The equations of thermoelasticity with time-dependent coefficients

The equations of thermoelasticity with time-dependent coefficients

Olaf Weinmann

Konstanzer Schriften in Mathematik und Informatik Nr. 221, Dezember 2006

ISSN 1430-3558

© Fachbereich Mathematik und Statistik

© Fachbereich Informatik und Informationswissenschaft Universität Konstanz

Fach D 188, 78457 Konstanz, Germany E-Mail: preprints@informatik.uni-konstanz.de

WWW: http://www.informatik.uni-konstanz.de/Schriften/

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/2292/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-22922

Olaf Weinmann

Abstract

We consider an inhomogeneous thermoelastic system with second sound in one space di- mension where the coecients are space- and time-dependent. For Dirichlet-Neumann type boundary conditions the global existence of smooth solutions is proved by using the theory of Kato. Then the asymptotic behavior of the solutions is discussed.

1 Introduction

The equations of thermoelasticity describe the elastic and the thermal behavior of elastic, heat conductive media, in particular the reciprocal actions between elastic stresses and temperature dierences. This paper is concerned with global existence, uniqueness, and asymptotic behavior of solutions to the linear inhomogeneous equations of one-dimensional thermoelasticity that model the second sound eect. Letu≡u(t, x),θ≡θ(t, x)andq ≡q(t, x)denote the unknown functions representing the displacement, the temperature dierence to a xed reference temperature, and the heat ux. Then the dierential equations foru, θ, q are represented as

u_tt−au_xx+bθ_x = f₁, (1)

θ_t+gq_x+du_tx = f₂, (2)

τ q_t+q+kθ_x = f₃. (3)

We emphasize that the coecients are space- and time-dependent, i.e., a ≡ a(t, x), b ≡ b(t, x), g ≡ g(t, x), d ≡ d(t, x), τ ≡ τ(t, x), and k ≡ k(t, x). Initial data and boundary conditions are given by

u(0,·) =u₀, u_t(0,·) =u₁, θ(0,·) =θ₀, q(0,·) =q₀, (4) and

au_x(t,0) +bθ(t,0) = 0, θ_x(t,0) = 0, u(t, L) =θ(t, L) = 0. (5) The boundary conditions(5)arise in the pulsed laser heating of solids, for instance in laser assisted particle removal from silicon wafers, cf. references [7],[13]and[15].

Note that the time-dependent coecientsaand b also appear in the rst boundary condition. It is dicult to deal with such time-dependent boundary conditions because in general they lead to a time-dependent domain of an associated evolution operator. We succeed in nding a transfor- mation of our problem into an evolution system

1

V_t+A(t)V =F(t), 0≤t≤T, V(0) =V₀

where the domain D(A(t)) of A(t) is independent of t. In view of the fact that we have to introduce certain Sobolev spaces to present D(A(t)) in detail, we refer to Section 3.2. Utilizing our transformation, we prove the existence of a unique, global solution to our problem using the classical theory of Kato.¹ After that we discuss the asymptotic behavior of our solution. In particular we prove that the solution to (1)−(5)decays exponentially if

Λ(t) :=¡

kf₁k²_L2+k(f₁)_tk²_L2+kf₂k²_L2+k(f₂)_tk²_L2 +k(f₂)_xk²_L2+kf₃k²_L2+k(f₃)_tk²_L2 +k(f₃)_xk²_L2

¢

decays exponentially. It will be necessary to construct a certain Lyapunov function and to combine techniques from energy methods and boundary control cf. [5], [9], [10], and [11].

For the classical homogeneous equations of thermoelasticity with constant coecients u_tt−αu_xx+βθ_x = 0,

θ_t−κθ_xx+δu_tx = 0,

it is well known that their solutions are exponentially stable for various types of boundary condi- tions. The latter equations result from replacing Cattaneo's law(3)by Fourier's law

q+kθ_x= 0, κ=gk. (6)

This classical model for example is treated in [2] and [5]. For a discussion of the second sound model see References [3], [4], [12], and [14]. In[9]Racke gives a detailed discussion of the problem (1)−(4) with one of the following boundary conditions in the homogeneous case with constant coecients.

