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Global Regular Solutions to a Kelvin-Voigt Type Thermoviscoelastic System

A classical 3-D thermoviscoelastic system of Kelvin-Voigt type is considered. The existence and uniqueness of a global regular solution is proved without small data assumption. The existence proof is based on the successive approximation method. The crucial part constitute a priori estimates on an arbitrary finite time interval, which are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm.


Motivation and goal.
This article is concerned with the existence and uniqueness of global regular solutions to a classical 3-D thermoviscoelastic system at small strains. The system describes materials which have the properties both of elasticity and viscosity. Such materials are usually referred to as Kelvin-Voigt type.
As noted in the recent paper on this subject by Roubíček [21] -and according to our best knowledge as well -the existence of global solutions to a thermoviscoelastic system with constant both specific heat and heat conductivity is, in spite of great effort through many decades, still open in dimensions n ≥ 2. In dimension n = 1 it was established in the pioneering papers by Slemrod [22], Dafermos [5] and Defermos-Hsiao [6].
The local in time existence and global uniqueness of a weak solution to 3-D thermoviscoelastic system with constant specific heat and heat conductivity has been proved by Bonetti-Bonfanti [3]. Other known results on multidimensional thermoviscoelasticity deal with a modified energy equation. Modifications involve either nonconstant specific heat or nonconstant heat conductivity. A thermoviscoelastic system with temperature-dependent specific heat has been addressed by Blanchard-Guibé [2] where the existence of global, weak-renormalized solutions has been proved, and recently in [21] where the existence of a very weak solution has been established. We mention also that the framework of renormalized solutions has been applied in [26] for 3-D thermoviscoelastic system arising in structural phase transitions.
In a more general setting allowing for large strains a 3-D thermoviscoelastic system has been studied under small data assumption by Shibata [23] and recently by Gawinecki-Zajaczkowski [11].
For thermoviscoelastic problems with a modified heat conductivity we refer to Eck-Jarušek-Krbec [8] and the references therein.
In the present paper we consider a thermoviscoelastic system with specific heat linearly increasing with temperature and with constant heat conductivity. Such setting is a particular case of systems addressed in [2] and [21].
The novelty of the existence result presented in this paper concerns the regularity of a 3-D global solution corresponding to sufficiently smooth but arbitrary in size initial data. The proof of the existence theorem is based on the successive approximation method. The key regularity estimates are derived with the help of the parabolic theory in anisotropic Sobolev spaces W 2.1 p,p0 (Ω T ), Ω T = Ω × (0, T ), p, p 0 ∈ (1, ∞), with a mixed norm with respect to space and time variables. Such framework has been previously applied by the authors [19] to the thermoviscoelastic system arising in shape memory alloys. It allowed to generalize the former results on this subject in [27].
As known, in deriving a priori estimates for a solution of a system of balance laws it is common to begin with estimates arising from the conservation of a total energy. Such estimates provide L ∞ -time regularity for the conserved quantities. To take advantage of such time regularity in deriving subsequent regularity estimates it is desirable to work in Sobolev spaces with a mixed norm, for example W 2,1 p,p0 (Ω T ), where the space exponent p is determined by the energy structure and the time exponent p 0 may be arbitrarily large. This is the idea behind using the framework of Sobolev spaces with a mixed norm to the thermoviscoelastic system under considerations. The theory of IBVP's in Sobolev spaces with a mixed norm is the subject of recent theoretical studies. We apply the general results due to Krylov [13] and Denk-Hieber-Prüss [7].
Here Ω ⊂ R 3 is a bounded domain occupied by a body in a fixed reference configuration, and (0, T ) is the time interval. The system is completed by appropriate boundary and initial conditions. Here we assume (1.3) u = 0, n · ∇θ = 0 on S T = S × (0, T ), where S is the boundary of Ω and n is the unit outward normal to S. The field u : Ω T → R 3 is the displacement, θ : Ω T → R + = (0, ∞) is the absolute temperature, the second order tensors ε = (ε ij ) i,j=1,2,3 and ε t = ((ε t ) ij ) i,j=1,2,3 denote respectively the linearized strain and the strain rate.
