UNIFORM ATTRACTORS FOR A PHASE TRANSITION MODEL COUPLING MOMENTUM BALANCE AND PHASE DYNAMICS

. In this paper we are concerned with the uniform attractor for a nonautonomous dynamical system related to the Fr´emond thermo-mechanical model of shape memory alloys. The dynamical system consists of a diﬀusive equation for the phase proportions coupled with the hyperbolic momentum balance equation, in the case when a damping term is considered in the latter and the temperature ﬁeld is prescribed. We prove that the solution to the related initial-boundary value problem yields a semiprocess which is continuous on the proper phase space and satisﬁes a dissipativity property. Then we show the existence of a unique compact and connected uniform attractor for the system.


1.
Introduction. Let us fix a bounded and regular subset Ω of R 3 and consider the following system (VSMA) of partial differential equations and relations in terms of the unknown functions χ 1 , χ 2 and u, in the space-time domain Q = Ω × (0, +∞) u tt + cu t − div (−ν∆(div u) + λ div u)I + 2µε(u) + α(ϑ) χ 2 I = G , (1.2) χ 1 (·, 0) = χ 0 1 , χ 2 (·, 0) = χ 0 2 , u(·, 0) = u 0 , u t (·, 0) = v 0 in Ω, (1.3) ∂ n χ j = 0 on ∂Ω × (0, +∞), j = 1, 2, (1.4) u = 0 on ∂Ω × (0, +∞), (1.5) ∂ n (ν div u) = 0 on ∂Ω × (0, +∞). (1.6) Here, k, η, ℓ, ϑ * , c, λ, µ are strictly positive parameters, while the coefficient ν is allowed to be greater than or equal to 0. Note that u = (u 1 , u 2 , u 3 ) ∈ R 3 , ε(u) := (ε ij ) is the strain tensor with ε ij := 1 2 ∂u i ∂x j + ∂u j ∂x i , i, j = 1, 2, 3, The system (VSMA) arises in the study of the behaviour of a viscoelastic shape memory body subjected to mechanical deformations when the temperature field is prescribed. A shape memory material is a metallic alloy which exhibits this peculiar and surprising behaviour: it can be permanently deformed (avoiding fractures) up to 8% of its strain and subsequently forced to recover its original shape just by thermal means. This unusual property is used nowadays in a variety of engineering applications. In particular, the field of applications of shape memory technologies ranges from bio-engineering, to structures-engineering and aerospace sciences (see [9]). The shape memory phenomenon has been interpreted as the effect of a thermo-elastic solid-solid phase transition between two different crystallographic configurations, the austenite, which is stable at higher temperatures and variants of martensite, stable at lower temperatures (see [2] and [16]). Here, we are interested in the macroscopic modelling proposed by Frémond in [16]. In this connection, ϑ has to be regarded as the absolute temperature (assumed to be known) of the shape memory sample, while u stands for the vector of small displacements. Hence, the 2-tensor ε(u) represents the linearised strain tensor. Finally, χ 1 , χ 2 are quantities related to the pointwise proportions of the phases. In particular, if β 1 , β 2 , β 3 denote the local proportions of the two martensitic variants and of the austenite, respectively, we point out that the following conditions have to be fulfilled 0 ≤ β i ≤ 1 for i = 1, 2, 3, and β 1 + β 2 + β 3 = 1.
Finally, G stands for the density of the body forces while α is a rather smooth function related to the thermal expansion of the system (see [16] for further details): in fact, let us refer to [16] for the physical derivation of the model and the related comments. However, we point out that, although the full Frémond's model comprises the evolution of an unknown temperature field as well, our setting, in which ϑ is a given datum, is physically justified and interesting for applications. We recall that also the positive damping cu t (which is not present in the original Frémond's model) in (1.2) has a physical motivation. In particular, this element can be understood as a friction term, hence it serves as a dissipation mechanism.
The mathematical analysis of Frémond's model was initiated in [12] and then extended in various directions. In particular, the reader is referred to [4] and [5] (and references therein) for an updated and minute presentation of the analytical results concerning Frémond's model. We note however, that [12], [4], [5] and most of the quoted references deal with the quasistationary situation, in which the macroscopic accelerations are not taken into account. On the other hand, the case in which the macroscopic accelerations are retained in the momentum balance has been studied in [10] (in the simple setting of a linearized energy balance and without diffusive effects for the phase variables) and, more recently, in [26] (for the system (1.1)-(1.6) when c = 0). Let us point out that, to our knowledge, the initial-boundary value problem for the full Frémond's model with macroscopic accelerations and the energy balance equation looks rather difficult and is still open even for the existence of solutions.
