GLOBAL REGULAR SOLUTIONS TO TWO-DIMENSIONAL THERMOVISCOELASTICITY

. A two-dimensional thermoviscoelastic system of Kelvin-Voigt type with strong dependence on temperature is considered. The existence and uniqueness of a global regular solution is proved without small data assumptions. The global existence is proved in two steps. First global a priori esti- mate is derived applying the theory of anisotropic Sobolev spaces with a mixed norm. Then local existence, proved by the method of successive approxima- tions for a suﬃciently small time interval, is extended step by step in time. By two-dimensional solution we mean that all its quantities depend on two space variables only.


1.
Introduction. This article is devoted to the problem of global existence and uniqueness of regular solutions to a two-dimensional (2d) thermoviscoelasticity system for small strains. The system describes homogeneous isotropic linearlyresponding viscoelastic materials in the Kelvin-Voigt rheology at small strains. We consider such thermoviscoelasticity system that the specific heat and the elasticity tensor depend on temperature in a very special relation.
Lately in [15] global existence of regular solutions to three-dimensional thermoviscoelasticity with specific heat linearly increasing with temperature and with constant heat conductivity is proved. Such setting is a particular case of systems presented in [2,18]. Existence of weak solutions for generalized thermoviscoelastic materials with various kinds of boundary conditions can be found in [17,19]. Moreover, papers of Roubíček [18,19,20] and Rossi-Roubíček [17] present a deep physical background of thermoviscoelastic materials.
Pioneering papers on global regular solutions to one-dimensional thermoviscoelasticity are [4,5,21] and the spherical case is considered in [7]. Lately, global existence of large solutions to spherically symmetric nonlinear viscoelasticity is proved in [8,9].
In this paper we consider a two-dimensional thermoviscoelastic system with the temperature dependent specific heat of the form c * = c v θ σ with 1 2 < σ < 1, c v positive constant and with constant heat conductivity. Such setting is a particular case of systems addressed in [18]. Moreover, the stress tensor is given by a linear thermoviscoelastic law of the Kelvin-Voigt type (cf. [6,Chapter 5.4]). The aim of this paper is to prove existence of global regular solutions to the 2d-thermoviscoelastic system without smallness assumptions on data and for σ as small as possible. In 3d-case (see [15]) global regular solutions for large data is only proved for σ = 1. Restricting in this paper our considerations to the 2d case we were able to use the specific heat c * = c v θ σ , σ ∈ (1/2, 1).
The proof of global existence basis on two main steps. First we prove global a priori estimate is Sobolev spaces W 2,1 p,p0 (Ω T ) with the mixed norm. This is possible because equations for displacement and temperature are parabolic. The idea was strongly developed in [15]. Since c * = c v θ σ is the coefficient heat near θ ,t we need continuity of θ to apply the theory for parabolic equations, so p and p 0 must be sufficiently large. Next, we prove local existence in W 2,1 p,p0 (Ω T )-spaces by the method of successive approximations. Combining these two steps we prove the main result: global existence of regular solutions with large data.
Then we consider the following thermoviscoelasticity system where Ω ⊂ R 2 , with a boundary S, is bounded, σ is a positive constant. We add the boundary conditions wheren is the unit outward normal vector to S, and the initial conditions The field u : Ω T → R 2 is the displacement, θ : Ω T → R + is the absolute temperature. The second order tensors ε = {ε ij } i,j=1,2 and ε ,t = {ε ij,t } i,j=1,2 denote, respectively, the fields of the linearized strain and the strain rate, which are defined by The fourth order tensors A 1 = {A 1ijkl } i,j,k,l=1,2 and A 2 = {A 2ijkl } i,j,k,l=1,2 are respectively, the linear viscoelasticity and the elasticity tensors, defined by the Hook law ε → A i ε = λ i trεI + 2µ i ε, i = 1, 2, (1.7) where λ 1 , µ 1 are the viscosity constants and λ 2 , µ 2 are the Lamé constants, both λ 1 , µ 1 and λ 2 , µ 2 with values within the elasticity range and I is the unit matrix. We define linear viscosity and elasticity tensors Q 1 and Q 2 by (1.9) In view of (1.9) equation (1.1) takes the form (1.10) We assume that tensors A m satisfy the following symmetry conditions coercivity and boundedness where a m * , a * m are positive constants. Main Theorem. Let (1.7), (1.8), g > 0 and the assumptions of Lemma 4.1 with (Ω), p, p 0 , q, q 0 ≥ 4, 1/2 < σ ≤ 1. Let S ∈ C 2 . Then there exists a global solution to problem (1.8)-(1.11) such that The paper is organized in the following way. In Section 1 the considered problem is formulated and its properties are listed. In [15] we showed that the property g ≥ 0 implies that the second law of thermodynamics holds. In Section 3 we define used in this paper spaces with corresponding imbeddings and interpolations and also solvability results for some parabolic initial-boundary value problems (3.1) and (3.4) are presented. Section 4 is devoted to show a positive infimum of temperature. In Section 5 we derive some global a priori estimates. The main estimate is the Hölder continuity of temperature which implies that W 2,1 q,q0 -theory can be applied to equation (1.2). This is compatible with results of Section 6. Applying the method of successive approximations we prove in Section 6 local existence of solutions to problem (1. 2. Physical and thermodynamical background. In the case of (1.1), (1.2) the free energy is specified by is the caloric energy, and The dissipation potential corresponding to (1.1), (1.2) is given by where k > 0 and the Fourier law for a heat flux is used.