Above and hereafer the summation convention over the repeated indices is used, vectors and tensors are denoted by bold letters, and the dot denotes the inner product of tensors, e.g.

Moreover,
Equation (1.2) is the energy balance in which the linear Fourier law for the heat flux, q = −k∇θ with constant heat conductivity k > 0, and temperature-dependent specific heat, c v θ with c v > 0, have been adopted. The first two terms on the righthand side of (1.2) represent heat sources created by the deformation of the material and by the viscosity. The field g : Ω T → R is the external heat source.
The boundary conditions in (1.3) mean that the body is fixed at the boundary S and thermally isolated. The initial conditions (1.4) prescribe displacement, velocity and temperature at t = 0.
1.3. Linear elasticity and viscosity operators. Assumptions. For further analysis we formulate problem (1.1)-(1.4) in terms of the linear viscosity and elasticity operators, Q 1 and Q 2 , defined by with boundary and initial conditions (1.3), (1.4). Throughout we shall assume that (A1) Ω ⊂ R 3 is a bounded domain with the boundary S of class at least C 2 ; T > 0 is an arbitrary finite number; (A2) α = (α ij ) i,j=1,2,3 is a second order symmetric tensor with constant α ij ; (A3) The fourth order tensors A 1 and A 2 are defined by (1.5) with the coefficients µ p , λ p , p = 1, 2, satisfying (1.6).
We list the implications of assumption (A3) which are used in further analysis. The condition (1.6) ensures the symmetry of tensors A p : and their coercivity and boundedness Moreover, (1.6) ensures the following properties of operators Q p , p = 1, 2: -Q p are strongly elliptic (property holding true under weaker assumption µ p > 0, λ p + 2µ p > 0, (see [20], Sect. 7)) and satisfy the estimate [17], Lemma 3.2: with positive constants c p depending on Ω. Since clearly, it follows that the norms Q p u L2(Ω) and u H 2 (Ω) are equivalent on D(Q p ); -the operators Q p are self-adjoint on D(Q p ): -the operators −Q p are positive on D(Q p ): Hence, there exist fractional powers Q 1/2 p with the domains D(Q 1/2 p ) = H 1 0 (Ω), satisfying (1.14) Let us also notice that by (1.10) and the Korn inequality Thus, the norms Q 1/2 p u L2(Ω) and u H 1 (Ω) are equivalent on D(Q 1/2 p ).

Main result.
Theorem A. (Existence) Let the assumptions (A1)-(A3) hold, S ∈ C 2 , T > 0 finite and where θ is a constant. Then there exists a solution to problem (1.
Corollary 1.1. The regular solution in Theorem A is uniquely defined.
1.5. Relation to other results. We comment on the connections of our result to the two other global existence results in three space dimensions. Firstly, we mention the result by Roubíček [21] who proved the existence of a very weak solution to the thermoviscoelasticity system (1.1)-(1.2) involving monotone viscosity of a p-Laplacian type, (A 1 ε t ) · ε t ∼ |ε t | p , and the specific heat having (ω − 1)-polynomial growth, c * (θ) ∼ c v θ ω−1 . This result, based on the Galerkin method, was obtained for L 1 -data under the conditions p ≥ 2, ω ≥ 1 and p > 1 + 3 2ω (in 3-D). In the case of linear viscosity, p = 2, the latter condition implies that ω > 3/2, that is the growth of the specific heat is greater than 1/2.
Our result concerns the case p = 2 and ω = 2. We have to restrict ourselves to the linear viscosity, p = 2, because the proof relies on the results by Krylov [13] and Solonnikov [24] on the solvability of the linear problem with the boundary and initial conditions (1.3), (1.4) (see Lemma 3.4). Concerning the specific heat growth exponent, ω − 1, it seems that after some additional technical effort it would be possible to admit ω < 2. However, in the case of a constant specific heat, i.e., ω = 1, we have been faced with a serious mathematical obstacle.