(i) u(t,0) =u(t, L) =q(t,0) =q(t, L) = 0 for t≥0; (ii) u(t,0) =u(t, L) =θ(t,0) =θ(t, L) = 0 for t≥0;

(iii) αu_x(t,0) +βθ(t,0) = 0,θ_x(t,0) = 0,u(t, L) =θ(t, L) = 0 fort≥0.

The organization of this work is as follows: In Section 2 we will give some technical results. In particular we summarize some of the main results of the theory of Kato. Section 3 is dedicated to the well-posedness of our problem (1)−(5). In Section 4 we discuss the asymptotic behavior of the solutions.

This work extends a diploma thesis at the University of Konstanz [16] where the homogeneous case of(1)- (5) is discussed.

Acknowledgement The Author is grateful to Prof. Dr. Reinhard Racke for attracting his at- tention to this problem and for many fruitful discussions.

1Note that these ideas can be generalized to three space-dimensions. This will be the content of a forthcoming paper.

2 Some technical results

In this section, we summarize some technical results that we need to prove the well-posedness of our problem (1)−(5). In particular, we present a theorem of Kato concerning the existence and regularity of solutions to the following abstract linear evolution system:

V_t+A(t)V =F(t), 0≤t≤T with given initial data

V(0) =V₀, whereT >0is an arbitrary but xed constant.

Lemma 2.1 Let (X,k · k) be a Banach space. Furthermore let k · k_t (t ∈ [0, T]) be equivalent norms to the given norm onX such that

∃c >0∀s, t∈[0, T]∀x6= 0 : kxk_t

kxk_s ≤e^c|t−s|

Let A(t) : D(A(t)) ⊂X_t −→ X_t for t∈ [0, T] be an operator with A(t) ∈ G(X_t,1, β). Then the family(A(t))_t is stable on (X,k · k) and also stable in (X,k · k_t) for arbitrary t∈[0, T].

A proof for this result can be found in [6].

Lemma 2.2 Lets_i(t,·) (i= 1, ...,4;t∈[0, T])be real valued functions dened onΩsuch that the following properties hold:

(1) ∀t∈[0, T] :s_i(t,·)∈L¹(Ω),

(2) ∃C₁, C₂>0 :∀t∈[0, T],∀x∈Ω :C₁² ≤s_i(t, x)≤C₂².

Dene S(t, x) := diag(s₁,s₂,s₃,s₄) and forV, W ∈(L²(Ω))⁴ the inner producthU, Vi_t:=hU, SVi. Then we have

(i) C₁kVk ≤ kVk_t ≤C₂kVk for t∈ [0, T] and V ∈ (L²(Ω))⁴. In particular ¡

L²(Ω))⁴,h·,·i_t¢ a Hilbert space. is

(ii) (L²(Ω))⁴ = (C₀^∞(Ω),k · k_t)^∼ for t∈[0, T].

Proof: Lett∈[0, T]be xed andV ∈(L²(Ω))⁴ then we have C₁²kVk² = C₁²

X4

i=1

­Vⁱ, Vⁱ®

= X4

i=1

Z

Ω

C₁²|Vⁱ(x)|²dx≤ X4

i=1

Z

Ω

c_i(t, x)|Vⁱ(x)|²dx

= kVk²_t ≤ X4

i=1

Z

Ω

C₂²|Vⁱ(x)|²dx=C₂² X4

i=1

­Vⁱ, Vⁱ®

≤C₂²kVk²

in detailC₁²kVk²≤ kVk²_t ≤C₂²kVk². The second claim is obvious. ¤ In [8] Pazy gives a proof for the following

Theorem 2.3 Let (A(t))_t∈[0,T] be a stable family of innitesimal generators with stability con- stants M and ω. Let (B(t))_t∈[0,T] be bounded linear operators on X. If kB(t)k ≤ K for all 0 ≤ t ≤ T, then (A(t) +B(t))_t∈[0,T] is a stable family of innitesimal generators with stability constantsM andω+KM.

