) can be extended in (0, ∞). As a consequence of last inequality, we can prove some a priori estimates of the solutions. In particular, if the initial data are chosen such that the energy is finite, we have kfk2 H1 + kAk2 H1 + kf∇θk2 + kuk2 ≤ C . (6.25) Moreover, by integrating the relation (6.24) in (0, t) we obtain the further estimate Z t 0 [kftk2 + kAtk2 + k∇φk2 + kfθtk2 + k∇uk2 ]ds ≤ C . (6.26) The inequalities (6.25) and (6.26) lead to an estimate for the variable ψ kψk2 H1 + Z t 0 kψtk2 ds ≤ C . (6.27) It can be proved ([3]) that if f0(x) ≤ 1 almost everywhere in Ω, then f(x, t) ≤ 1 , (6.28) for all t > 0. Accordingly, the relations (6.1), (6.25), (6.26) and (6.27) yield Z t 0 k4ψk2 ds ≤ C . (6.29) THEOREM 6.2 The solution (ψ, A, u) of the system (6.1)–(6.5), with ini- tial data (ψ0, A0, u0) ∈ H1 (Ω) × D(Ω) × L2 (Ω) is unique. Proof. Let (ψ1, A1, u1), (ψ2, A2, u2) be two solutions of the problem (6.6)– (6.10) with the same initial data (ψ0, A0, u0) and sources Aex, e u. By denoting by ψ = ψ1 − ψ2, A = A1 − A2, φ = φ1 − φ2 and u = u1 − u2, from the equations (6.6)–(6.8) and the inequalities (6.25)–(6.29) we deduce ([4]) 1 2 d dt · γkψk2 + 1 κ2 k∇ψk2 + kAk2 + k∇ × Ak2 + αkuk2 ¸ ≤ ϕ1(t)kψk2 H1 + ϕ2(t)kAk2 H1 + Ckuk2 Copyright © 2006 Taylor & Francis Group, LLC
- 46. 32 V. Berti and M. Fabrizio where ϕ1, ϕ2 are L1 -functions of time. Therefore, an application of Gronwall’s inequality proves ψ = 0, A = 0, u = 0. References [1] J. Bardeen: Theory of superconductivity. In Handbuch der Physik XV (Edited by S. Flügge), (Springer, 1956), 274–369. [2] J. G. Bednorz and K. A. Müller: Earlier and Recent Aspects of Super- conductivity, (Springer, 1990). [3] V. Berti and M. Fabrizio: A nonisothermal Ginzburg-Landau model in superconductivity: existence, uniqueness and asymptotic behaviour, sub- mitted. [4] V. Berti and M. Fabrizio: Existence and uniqueness for a nonisothermal dynamical Ginzburg-Landau model of superconductivity, submitted. [5] R. de Bruyn Oubouter: Superconductivity: Discoveries during the early years of low temperature research at Leiden 1908-1914, IEEE Transac- tions on Magnetics Mag-23 (1987), 355–370. [6] B. S. Chandrasekhar: Early experiments and phenomenological theories, Superconductivity (I), edited by R.D. Parks, Marcel Dekker, 1969. [7] S. J. Chapman, S. D. Howison, and J. R. Ockendon: Macroscopic models for superconductivity, SIAM Rev. 34 (1992), 529–560. [8] E. Coskun, Z. Cakir and P. Takac: Nucleation of vortices with a tempera- ture and time-dependent Ginzburg-Landau model of superconductivity, Euro. J. Appl. Math. 14 (2003), 111–127. [9] M. Fabrizio, G. Gentili and B. Lazzari: A nonlocal phenomenological theory in superconductivity, Math. Models Methods Appl. Sci. 7 (1997), 345–362. [10] V. L. Ginzburg and L. D. Landau: On the theory of superconductivity, Zh. Eksp. Teor. Fiz. 20 (1950), 1064–1082. [11] L. P. Gor’kov and G.M. Éliashberg: Generalization of the Ginzburg- Landau equations for nonstationary problems in the case of alloys with paramagnetic impurities, Soviet Phys. JETP 27 (1968), 328–334. [12] C. J. Gorter and H. Casimir: Zur thermodynamik des supaleitenden Copyright © 2006 Taylor & Francis Group, LLC
- 47. A nonisothermal dynamical Ginzburg-Landau model 33 zustantles, Phys. Zs. 35 (1934), 963–966. [13] H.G. Kaper and P. Takac: An equivalence for the Ginzburg-Landau equations of superconductivity, ZAMP 48 (1997), 665–675 [14] L. D. Landau: Unbranched model of intermediate state, Phys. Z. Sowjet. 11 (1937), 129–138. [15] F. London: Superfluids I, Wiley, 1950. [16] H. London: An experimental examination of electrostatic behaviour of superconductors, Proc. Roy. Soc. A 155 (1936), 102–108. [17] A. B. Pippard: An experimental and theoretical study of relations be- tween magnetic fields and current in superconductors, Proc. Roy. Soc. London A 216 (1953), 547–568. [18] A. Schmid: A time dependent Ginzburg-Landau equation and its appli- cation to the problem of resistivity in the mixed state, Phys. Kondeus. Mater. 5 (1966), 302–317. [19] M. Tinkham: Introduction to superconductivity, McGraw-Hill, 1975. Valeria Berti Mauro Fabrizio Department of Mathematics Department of Mathematics University of Bologna University of Bologna P.zza Porta S.Donato 5 P.zza Porta S.Donato 5 40126 Bologna 40126 Bologna Italy Italy berti@dm.unibo.it fabrizio@dm.unibo.it Copyright © 2006 Taylor & Francis Group, LLC