As already mentioned in Subsection 1.1, the local existence result in such a case was obtained by Bonetti and Bonfanti [3].
Secondly, we recall the multidimensional result by Blanchard and Guibé [2] who addressed problem (1.1)-(1.2) equally in the prototype case p = 2 and ω = 2, and in the more general setting involving linear viscosity, p = 2, specific heat with (ω − 1)polynomial growth and a nonlinear thermoelastic coupling; more precisely, the term ∇θ in (1.1) was replaced by ∇f (θ) with f hawing an α-polynomial growth. The existence of solutions in the weak-renormalized sense was proved there by the Schauder fixed point theorem. It is worth to remark that in the case of the linear thermoelastic coupling, α = 1, the result in [2] requires the specific heat to have growth of the order greater than 1/2, as in the result by Roubíček [21].
1.6. Outline. In Section 2 we present a thermodynamic basis of system (1.1), (1.2). Section 3 recalls basic results on the Sobolev spaces with a mixed norm and on the solvability of boundary-value problems for linear parabolic equations in such spaces. In Section 4 we derive a priori estimates for problem (1.1)-(1.4). The procedure consists in a recursive improvement of the basic energy estimates. The main tool in this procedure are the results on the solvability of linear parabolic problems in Sobolev spaces with a mixed norm. Section 5 presents the proof of Theorem A, which is based on the successive approximation method. The proof of the uniqueness, stated in Theorem B, is given in Section 6.
The norms of θ will be later removed by some interpolation inequalities based on estimate (1.18). In Lemma 4.7 we obtain the estimate and next in Lemma 4.8, To deduce the boundedness of θ we first prove in Lemma 4.10 the estimate and in Lemma 4.17, To get (1.21) we make use of the important inequality (see Lemma 4.9) ε t Lp,σ(Ω T ) ≤ c(t)( θ Lp,σ(Ω T ) + data), p, σ ∈ (1, ∞), and the fact that the coefficient near θ t in (1.2) is proportional to θ.
2. Thermodynamic basis. System (1.1), (1.2) represents balance laws for the linear momentum and energy in a referential description, with the referential mass density assumed constant, normalized to unity, ρ 0 = 1: where S is the stress tensor, q -the referential heat flux, and e -the specific internal energy.
In the case of (1.1), (1.2) the free energy is specified by is the caloric energy, and is the elastic energy; we recall that A 2 stands for the fourth order elasticity tensor and α for the second order thermal expansion tensor. The caloric energy (2.3) is associated with temperature-dependent caloric specific heat which gives rise to the term c v θθ t in energy equation (1.2).
We remark that in the case f * is given by the standard formula where c v , θ 1 ,c are positive constants, the caloric specific heat is constant This gives rise to the usual parabolic term c v θ t in (1.2) in place of c v θθ t . As mentioned in Section 1, in the case of a constant caloric heat there are serious mathematical obstacles in the proof of the global existence.
A thermoviscoelastic system with the specific heat c * (θ) given by (2.5) has been considered in [9], [2] and [21], where also more general forms of c * (θ) have been analysed. Moreover, we mention that a fourth order thermoviscoelastic systems with temperature-dependent specific heat, arising in shape memory materials, have been studied in [27] and [19].
The dissipation potential corresponding to system (1.1), (1.2) is given by where A 1 is the viscosity tensor and k > 0 the constant heat conductivity.
In accord with the basic thermodynamic relations the internal energy e and the entropy η are related to the free energy f by the equations For the free energy f defined by (2.2)-(2.4) this gives As a consequence of the second law of thermodynamics expressed by the Clausius-Duhem inequality, the stress tensor S and the heat flux q satisfy the following relations: For f given by (2.2)-(2.4) and D by (2.8) the formulas (2.11) yield the standard forms of the stress tensor and the heat flux Thus, S consists of two terms: the nondissipative equilibrium term determined by f , and the dissipative one determined by D. The dissipative heat flux q is entirely determined by D.