Theorem 2.4 Suppose that X₀, Y₁ are real, separable Hilbert spaces. Let (A;X₀, Y₁) be a CD- system. Let V⁰ ∈ Y¹, F ∈ C⁰([0, T], X₀) and F_t ∈ L¹([0, T], X₀). Then there exists a unique solution

V ∈ C⁰([0, T], Y¹)∩ C¹([0, T], X₀), V(0) =V⁰ for the initial value problem

V_t+A(t)V =F(t), 0≤t≤T, V(0) =V⁰. (7)

A proof for this result is given in [6]. Next we want to gain more regularty of the solution given in Theorem2.4. Therefore we introduce a double scale of real Banach spacesX_j,Y_j (0≤i, j≤s−1) of the following structure

X₀ ⊃ X₁ ⊃ X₂ ⊃ · · · ⊃ X_s−1 X₀=Y₀ ⊃ Y₁ ⊃ Y₂ ⊃ · · · ⊃ Y_s−1

Here it is assumed that all the inclusions are continuous and dense and that, ifs≥2,Y₁is a closed subspace of X₁ andY_j =Y₁∩X_j for1≤j ≤s−1. We introduce the following assumptions:

(L1) (Stability): The triple(A;X₀, Y₁) is a CD-system with stability constantsM and β. (L2) (Smoothness): We have

∂_t^rA∈Lip([0, T], L(Y_j+r+1;X_j)), 0≤j≤s−r−1,

for 0≤r ≤s−1. This implies that ∂_t^r+1A∈L^∞([0, T], L(Y_j+r+1;X_j))for the same range of r andj.

(L3) (Ellipticity): for a.e. t∈[0, T]and 0≤j≤s−1,

φ∈Y₁, A(t)φ∈X_j =⇒φ∈Y_j+1, kφk_Yj+1 ≤K¡

kA(t)φk_Xj +kφk_X0¢ , whereK >0is a constant.

(L4) Let∂_t^kF ∈ C⁰([0, T], X_s−1−k), k = 0, ..., s−1;∂_t^sF ∈L¹([0, T], X₀) (A1) (compatibility condition)

V^r :=∂_t^r−1F(0)− Xr−1

k=0

µr−1 k

¶

(∂^k_tA)(0)V^r−1−k∈Y_s−r 0≤r≤s A proof for the following result is given in [5].

Theorem 2.5 LetX₀ andY₁ be real separable Hilbert spaces. Let the triple (A, X₀, Y₁) be a CD- System such that the conditions (L1)- (L4)hold. If V⁰ ∈Y_s, then the solution given by Theorem 2.4 belongs to C⁰([0, T], Y_s) (hence ∂_t^kV ∈ C⁰([0, T], Y_s−k), k = 0, ..., s−1) if and only ifV⁰ and F satisfy the compatibility condition(A1)with respect to the family A and F.

3 Well-posedness

We consider the system of hyperbolic thermoelasticity

u_tt−a(t, x)u_xx+b(t, x)θ_x = f₁, (8) θ_t+g(t, x)q_x+d(t, x)u_tx = f₂, (9) τ(t, x)q_t+q+k(t, x)θ_x = f₃, (10) together with initial conditions

u(0,·) =u₀, u_t(0,·) =u₁, θ(0,·) =θ₀, q(0,·) =q₀, (11) and boundary conditions

(au_x)(t,0)−(bθ)(t,0) = 0, u(t, L) =θ(t, L) = 0, q(t,0) = 0. (12) Here a≡a(t, x), b ≡b(t, x), g =g(t, x), d≡d(t, x), τ ≡τ(t, x), and k≡k(t, x) are real-valued functions dened on[0, T]×Ω. The given functions f₁ ≡f₁(t, x),f₂≡f₂(t, x), and f₃ ≡f₃(t, x) are also dened on[0, T]×Ω.

3.1 Assumption Lets≥1and

∂_t^ra(t,·), ∂_t^rb(t,·), ∂_t^rg(t,·), ∂_t^rd(t,·), ∂_t^rk(t,·), ∂_t^rτ(t,·)∈L^∞([0, T], H^s−r+1(Ω)) for 0≤r≤s+ 1as well as

∂_t^r∂_xa(t,·), ∂_t^r∂_xb(t,·), ∂^r_t∂_xg(t,·), ∂_t^r∂_xd(t,·)∈L^∞([0, T], H^s−r(Ω)) for 0≤r≤s.

