ASYMPTOTIC ANALYSIS FOR VANISHING ACCELERATION IN A THERMOVISCOELASTIC SYSTEM

We have investigated a dynamic thermoviscoelastic system (2003), establishing existence and uniqueness results for a related initial and boundary values problem. The aim of the present paper is to study the asymptotic behavior of the solution to the above problem as the power of the acceleration forces goes to zero. In particular, well-posedness and regularity results for the limit (quasistatic) problem are recovered.


Introduction
A three-dimensional model for thermoviscoelastic phenomena proposed by Frémond has been investigated in [2] from the point of view of existence and uniqueness of solutions.In the present paper, we examine the asymptotic behavior of the solution to the system considered in [2], as the power of the acceleration forces vanishes.As we are mainly interested in the analytical investigation of the model, we just recall the evolution equations and refer to [2,9], and the references therein, for a detailed thermomechanical derivation and justification.
Thus, we consider a viscoelastic body (see, e.g., [7]) located in a smooth bounded domain Ω ⊂ R 3 during a finite time interval [0,T], T > 0. Its thermomechanical evolution is described by the variables θ (absolute temperature) and u (small displacement).The latter is considered as a scalar quantity, in regard of simplicity.The equations of the model are the universal balance laws of the thermomechanics, that is, the energy balance and the momentum balance; the constitutive relations are derived from a free energy functional and a pseudo-potential of dissipation (see [9]).In particular, the energy balance law (considering the Fourier heat flux law) leads to the equation c s ∂ t θ − k 0 ∆θ = θ∇∂ t u • a + µ ∇∂ t u 2 + r a.e. in Ω×]0,T[, (1.1) and the momentum balance law (accounting for the accelerations) takes the form where c s > 0 denotes the heat capacity of the system, k 0 > 0 denotes the thermal conductivity, ε > 0 is proportional to the mass density, µ > 0 and ν > 0 are respectively a dissipation and an interaction coefficient, r stands for an external heat source, and f stands for an external volume force.Moreover, a is a three-dimensional constant vector related to the thermal expansion coefficient (we will assume, e.g., a = (1,1,1)).Finally, ∂ t denotes ∂/∂t, ∂ tt indicates ∂ 2 /∂t 2 , ∆ is the Laplacian, and ∇ is the gradient (in space variables).
For the sake of simplicity, we will take r = f = 0 and we will get rid of the physical constants.However, we point out that our results apply also to the more general case of nonzero applied volume force f and exterior heat source r, provided f and r are taken with sufficient regularity.
We remark that in (1.1) we deal with the complete energy balance equation, as we retain all the mechanically induced heat sources at the right-hand side, coming from the power of interior forces written for the actual velocities, even if they are higher-order nonlinearities.Note that, from the analytical point of view, the presence of quadratic terms in (1.1) involving strain rate introduces strong technical difficulties.Thus, it seems very hard to obtain global well-posedness results for the above system.Similar difficulties arise in the investigations carried out, for example, in [2,3,4,5].
The aim of the present paper is to study the asymptotic behavior of the solution to (1.1), (1.2), and (1.3), as ε vanishes in (1.2).In [2], we provide a local-in-time existence result, as well global uniqueness of sufficiently regular solutions.Nonetheless, such results cannot be considered as the starting point of our asymptotic analysis because the lifetime of the solution depends on the parameter ε and it can vanish when ε tends to 0. Thus, we derive here an alternative proof which provides existence and uniqueness of solution to (1.1), (1.2), and (1.3) in a time interval independent of ε.Next, by a careful derivation of suitable a priori estimates, we prove convergence and regularity results.Concerning applications, we point out that our asymptotic analysis implies, in particular, the local well posedness for a quasistatic viscoelastic problem (see, e.g., [11] for related results) as a limit of the dynamical one, as the mass coefficient in the inertial term tends to zero.
The rest of the present work is as follows.The next Section is devoted to the notation, the assumptions, and the statements of the main results.Section 3 contains the proof of the existence and uniqueness result given by Theorem 2.2.Section 4 is concerned with the proofs of our first convergence theorem (Theorem 2.3) and of the uniqueness result for the limit problem (Theorem 2.4).Finally, Section 5 is devoted to the proof of the regularity result given by Theorem 2.5 (see also Remarks 2.6-2.7 and Proposition 2.8).Throughout the paper some comments and remarks are given.