For further purposes (see Lemma 4.2) it is of interest to notice that on account of the identity along with the relation (2.11) 1 , the energy balance (2.1) 2 admits the form For D given by (2.8) this leads to the following equivalent form of equation (1.2): (2.14) Let us also notice that assuming θ > 0 and using (2.11) 2 , the equation (2.13) may be expressed as is the specific entropy production. From (2.15) it follows that system (1.1), (1.2) complies with the Clausius-Duhem inequality 3. Notation and auxiliary results.

Notation.
Let Ω ⊂ R n be an open bounded subset of R n , n ≥ 1, with a smooth boundary S, and Ω T = Ω × (0, T ), S T = S × (0, T ), T > 0 finite. We introduce the following spaces: -the Sobolev space with a mixed norm, which is a completion of C ∞ (Ω T )-functions under the finite norm is the integer part of s. For s odd the last term in the above norm vanishes whereas for s even the two last terms vanish.
In [12] it has been proved 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 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
We recall from [4] the trace and the inverse trace theorems for Sobolev spaces with a mixed norm.
(Ω), and where constant c does not depend onũ.
We recall also (see [1]) that if l > 1/p then every function from B l p,p0 (Ω) has a trace on the boundary S belonging to B l−1/p p,p0 (S), and We apply the following imbeddings between Besov spaces.
We recall now from [19] a result on the solvability of a linear parabolic system with elasticity operator Q in Sobolev space with a mixed norm. This result will be repeatedly used in Section 4 in deriving a priori estimates for viscoelasticity system (1.1). It generalizes the result by Krylov [13] from the single parabolic equation to the following parabolic system with µ > 0, ν > 0. Let us notice that letting the assumption (1.6) implies that µ > 0 and ν > 0.
(Ω). Assume the compatibility condition Then there exists a unique solution to (3.1) such that u ∈ W 1,1/2 p,p0 (Ω T ) and with a constant c depending on Ω, S, p, p 0 .
In the proof of Theorem A we shall apply also the following regularity result for a linear parabolic equation. This result is the special case of a more general theorem due to Denk-Hieber-Prüss [7, Theorem 2.3].

A priori estimates.
In this section we derive a priori estimates for solutions of problem (1.1)-(1.4) on an arbitrary finite time interval (0, T ). The estimates are essential for the existence proof by the successive approximation method, presented in Section 5.
The procedure of deriving a priori estimates consists in a recursive improvement of the basic energy estimates. The main tool used in this procedure is Lemma 3.4 which provides the solvability of the viscoelasticity system (1.8) 1 in the Sobolev space W 2,1 p,p0 (Ω T ). The applied procedure is aimed to establish the continuity of temperature θ and finally to apply Lemma 3.5 on the solvability of parabolic equation in Sobolev space W 2,1 q,q0 (Ω T ). Throughout this section we assume that assumptions (A1)-(A3) (see Sect. 1.3) hold, and where θ is a positive constant.