Furthermore, letC_a,C^a,C_b, C^b,C_g, C^g,C_d, C^d, C_τ, C^τ, C_k, C^k be positiv constants such that for all (t, x)∈[0, T]×Ωthe following inequalities hold:

C_a ≤ a(t, x) ≤ C^a, C_b ≤ b(t, x) ≤ C^b, C_g ≤ g(t, x) ≤ C^g, C_d ≤ d(t, x) ≤ C^d, C_k ≤ k(t, x) ≤ C^k, C_τ ≤ τ(t, x) ≤ C^τ.

(13)

3.2 Existence

Let(u, θ, q)be a solution to(8)−(12) and let

V ≡V(t, x) :=







a bu_x

u_t

gθ

dq





(t, x), V₀ ≡V₀(x) :=







¡_a

b

¢(0, x)u_0,x(x) u₁(x) θ₀(x)

¡_g

d

¢(0, x)q₀(x)





.

Then V satises

V_t+AV =F, V(0) =V₀, (14)

whereA≡A(t, x) is dened asA=Q⁻¹(N₀+N₁). HereF, Q, N₀, andN₁ are dened as follows:

F ≡F(t, x) =





 0 f₁ f₂

g dτf₃





, N₀ ≡N₀(t, x) =







−¡_a

b

¢

t b²

a² 0 0 0

¡_a

b

¢

x b

a 0 0 0

0 0 0 −¡_g

d

¢

xd g

0 0 0 −¡_g

d

¢

td²τ g²k





(t, x),

Q⁻¹≡Q⁻¹(t, x) =







ab 0 0 0 0 b 0 0 0 0 d 0 0 0 0 ^gk_dτ





(t, x), N₁ ≡N₁(t, x) =







0 −∂_x 0 0

−∂_x 0 ∂_x 0 0 ∂_x 0 ∂_x 0 0 ∂_{x gk}^d





(t, x).

In the following we will prove an existence theorem for (14)under Assumption3.1. In view of(13) we can chooseC₁ >0 andC₂ >0such that

C₁ ≤ b a,1

b,1 d,dτ

gk ≤C₂

holds for arbitrary (t, x)∈[0, T]×Ω. Now we dene for U, V ∈(L²(Ω))⁴ the inner product hU, Vi_t:=hU, Q(t,·)Vi.

Lett∈[0, T]be xed. With the help of Lemma2.2we conclude that((L²(Ω))⁴,h·,·i_t)is a Hilbert space, which we denote by H_t. By utilizing the matrices Q⁻¹ and N₁ we dene the Operator

A₁(t) :D(A₁(t))⊂ H_t−→ H_t with domain

D(A₁(t)) :={(V¹, V², V³, V⁴)∈ H_t:V¹−V³ ∈H_l¹(Ω), V², V³∈H_r¹(Ω), V⁴ ∈H_l¹(Ω)}

by

A₁(t)f :=Q⁻¹(t,·)N₁(t,·)f. (15) Our next aim is to show that the operator−A₁(t)generates aC₀semigroup of contractions onH_t for every xedt∈[0, T]. Then we conclude that −A(t) generates aC₀ semigroup of contractions on(L²(Ω))⁴for every xedt∈[0, T]. Finally we show that the family(−A₁(t))_tis a stable family of generators of aC₀ semigroup on(L²(Ω))⁴.

In order to show that the operator−A₁(t) generates aC₀ semigroup of contractions we show that

−A₁(t) is densely dened and closed. Furthermore, we show that both −A₁(t) and its adjoint operator are dissipative.

Lemma 3.1 LetA₁(t) be dened as in (15). Then−A₁(t) is densely dened.

Proof: It is easy to see that C^∞₀ (Ω)⊂D(A₁(t)). Thus the density of −A₁(t) is a consequence of

Lemma2.2. ¤

Lemma 3.2 LetA₁(t) be dened as in (15). Then−A₁(t) is closed.