Statement of the problem and main results
We start by fixing some notation.Let Ω ⊂ R 3 be a bounded domain with smooth boundary ∂Ω.We set Q t := Ω×]0,t[ for t ∈]0, T[ and Q := Ω×]0,T[.Then, we introduce the notation ) and identify H with its dual space H , so that W V H V W with dense and compact embeddings.We use the same symbol for the norm of a space of scalar functions and the norm of the space of corresponding vector-valued functions.For instance, • V means the norm of both V and V 3 .Besides, let the symbol • indicate the norm of H (or H 3 ).Henceforth, we denote by •, • the duality pairing between V and V , by (•,•) the scalar product in H, and by ((•,•)) the scalar product in V .Then, the associated Riesz isomorphism J : V → V and the scalar product in V , denoted by ((•,•)) * , can be specified by For the reader's convenience, we recall the Poincaré inequality as well as the following inequality (which can be obtained, e.g., applying the Green formula, the Hölder inequality, and (2.3)): We start by considering Problem 2.1 below, that is, the system resulting from (1.1), (1.2), and (1.3), by getting rid of some physical constants and taking external sources and forces equal to zero.
Let ε > 0 be given.We introduce the families of initial data {θ 0ε } ε>0 , {u 0ε } ε>0 , {u 1ε } ε>0 satisfying ) ) ) (2.12) ∂ t u ε (•,0) = u 1ε a.e. in Ω. (2.13) In order to perform an asymptotic analysis on Problem 2.1 as ε tends to 0, from now on, we let ε vary, say, in ]0,1[.As mentioned in the introduction, as a starting point of our asymptotic study, we need here a result (see Theorem 2.2 below) which provides the existence and uniqueness of the local solution to Problem 2.1 during a time interval independent of ε.To obtain such well-posedness result, we suppose the following bounds on the families of the initial data (in agreement with (2.5)-(2.6)): (2.14)  2.15) hold.Then, there exists a final time τ ∈]0, T] such that for any ε ∈]0, 1[ there exists a unique pair (θ ε ,u ε ) solving Problem 2.1 during the time interval ]0,τ[.After proving the above existence and uniqueness result, we are allowed to investigate the asymptotic behavior of the pair (θ ε ,u ε ) given by Theorem 2.2, as ε 0. To this aim, we require, in addition, the following convergence conditions on the sequences of the initial data allowing us to pass to the limit in (2.9), (2.10), (2.11), and (2.12): ) as ε 0.
Our first asymptotic result reads as follows.
Remark 2.6.In view of Theorems 2.3 and 2.4, from (2.26), we deduce the further convergences (2.29)Moreover, from (2.27), (2.28), and (2.29), using a classical compactness argument (see [10, page 58]) and the generalized Ascoli theorem (see [12,Corollary 4]), we can deduce the following strong convergences: (2.30) (2.31) Thanks to the above convergences, the pair (θ,u) fulfilling as ε 0. From a mechanical point of view, this corresponds to having for the limit problem that the divergence of the stress is 0 at time t = 0, as no volume exterior forces are applied.
In view of Theorem 2.5 (cf.Remark 2.6), we have in addition the following regularity result for the limit problem.