Notation.
Let Ω ⊂ R n , n ≥ 1, be a domain in R n with boundary S. Let are Sobolev spaces with a mixed norm, which are completion of C ∞ (Ω T )-functions under the finite norm , we denote the Sobolev-Slobodetskii space with the finite norm where a ∈ N 0 and [s] is the integer part of s. For s odd the middle term in the above norm vanishes whereas for s even the two last terms vanish. We use also the notation is the Besov space of functions making the following norm finite [11] it is known that the norms of the Besov space B l p,p0 (Ω) are equivalent for different m and k satisfying the condition m > l − k > 0.
By V 2 (Ω T ) we denote the space L ∞ (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 (Ω)) of functions making the following norm finite ). By C α,α/2 (Ω T ), α ∈ (0, 1), we denote the anisotropic Hölder space of functions making the following norm finite By c we denote a generic positive constant which changes its value from formula to formula and depends at most on the imbedding constants, constants of the considered problem, and the regularity of the boundary. By ϕ = ϕ(σ 1 , . . . , σ k ), k ∈ N, we denote a generic function which is a positive increasing function of its arguments σ 1 , . . . , σ k , and may change its form from formula to formula.

Auxiliary results. We need the following interpolation lemma
. As a special case of Lemma 3.1 we need . We also need the following interpolation result Then there exists a constant c such that We recall from [3] the trace and the inverse trace theorems for Sobolev-Slobodestskii spaces with a mixed norm (Ω) and (Ω) , where constant c does not depend on u.
Let us consider the problem ∈ (0, 1). Then there exists a solution to problem (3.4) such that θ ∈ W 2,1 p,p0 (Ω T ) and the estimate holds (Ω) ). (3.5) 4. Lower bound for temperature. The existence of the lower positive bound on temperature is very important for getting an a priori global estimate in this paper. However, we follow the arguments from the proof of Lemma 4.1 in [15], but our proof is essentially different. where a 1 * is introduced in (1.12). Then for sufficiently regular solutions to problem (1.1)-(1.4) we have Let σ < 1 and let a 2 (t) = 1 2a1 * lim →∞ A 2 (Ω) . Then for a sufficiently regular solution to (1.1)-(1.4) it follows Proof. Multiplying (1.2) by −θ − and integrating over Ω yields Now we examine the particular terms in (4.4). The first term is equal to The second term equals 4κ In view of (1.12) the third term is bounded from below by The fourth term is positive because g > 0.
Applying the Cauchy inequality the last term in (4.4) it is bounded by In view of the above considerations (4.4) takes the form where is assumed to be large. Let us introduce the notation . (4.6) Using (4.6) and assuming that a 1 (t) = 1 2a1 * sup Ω |A(ε)| 2 we obtain (4.7) Setting σ = 1 we obtain from (4.7) the inequality Hence passing with → ∞ yields which yields (4.2). Let us consider the case σ < 1. Then (4.5) takes the form Passing with → ∞ we get (4.10) This implies (4.3) and concludes the proof.

5.
A priori estimates. We prove estimates in this section under the existence of the lower bound on temperature proved in Lemma 4.1 such that where T is the time of local existence.
where c(t) is an increasing function.
Proof. Multiplying (1.1) by u ,t and integrating over Ω yields Integrating (1.2) over Ω implies In view of (1.9) we have Integrating (5.6) with respect to time with applying the Gronwall inequality and the Poincaré inequality yields (5.2). This concludes the proof.
Proof. From (5.7) and (5.11) we have Hence (5.20) holds for finite p 1 and r 1 such that For at least one of p 1 , r 1 equal to infinity we have the strong inequality in (5.21). This concludes the proof.
where c 8 depends on all norms from the assumption.
Corollary 5.10. In view of (5.24) and (5.29) we have that θ, ε t belong to L p,r (Ω T ), p, r ∈ (1, ∞). Using this and the Hölder continuity of θ we get for solutions to problem (1. where p, p 0 ∈ (1, ∞). We are not interested to increase regularity of solutions to problem (1.1)-(1.6) as much as possible. We want to have only such regularity that the existence of local solutions holds and that the local solution might be extended in time to get the global existence.