The problem of the long-time behaviour of the solutions to Frémond's model has been first considered in [14]. In that paper, the authors investigate the convergence to the steady state solutions for the full model (but without the inertial term in the momentum balance equation) in the one-dimensional situation. The structure of the ω-limit set has been further analysed in [30]. There, still in the 1-D case it is shown how, in a prescribed temperature range, the set of solutions to the stationary problem contains elements that present a deeply structured alternance of martensitic variants. This fact is in complete agreement with experimental evidence. The study of the asymptotic stability from the point of view of global attractors has been tackled in [15] in the one-dimensional setting and then extended to the three-dimensional situation in [11]. The analysis of [15] relies on the crucial observation that, in the absence of inertial terms, the momentum balance equation (i.e., our (1.2) without the part u tt + cu t ), along with the boundary conditions (1.5) and (1.6), allows to completely determine the displacement u in terms of data of the problem and other unknowns. Thus, the original system for three unknowns ( χ 1 , χ 2 ), u, ϑ reduces to a system for the two unknowns ( χ 1 , χ 2 ), ϑ, in which u (that is now a function of ( χ 1 , χ 2 ), ϑ) plays the role of a driving force depending on time. Consequently, the system is intrinsically nonautonomous. The long-time dynamics of the related dynamical system has been characterised with the aid of the study of a proper limiting autonomous system. In [11] the same type of result has been extended to the three-dimensional situation.
In this paper, we aim to analyse the long-time behaviour in terms of global attractors for (1.1)-(1.2). Our situation differs from the one studied in [15] and [11], since we retain the macroscopic acceleration in the momentum balance equation (1.2), thus obtaining a hyperbolic equation for the vector u. Moreover, we assume that the evolution of the temperature field ϑ is known. In this regard, the function ϑ becomes a forcing term depending on time. In particular, our system (VSMA) is nonautonomous. The problem of existence, uniqueness and continuous dependence of solutions to the full system (1.1)-(1.6) has been investigated in [26] when c = 0 in (1.2). The presence of this damping term is however mandatory from the point of view of the long-time behaviour: in fact, it provides some dissipation to the system. The existence and uniqueness analysis of [26] refers to the situation in which c = 0, but let us note that all the results established in [26] extend to the case c = 0. In this paper, we rely on the concept of uniform attractor to handle the fact that the system is nonautonomous (see Subsection 2.2). In particular, we will prove that the solution operator to (a suitable weak formulation of) (1.1)-(1.6) is a semiprocess which is continuous on the proper phase space (see Theorem 2.8). Then, we show the dissipativity of the system in Theorem 3.1 and finally the existence of a unique compact and connected uniform attractor in Theorem 3.5. The crucial step in proving the existence of the uniform attractor relies on the proof of some form of compactness for the solution operator. The simplest and, by the way, the strongest form of compactness one could expect is that the solution operator itself becomes a compact operator after some finite time. Unfortunately, this form of compactness is not usually available for hyperbolic equations (as our (1.2)). Thus, we rely here on the concept of uniform asymptotic compactness, which is well known for autonomous and also nonautonomous systems (see [24]). In the simpler autonomous setting, the uniform asymptotic compactness (there called asymptotic compactness) means, roughly speaking, that {u n (t n )} is convergent (up to a subsequence) with respect to the topology of the phase space, when t n ր +∞ and {u n } is a sequence of solutions corresponding to initial data bounded in the phase space. Now, if the underlying phase space is a Hilbert space (but a uniformly convex Banach space would be sufficient), one can try to prove the abovementioned convergence by first checking the weak convergence of the sequence and then showing the convergence of the correspondig norms, by means of the energy identity provided by the equation. Hence, the so-called energy method introduced by J.M. Ball in [3] provides an efficient and elegant way to establish the asymptotic compactness.
The same strategy works also in the nonautonomous case. In fact, following [24], once the concept of uniform asymptotic compactness has been introduced, one can prove the uniform asymptotic compactness of the process by using a proper extension of the energy method. In the present paper, we follow exactly this approach. However, it is worthwhile noting that there exists at least another method to prove the asymptotic compactness of the system. As in the autonomous case, one can try to decompose the solution operator into two parts: a (uniformly) compact part and a part which decays to zero as time goes to infinity (see, e.g., [7], [17], [18]). This method could be in principle successfully applied to our system (1.1)-(1.6), under some extra regularity with respect to time for the forcing function G than the one we use to prove the mere existence of solutions and the continuity of the semiprocess (cf. Theorems 2.6 and 2.8). In this concern, we can say that the energy method, essentially relying on the standard energy estimate for hyperbolic problems, is optimal with respect to the regularity of the data. This is the plan of the paper. In Section 2 we introduce the weak formulation of (1.1)-(1.6). Moreover, we summarize some preliminary machinery on the long-time behaviour of nonautonomous dynamical systems. Section 2.3 contains the results on the well-posedness of the weak formulation of (1.1)-(1.6). Finally, in Section 3 we prove the existence of the uniform attractor for (VSMA).

Mathematical setting and preliminaries.