Proof of Theorem 2.2
We aim here to apply the Schauder fixed point theorem to a suitable operatorconstructed as follows.
For R > 0, we consider Y (τ,R) the closed ball of H 1 (0,τ;W 1,4 0 (Ω)) with center 0 and radius R, that is, where τ ∈]0, T] will be determined later in such a way that -: Y (τ,R) → Y (τ,R) is a compact and continuous operator.We consider the following auxiliary problems whose well-posedness is guaranteed by standard arguments.Thus, for the sake of brevity, we omit any details (cf., e.g., [1,8]).
E. Bonetti and G. Bonfanti 111 Now, given such θ, let u be the unique solution of the following problem. ) We have actually introduced a mappingsuch that -( u) = u.Our aim is to show that, at least for small times, the Schauder theorem applies to the mapfrom Y (τ,R) into itself.To do this, we start by deriving some a priori bounds on θ and u.In the sequel, c will denote any positive constant possibly dependent on data of the problem but not on ε.Of course, c may vary from line to line.
Moreover, we recall here the Young inequality which will be useful in the sequel: for all a,b ∈ R + , δ > 0 and p > 1, q < ∞ such that 1/ p + 1/q = 1.Now, in order to obtain a priori bounds on θ, we choose v = θ in (3.3) and we integrate from 0 to t, with 0 < t < τ.Owing to the Hölder inequality and the continuous embedding V L 4 (Ω), we have θ(s) ds. (3.10) Next, in order to recover the full V-norm of θ in the left-hand side, we add to (3.10) θ 2 L 2 (0,t;H) .Then, we use (3.9) and we get Recalling the definition of Y (τ,R) and (2.14), we apply to (3.11) a generalized version of the Gronwall lemma introduced in [1] and we deduce that there exists a positive constant c 4 depending on T, Ω, c 1 , and R such that θ L ∞ (0,τ;H)∩L 2 (0,τ;V ) ≤ c 4 . (3.12) Next, in order to obtain a priori bounds on u, we multiply (3.6) by −∆∂ t u and we integrate over Q t , with 0 < t < τ.Applying the Hölder inequality, we have In account of (2.14), (2.15), and (3.12), using (3.9), we deduce for some positive constant c 5 with the same dependence of c 4 and depending in addition on c 2 .
Next, proceeding formally, we multiply (3.6) by ∂ tt u and we integrate over Q t (the procedure can be made rigorous using, e.g., the tools in [6, appendix]).We have We estimate the right-hand side in (3.16) integrating by parts (in time and in space).Using also (3.9), we can write Qt (3.17) Similarly arguing, we have (3.20) Then, we recall that, due to (3.1) and (3.12), θ ∇∂ t u • a + |∇∂ t u| 2 belongs to L 1 (0,τ) and the Gronwall lemma enables us to deduce u W 1,∞ (0,τ;H 1 0 (Ω)) ≤ c 6 , (3.21) for some positive constant c 6 with the same dependence of the previous constants.Now our aim is to find τ (independent of ε) such thatmaps Y (τ,R) into itself.Using standard interpolation inequalities for L p -norms, from (3.14) and (3.21), we can deduce for some positive constant c 7 .
By the Hölder inequality we have Thus, to ensure that u ∈ Y (τ,R), that is it is enough to choose τ ∈ (0,T] such that, for example, τ ≤ R 8 /c 8  8 .Note that τ can be chosen independently of ε. Moreover, the above arguments (cf.for a positive constant k independent of the choice of u ∈ Y (τ,R) (depending on ε and the previous constants), which ensures thatis a compact operator.Hence, to achieve the proof of the Schauder theorem, it remains to show thatis continuous with respect to the natural strong topology induced in Y (τ,R) by H 1 (0,τ;W 1,4 0 (Ω)).To this aim, we consider a sequence { u n } n∈N ⊂ Y (τ,R) such that u n −→ u strongly in H 1 0,τ;W 1,4  0 (Ω) , (3.27) as n → +∞.Denote now by θ n (resp., θ) the solution to Problem 3.1 corresponding to the datum u n (resp., u) and by u n = -( u n ) (resp., u = -( u)) the solution to Problem 3.2 corresponding to θ n (resp., θ).Let finally θ = θ n − θ and ũ = u n − u.We consider the difference between the corresponding equations (3.3), we test by θ, and we integrate over (0,t).Arguing as in the derivation of (3.10), we get for some positive constant c independent of n.Applying (3.9), we have for some positive constant c independent of n.Then, applying the Gronwall lemma to (3.29), in account of (3.12) and (3.27), we infer as n → +∞.Now, we consider the difference between the corresponding equations (3.6), we multiply by −∆∂ t ũ, and we integrate over Q t (cf.(3.13)).Mainly exploiting (3.30), we readily deduce as n → +∞.