Proof: Let (V_n)_n∈N ⊂ D(A₁(t)) be a sequence with V_n → V ∈ H_t and A₁(t)V_n → W ∈ H_t as n→ ∞. Then we have that

∀Φ∈ H_t: h−A₁(t)V_n,Φi_t→ hW,Φi_t (16) asn→ ∞. In other words,

­∂_xV_n²,Φ¹® +­

∂_xV_n¹−∂_xV_n³,Φ²®

−­

∂_xV_n²+∂_xV_n⁴,Φ³®

−

¿

∂_xV_n³+ d

gkV_n⁴,Φ⁴ À

→

¿ W¹,b

aΦ¹ À

+

¿ W²,1

bΦ² À

+

¿ W³,1

dΦ³ À

+

¿ W⁴,dτ

gkΦ⁴ À

(17) asn→ ∞.

(a) ChoosingΦ = (Φ¹,0,0,0)withΦ¹ ∈L²(Ω)we obtain with the help of (17):

­∂_xV_n²,Φ¹®

→

¿ W¹,b

aΦ¹ À

asn→ ∞. Now, choosingΦ¹∈ C₀^∞(Ω), we conclude that

−­

V², ∂_xΦ¹®

=− lim

n→∞

­V_n², ∂_xΦ¹®

= lim

n→∞

­∂_xV_n²,Φ¹®

=

¿b

aW¹,Φ¹ À

.

So, we haveV² ∈H¹(Ω)with∂_xV²= _a^bW¹. ChoosingΦ¹ ∈H_l¹(Ω)we obtainV² ∈H_r¹(Ω). Summarizing results in

V² ∈H_r¹(Ω) and ∂_xV² = b aW¹.

(b) ChoosingΦ := (0,0,Φ³,0)withΦ³∈L²(Ω)we obtain with the help of(17) that

−­

∂_xV_n²+∂_xV_n⁴,Φ³®

→

¿ W³,1

dΦ³ À

Assuming thatΦ³∈ C^∞₀ (Ω)yields

­V⁴, ∂_xΦ³®

=

¿b

aW¹+1

dW³,Φ³ À

. Now, choosing Φ³ ∈H_r¹(Ω)we get that

V⁴ ∈H_l¹(Ω) and −∂_xV⁴= 1

dW³+∂_xV².

(c) Now, choosing Φ := (0,0,0,Φ⁴) withΦ⁴ ∈L²(Ω)we obtain with the help of (17)that

−

¿

∂_xV_n³+ d

gkV_n⁴,Φ⁴ À

→

¿ W⁴,dτ

gk,Φ⁴ À

. Similarly to(a) and(b) we deduce

V³ ∈H_r¹(Ω) and −∂_xV³ = dτ

gkW⁴+ d gkV⁴.

(d) Finally, choosing Φ := (0,Φ²,0,0)withΦ² ∈L²(Ω)we get with the help of(17) that

­∂_xV_n¹−∂_xV_n³,Φ²®

→

¿ W²,1

bΦ² À

.

Summarizing then yields

V¹−V³ ∈H_l¹(Ω) and ∂_x(V¹−V³) = 1 bW².

Overall we have V ∈D(A₁(t))and−A₁(t)V =W. This completes the proof. ¤ Lemma 3.3 LetA₁(t) be dened as in (15). Then we have that D(−A₁(t)) =D((−A₁)^∗(t)) and that

(−A₁)^∗(t) =Q⁻¹







0 −∂_x 0 0

−∂_x 0 ∂_x 0 0 ∂_x 0 ∂_x 0 0 ∂_x −_gk^d







Proof: Similarly to the proof of Lemma 3.2 we obtain this claim. ¤ Lemma 3.4 LetA₁(t) be dened as in (15). Then both −A₁(t) and (−A₁)^∗(t) are dissipative.

Proof: LetV ∈D(A₁(t)).

h−A₁(t)V, Vi_t = ­

−Q⁻¹(t)N₁(t)V, V®

t=h−N₁(t)V, Vi

= − Ã

−­

∂_xV², V¹® +­

V¹, ∂_xV²® +­

V³, ∂_xV²®

−­

∂_xV², V³®

−­

V⁴, ∂_xV³® +­

∂_xV³, V⁴® +

¿ d

gkV⁴, V⁴ À !