- 48. Some global in time results for integrodifferential parabolic inverse problems Fabrizio Colombo, Davide Guidetti and Vincenzo Vespri Abstract We discuss a global in time existence and uniqueness result for an inverse problem arising in the theory of heat conduction for materials with mem- ory. The novelty lies in the fact this is a global in time well posed problem in the sense of Hadamard, for semilinear parabolic inverse problems of integrodifferen- tial type. 1 Introduction In this paper we discuss some strategies we can use in the study of parabolic integrodifferential inverse problems. The choice of the strategy depends on what type of nonlinearities are involved. We consider the heat equation for materials with memory since it is one of the most important physical examples to which our methods apply. Other models, for instance in the theory of population dynamics, can also be considered within our framework. We recall, for the sake of completeness, the heat equation for materials with memory. Let Ω be an open and bounded set in R3 and T be a positive real number. The evolution equation for the temperature u is given, for (t, x) ∈ [0, T] × Ω, by Dtu(t, x) = k∆u(t, x) + Z t 0 h(t − s)∆u(s, x) ds + F(u(t, x)), (1.1) where k is the diffusivity coefficient, h accounts for the memory effects and F is the heat source. In the inverse problem we consider, besides the temperature u, also h as a further unknown, and to determine it we add an additional measurement on u represented in integral form by Z Ω φ(x)u(t, x) dx = G(t), ∀t ∈ [0, T], (1.2) 35 Copyright © 2006 Taylor & Francis Group, LLC
- 49. 36 F. Colombo, D. Guidetti and V. Vespri where φ and G are given functions representing the type of device used to mea- sure u (on a suitable part of the body Ω) and the result of the measurement, respectively. We associate with (1.1)–(1.2) the initial-boundary conditions, for example of Neumann type: ( u(0, x) = u0(x), x ∈ Ω, Dνu(t, x) = 0, (t, x) ∈ [0, T] × ∂Ω, (1.3) ν denoting the outward normal unit vector. So one of the problems we are going to investigate is the following. PROBLEM 1.1 (The Inverse Problem with two types of nonlinearities): determine the temperature u : [0, T] × Ω −→ R and the convolution kernel h : [0, T] × Ω −→ R satisfying (1.1)–(1.3). In the case when F is independent of u, but depends only on x and on t, we assume that the heat source is placed in a given position, but its time dependence is unknown, so we can suppose that F(t, x) = f(t)g(x), where f has to be determined and g is a given datum. Then we also assume that the diffusion coefficient k is unknown. The second inverse problem we will study is as follows. PROBLEM 1.2 (An inverse problem with a nonlinearity of convolution type): determine the temperature u : [0, T] × Ω −→ R, the diffusion coefficient k and the functions h : [0, T] −→ R, f : [0, T] −→ R satisfying the system Dtu(t, x) = k∆u(t, x) + R t 0 h(t − s)∆u(s, x) ds + f(t)g(x), u(0, x) = u0(x), x ∈ Ω, ∂u ∂ν (t, x) = 0, (t, x) ∈ [0, T] × ∂Ω, (1.4) with the additional conditions Z Ω u(t, x)µj(dx) = Gj(t), ∀t ∈ [0, T], j = 1, 2, (1.5) where g, u0, G1, G2 are given data and µ1 and µ2 are finite Borel measures in C(Ω). REMARK 1.1 The additional conditions considered for Problem 1.2 (cf. (1.5)) is more general than the one considered for Problem 1.1 (cf. (1.2)). This is due to the fact that in Problem 1.2 we will choose the space of continuous Copyright © 2006 Taylor & Francis Group, LLC
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