E. Bonetti and G. Bonfanti 115
We conclude thathas a fixed point in Y (τ,R), that is, there exists a pair (θ ε ,u ε ) (hereafter we make explicit the dependence of the solution on ε, by using a subscript ε) solving Problem 2.1 during the time interval ]0,τ[.Actually, we have to complete the proof of the regularity for θ ε specified by (2.7).To this aim, after adding θ ε to both sides of (2.9), we test by J −1 ∂ t θ ε and we integrate over (0,t).Using the Hölder inequality, recalling the definition of J and the continuous embedding V L 4 (Ω), we obtain Hence, the further regularity specified by (2.7) easily follows.
Remark 3.3.The nonnegativity of θ ε a.e. in Q τ can be proved proceeding as in [2], using a maximum principle argument, and exploiting the nonnegativity of the initial datum θ 0ε (cf.(2.5)).For the sake of brevity, we just sketch the proof and refer to [2] for the details.Note that in the case of nonzero heat source r, one has to prescribe in addition r ≥ 0 (and analogously for a nonhomogeneous Neumann boundary condition for the temperature).Test (2.9) by −θ − ε (θ − ε is the so-called negative part of the function θ ε ), and integrate over (0,t).Thanks to the nonnegativity of θ 0ε , standard arguments show from which the Gronwall lemma implies that is, θ − ε = 0 a.e. in Q τ , from which it follows θ ε ≥ 0 a.e. in Q τ .Now, in order to complete the proof of Theorem 2.2, we have to show that the solution to Problem 2.1 is unique.This fact follows applying [2, Theorem 2.2].Nonetheless, we prove here directly the uniqueness result, providing an alternative proof.We stress that such a result holds in any interval where the solution exists.We proceed by contradiction.Let (θ ε1 ,u ε1 ) and (θ ε2 ,u ε2 ) be two solutions to Problem 2.1 during some interval ]0,τ[, τ ∈]0, T].Set θ ε = θ ε1 − θ ε2 and ũε = u ε1 − u ε2 .We consider the difference between the corresponding equations (2.10), we multiply it by ∂ t ũε , and we integrate over Q t .We get 116 Quasistatic thermoviscoelasticity Next, we consider the difference between the corresponding equations (2.9), we add (in both sides) θε , we test by J −1 θ ε , and we integrate from 0 to t; we obtain where the integrals I i (t), i = 1,2,3, are specified as follows: (3.38) We handle I 1 (t) using again the Hölder inequality, (3.9) and the Sobolev embeddings.

Proof of Theorems 2.3 and 2.4
In the present section, we will study the asymptotic behavior of the solution to Problem 2.1 given by Theorem 2.2, as ε 0. To this aim, we need some a priori estimates (independent of the parameter ε) concerning the pair (θ ε ,u ε ), holding in the interval ]0,τ[.Remark that such asymptotic analysis is meaningful, because τ can be chosen independently of ε.The a priori estimates allowing us to pass to the limit in (2.9)-(2.10)can be established arguing as in the previous section.In fact, the fixed point argument employed in the proof of Theorem 2.2 provides some uniform bounds on the solution according to the regularity specified by (2.7)-(2.8)(recall that u ε ∈ Y (τ,R), that is u ε H 1 (0,τ;W 1,4  0 (Ω)) ≤ R, for any ε ∈]0, 1[).Thus, we can reason as in the derivation of (3.12), (3.14), (3.21), (3.22), and (3.33) and deduce ) for some positive constant c independent of ε.
Our next goal is to pass to the limit in (2.9).The critical terms are the nonlinearities in the right-hand side.In order to handle them, we will show the further strong convergence To this aim, we consider the difference between (2.10) and (2.21); we multiply it by ∂ t (u ε − u) and we integrate over Q t , with 0 < t < τ.After an integration by parts in space, using (2.3) and (3.9), we obtain

Proof of Theorem 2.5
Owing to the better regularity assumed on the families of the initial data (cf.(2.24)-(2.25)),we derive here some further bounds (independent of the parameter ε) concerning the pair (θ ε ,u ε ).