We conclude thatReh−A₁(t)V, Vi_t=−R

Ω d

gk|V⁴|²dx and the proof is completed. ¤ This implies

Theorem 3.5 Lett∈[0, T] be xed and

A₁(t) :D(A₁(t))⊂ H_t−→ H_t with

D(A₁(t)) :={(V¹, V², V³, V⁴)∈ H_t:V¹−V³ ∈H_l¹(Ω), V², V³∈H_r¹(Ω), V⁴ ∈H_l¹(Ω)}

be dened as A₁(t)f :=Q⁻¹(t,·)N₁(t,·)f. Then the following statements hold:

(i) −A₁(t) is a generator of aC₀ semigroup of contractions on (H_t,h·,·i_t).

(ii) The family (−A₁(t))_t∈[0,T] is a stable family of generators of a C₀ semigroup on the Hilbert space³¡

L²(Ω)¢₄ ,h·,·i

´ . Proof:

(i) This statement is a direct consequence of the Lemmas 3.1−3.4. (ii) LetV ∈(L²(Ω))⁴ withV 6= 0. We consider the function

f_V : [0, T]−→R, t7−→ln(kVk²_t) Obviouslyf_V is continuously dierentiable. Furthermore, we have

¯¯f_V⁰ (t)¯

¯ ≤ R

Ω

³¯¯¡_b

a

¢

t

¯¯|V¹|²+¯

¯¡₁

b

¢

t

¯¯|V²|²+¯

¯¡₁

d

¢

t

¯¯|V³|²+

¯¯

¯

³dτ gk

´

t

¯¯

¯|V⁴|²

´ dx R

Ω

³b

a|V¹|²+¹_b|V²|²+¹_d|V³|²+^dτ_gk|V⁴|²

´ dx

≤ C

for a certain constantC >0. The existence of such a constant follows direct from Assumption 3.1. Now we conclude fors, t∈[0, T]withs < t by using the mean value theorem that

∃ξ∈[s, t] : |ln(kVk²_t)−ln(kVk²_s)|

|t−s| =

¯¯

¯¯f_V(t)−f_V(s) t−s

¯¯

¯¯=|f_V⁰ (ξ)|.

This implies

ln(kVk²_t)−ln(kVk²_s)≤ |ln(kVk²_t)−ln(kVk²_s)| ≤C|t−s|

for s, t∈[0, T]. Since the exponential function is monotone we deduce kVk²_t

kVk²_s ≤e^c|t−s|

for s, t∈[0, T]. Applying Lemma2.1 the claim follows.

¤ Our next aim is to show that(−A(t))_t∈[0,T]is also a stable family of generators of aC₀ semigroup.

For this purpose we intend to apply Theorem 2.3. Thus, we have to show that

A₀(t, x) :=Q⁻¹(t, x)N₀(t, x) (18) is a bounded linear operator for every xedt∈[0, T]and that the family(A₀(t))_t∈[0,T]is uniformly bounded in t.

Lemma 3.6 For every xedt∈[0, T]the operator A₀(t, x) as dened above is bounded. Further- more, there exists a constant K >0 such that kA₀(t)k ≤K for every t∈[0, T].

Proof: Lett∈[0, T]be xed. Obviously we have that

A₀(t, x) =







−¡_a

b

¢

tb

a 0 0 0

¡_a

b

¢

x b²

a 0 0 0

0 0 0 −¡_g

d

¢

xd² g

0 0 0 −¡_g

d

¢

td g





.

Choosingξ ∈(L²(Ω))⁴ we obtain kA₀(t,·)ξ(·)k² =

°°

°°

³a b

´

t

b aξ₁

°°

°°

2

+

°°

°°

³a b

´

x

b² aξ₁

°°

°°

2

+

°°

°°

³g d

´

x

d² g ξ₄

°°

°°

2

+

°°

°°

³g d

´

t

d gξ₄

°°

°°

2

≤ 4Ckξk².