Proof. For m ∈ R + let us define the truncation Multiplying (1.2) by −θ −̺ m with ̺ > 2 (admissible test function) and integrating over Ω m (t) gives The first term on the left-hand side of (4.3) is equal to The second term on the left-hand side of (4.3) equals because ∇θ m = ∇θ for x ∈ Ω m (t) and ∇θ m = 0 for x ∈ Ω \ Ω m (t). On account of (1.10) the third term on the left-hand side of (4.3) is bounded from below by (4.6) and the fourth one by (4.7) In view of the boundedness of A 2 and α, the integral on the right-hand side of (4.3) is estimated as follows: Now, setting δ/2 = a 1 * and incorporating (4.4)-(4.8) into (4.3) we arrive at where in the last inequality we used the fact that θ m > 0 in Ω. Hence, by the Gronwall inequality, it follows that that is, with a constant c independent of ̺ and m. Letting ̺ → ∞, (4.9) yields the bound Further, letting m → ∞ and noting that for sufficiently large m, θ m (0) = = max θ 0 , 1 m ≥ θ, we conclude the bound (4.2). Lemma 4.2. (Energy estimates) Let us assume that (4.1) holds, θ > 0, and Then a sufficiently smooth solution (u, θ) to (1.1)-(1.4) satisfies the estimate (4.10) Proof. Note that the positivity of θ is ensured by Lemma 4.1. Multiplying (1.1) by u t , integrating over Ω and integrating by parts using boundary condition (1.3) 1 , gives Further, integrating (1.2) over Ω and by parts using (1.3) 2 yields Next, multiplying (1.2) by 1/θ, integrating over Ω and integrating by parts leads to Adding by sides (4.11) and (4.12) gives (4.14) d dt Now, multiplying (4.13) by a positive constant β, and subtracting by sides from (4.14), we get Integrating (4.15) with respect to time, noting that by the boundedness of A 2 and α, and using the coercivity of A 1 (see (1.10)), we get for t ≤ T , with a constant c depending only on parameters. Now, applying the Young inequality to the second term on the right-hand side of the above inequality yields (4.10).
Remark 4.3. By integrating the identity (4.14) with respect to time one can immediately conclude that which is a part of inequality (4.10).
On the contrary to (4.10) this inequality does not require the assumption θ > 0. We point out that in deriving further estimates we shall use just the bounds in (4.16) loosing the information contained in the two dissipative terms of (4.10). This information may be of importance in the analysis of the long time behaviour of solutions. Remark 4.4. We complement Lemma 4.2 by some physical interpretations. In view of (2.10) 1 , (2.12) and boundary conditions (1.3), identity (4.14) represents the balance equation for the total energy d dt On the other hand, in view of (2.10) 2 , (2.12) 2 and the boundary condition (1.3) 2 , identity (4.13) represents the balance equation for the entropy with the entropy production Equation (4.15) represents the so-called availability identity where β = const > 0. Hence, since σ ≥ 0, it follows that if the external sources vanish and if the boundary conditions on S imply that This provides the Lyapunov functional Ω e + 1 2 |u t | 2 − βη dx, which is nonincreasing on solutions paths. Let us notice that the boundary conditions (1.3) ensure (4.18). The identity (4.17) has been used in deriving energy estimates in Lemma 4.2.
Our goal now is to derive further regularity properties from the energy estimates (4.10). To this purpose we use the regularity results for parabolic systems in Sobolev space with a mixed norm, stated in Lemmas 3.4 and 3.5.
For further purposes (see the proof of Lemma 4.7) we prepare now some inequalities between the norms of u and θ.
and the constant c(t) depends exponentially on t.
Inserting the above estimates into (4.59) leads to Hence, by the Gronwall inequality, we conclude that for t ∈ (0, T ), with A 6 defined by (4.57). This gives (4.56). The proof is completed.
Proof. Let us consider once more the identity (4.58). Regarding (4.81), we have (4.83) Taking into account that θ ≥ θ * > 0, and using the inequality where constant c is independent of r, we deduce from (4.83) that Hence, which in conjunction with (4.81) leads to Now, letting r → ∞ in (4.84) we get the assertion.
The next step consists in obtaining a "better" estimate for θ by means of the parabolic regularity result stated in Lemma 3.5. To apply this result to the quasilinear equation (1.2) we need to prove first that θ is a continuous function on Ω T . We are able to prove more, namely the Hölder continuity by means of the parabolic De Giorgi method in the same way as in [18, Lemma 6.1].
Since the above reference concerns much more general situation, we present here for reader's convenience a direct, simpler proof.
For τ sufficiently small there exists a positive constant A such that (5.28) X 0 (τ ) ≤ A and ϕ(τ a A, D) ≤ A.
Then (5.27) implies that X n (τ ) ≤ A for any n ∈ N.