Observe, that here the constant C can be chosen independently of t in view of 3.1. Setting

K := 2C, we obtainkA₀(t,·)ξk ≤Kkξk. ¤

We dene for

V, W ∈ D:={(V¹, V², V³, V⁴)∈ H_t:V¹−V³ ∈H_l¹(Ω), V², V³∈H_r¹(Ω), V⁴∈H_l¹(Ω)}

the inner product hV, Wi_D as

hU, Vi_D:=

X4

i=1

­Uⁱ, Vⁱ®

H¹(Ω). Lemma 3.7 (D,h·,·i_D) is a Hilbert space.

Proof: Note that forV ∈ D the norm kVk_D is given by

kVk²_D =kV¹k²_H1(Ω)+kV²k²_H1(Ω)+kV³k²_H1(Ω)+kV⁴k²_H1(Ω).

Let (V_n)_n∈N be a Cauchy sequence in D. Then the (V_nⁱ)_n∈N for i = 2,3 are Cauchy sequences in H_r¹(Ω), (V_n⁴)_n∈N is a Cauchy sequence in H_l¹(Ω), and (V_n¹−V_n³)_n∈N is a Cauchy sequence in H_l¹(Ω). This implies the convergence of(V_n)_n∈N inD. ¤ Employing Lemma 2.4 we obtain the existence of a solution of our problem, if we assume that A∈Lip([0, T];L(Y₁, X₀)). We will discuss this point later. First we want to gain more regularity for our solution. To this end we dene fors≥2,

X₀ := (L²(Ω))⁴, X₁ := (H¹(Ω))⁴, X₂ := (H²(Ω))⁴, ..., X_s−1:= (H^s−1(Ω))⁴, and

Y₀:=X₀, Y₁:=D as well as Y_j :=Y₁∩X_j for 1≤j≤s−1.

It is clear that

X_s−1 ⊂...⊂X₂ ⊂X₁ ⊂X₀ and

Y_s−1⊂...⊂Y₂⊂Y₁ ⊂Y₀=X₀ and that all the inclusions are continuous.

Lemma 3.8 For arbitrary i= 1...s−1 we have

X_i−1 = (X_i,k · k_Xi−1)^∼ and Y_i−1 = (Y_i,k · k_Yi−1)^∼ Proof: The rst claim is obvious. In order to see the second claim we dene

D^∞:={(V¹, V², V³, V⁴) :V¹−V³∈ C_l^∞(Ω), V², V³ ∈ C_r^∞(Ω), V⁴ ∈ C_l^∞(Ω)}.

We easily deduce thatD^∞⊂Y_i fori= 1...s−1. Let1≤i≤s−1be xed and(V¹, V², V³, V⁴)∈ Y_i. Obviously there exists a sequence (V_n¹)_n ⊂ C_l^∞(Ω) with V_n¹ →_Hⁱ V¹ −V³, and sequences (V_n²)_n ⊂ C_r^∞(Ω), (V_n³)_n ⊂ C_r^∞(Ω) and (V_n⁴)_n ⊂ C_l^∞(Ω) with V_n² →_Hi V², V_n³ →_Hi V³ and V_n⁴ →_Hi V⁴. For ((V_n¹+V_n³, V_n², V_n³, V_n⁴))_n we have that ((V_n¹ +V_n³, V_n², V_n³, V_n⁴))_n ⊂ D^∞, that (V_n¹+V_n³, V_n², V_n³, V_n⁴)→_Hⁱ (V¹, V², V³, V⁴). ¤ Lemma 3.9 Lets≥1 and0≤r ≤s−1. Then we have

∂_t^r+1A∈L^∞([0, T], L(Y_j+r+1;X_j)), for 0≤j≤s−r−1.

Proof: Observe that

∂_t^r+1A(t) = ∂_t^r+1







−¡_a

b

¢

t b

a −^a_b∂_x 0 0

¡_a

b

¢

xb²

a −b∂_x 0 b∂_x 0

0 d∂_x 0 −¡_g

d

¢

x d² g +d∂_x

0 0 ^gk_dτ −¡_g

d

¢

td g +¹_τ





.

In the following we prove that there is a constantC >0 such that

°°∂_t^r+1A(t)V°

°²

H^j ≤CkVk_Yj+r+1 (19)

for all t∈[0, T]and arbitraryV ∈Y_j+r+1. In the next estimate we order the terms with respect to the order of the derivatives, i.e., rst we write the ones with lowest order, then the ones with second lowest order, a.s.o. Note that the terms that are not written explicitly can be treated in a similar way.

°°∂^r+1_t A(t)V°

°²

H^j ≤ Xj

l=0

Xl

k=0

µl k

¶ °°

°°∂_x^k

·

∂_t^r+1

·³a b

´

t

b a

¸¸

∂_x^l−kV¹

°°

°°

2

+...

+ Xj

l=0

Xl

k=0

µl k

¶ °°

°°∂_x^k

·

∂_t^r+1

·1 τ

¸¸

∂_x^l−kV⁴

°°

°°

2

+ Xj

l=0

Xl

k=0

µl k

¶ °°°∂_x^k£

∂_t^r+1b¤

∂_x^l−k+1V¹

°°

°²+...

+ Xj

l=0

Xl

k=0

µl k

¶ °°°∂_x^k£

∂_t^r+1d¤

∂_x^l−k+1V⁴

°°

°²

No we distinguish the following two cases:

(i) The case r > 0. Here we have V_i ∈ H^j+2(Ω) for i = 1...4. Therefore in the terms above there appear only derivatives of order less or equal to j+ 1. So, relation (19) follows by Sobolev's imbedding theorem.

(ii) The caser= 0. In the case thatj6=s−1we can estimate the derivatives of the coecients again by Sobolev's imbedding theorem. In the case that j =s+ 1 the derivatives of Vⁱ in the terms with highest order derivatives of the coecients only have order one. Hence they also can be estimated by Sobolev's imbedding theorem.

¤ Lemma 3.10 For t∈[0, T]and 0≤j ≤s−1 we have the following statement: Let V ∈Y₁ and A(t)V ∈X_j. Then we have V ∈Y_j+1 and that

kVk_Yj+1 ≤K¡

kA(t)Vk_Xj+kVk_X0¢ .

Proof: The rst claim is obtained successively. We prove the estimate by induction over j. 1. j= 0. Note that by

A(t)V =







¡_a

b

¢

tb

aV¹−^a_b∂_xV² b∂_xV¹+¡_a

b

¢

x b²

aV¹+b∂_xV³ d∂_xV²+d∂_xV⁴−¡_g

d

¢

xd² gV⁴

gk

dτ∂_xV³−¡_g

d

¢

td

gV⁴+ ¹_τV⁴







we can estimate

°°

°−a b∂_xV²

°°

°² =

°°

°°

³a b

´

t

b

aV¹−a

b∂_xV²−

³a b

´

t

b aV¹

°°

°°

2

≤ 2kA(t)Vk²+ 2

°°

°°

³a b

´

t

b aV¹

°°

°°

2

This implies

∃K₁ >0 : k∂_xV²k² ≤K₁¡

kA(t)Vk²+kVk²¢

. (20)

Furthermore,

°°d∂_xV⁴°

°² =

°°

°°d∂_xV²+d∂_xV⁴−

³g d

´

x

d²

g V⁴−d∂_xV²+

³g d

´

x

d² g V⁴

°°

°°

2

≤ 2kA(t)Vk²+ 4kd∂_xV²k²+ 4

°°

°°

³g d

´

x

d² g V⁴

°°

°°

2

which yields

∃K₂>0 : k∂_xV⁴k² ≤K₂¡

kA(t)Vk²+kVk²¢

(21) Next we estimate

°°

°°gk dτ∂_xV³

°°

°°

2

=

°°

°°gk

dτ∂_xV³−

³g d

´

t

d

gV⁴+1 τV⁴+

³g d

´

t

d

gV⁴−1 τV⁴

°°

°°

2

≤ 2

°°

°°gk

dτ∂_xV³−

³g d

´

t

d

gV⁴+1 τV⁴

°°

°°

2

+

°°

°°

³g d

´

t

d

gV⁴− 1 τV⁴

°°

°°

2