2.1. Function spaces and weak formulation. We first introduce some notation. We set where the coefficient ν in the definition of V allows to consider at the same time both the ν = 0 case and the ν > 0 situation. As usual, we identify H and H with their respective dual spaces H ′ and H ′ , so that V ⊂ H ⊂ V ′ and V ⊂ H ⊂ V ′ may be regarded as classical Hilbert triplets. The spaces H, V, H will be endowed with usual norms, while for V we prescribe the equivalent norm In the sequel, we denote by (·, ·) Ω the scalar product in H or in H, by ·, · the duality pairing between V ′ and V or between V ′ and V . The symbol · E will indicate the norm in a generic normed vector space E. In addition, we introduce the continuous and symmetric bilinear form a(·, Recalling the well-known Korn inequality, the following property holds for some positive constant c a . Moreover, we have that Since the special triangular form of K in (1.7) is not needed in our analysis, we let K stand for any bounded, convex and closed subset of R 2 such that (0, 0) ∈ K. Consequently, we denote by K := {(γ 1 , γ 2 ) ∈ H × H : (γ 1 , γ 2 ) ∈ K a.e. in Ω} the realization of K in H×H, which is clearly bounded, convex and closed. In particular, it is straightforward to find a positive constant c K such that for all (γ 1 , γ 2 ) ∈ K and almost everywhere in Ω. The symbol I K will clearly indicate the indicator function of K, while ∂I K stands for its subdifferential, which is now a maximal monotone operator in H × H.
In order to describe the asymptotic behaviour of solutions, we need to introduce the Banach space of L p loc -translation bounded functions with values in a Banach space B. More precisely, for p ≥ 1 we set 6) (note that this is not the standard position) and consequently define We prescribe the following assumptions on data
for every t ≥ 0, solving almost everywhere in the time interval (0, +∞) and such that Long-time behaviour of nonautonomous evolution systems: the abstract setting. In this subsection, we present some known results on the long-time behaviour of nonautonomous systems, especially in connection with the construction of the so-called uniform attractor. The reader is referred to the seminal references [28,29,19,7] for the related proofs and further remarks. The basic concept in studying the long-time behaviour of a nonautonomous system is the notion of semiprocess. Let X and Σ be two complete metric spaces. We say that the set {U σ (t, τ )} t≥τ ≥0, σ∈Σ is a family of semiprocesses in X if the following properties are satisfied for each σ ∈ Σ. Σ is the so-called symbol space. As we shall see in the concrete case of system (2.19)-(2.21), the symbol space Σ will be a space of time-dependent functions, which collects all the forcing terms that depend on time.
be a semigroup of translations in Σ, that is (T h (σ))(t) := σ(t + h), and assume the following translation invariance condition The parameter σ is then termed the time symbol of the semiprocess U σ (t, τ ). The class of translation compact forcing functions will be of interest for us: we say that σ is translation compact if for all t ≥ 0 and h ≥ 0.
The class of translation compact functions is quite large. For example, it contains the constant B-valued funtions and the periodic, quasiperiodic and the almost periodic functions (in the Bochner-Amerio sense, see [1]).
The family of semiprocesses is said to be X × Σ−continuous if, for any t, τ with Now, let us recall the notions of absorbing set and attractor for the family of semiprocesses {U σ (t, τ )} t≥τ ≥0, σ∈Σ . We say that B ⊂ X is a uniformly absorbing set if for any τ ≥ 0 and any B ⊂ X bounded, there exists a time T = T (τ, B) ≥ τ such that U σ (t, τ )B ⊂ B for any t ≥ T and for all σ ∈ Σ. Then, we say that C ⊂ X is uniformly attracting denotes the semidistance of two sets A, B ⊂ X . Finally, we say that A is the uniform attractor for the family {U σ (t, τ )} if it is at the same time uniformly attracting and contained in every closed uniformly attracting set (minimality property). Then, it is unique by construction. Now, we quote a general abstract criterion providing sufficient conditions for the existence of the uniform attractor (see [ Then, {U f (t, τ )} possesses a compact uniform attractor.
Note that the above definition of uniform asymptotic compactness (taken from [24]) is different from the one given by Haraux [19]. More precisely, in [19] a semiprocess is said to be uniformly asymptotically compact if it possesses a compact uniformly attracting set in the sense of (2.29). However, it is not difficult to prove that if a semiprocess is uniformly asymptotically compact in the sense of (2.30) and possesses a bounded uniformly absorbing set, then it is uniformly asymptotically compact in the sense of Haraux. Furthermore, we can note that the notion of uniform asymptotic compactness introduced in [24] and used in this paper, is also completely in agreement with the corresponding definition for semigroups (see [21]) and seems easier to be verified using the energy method.
Starting from the semiprocess U f (t, τ ), f ∈ H(σ), we can define a semigroup S t acting on the extended phase space X × H(σ) as follows This construction is well known (see, e.g., [7]). It is also well known that the uniform attractor A could be equivalently defined in terms of the global attractor A ⊂ X × H(σ) of the semigroup S t , that is This construction will help us in proving the connectedness of the uniform attractor for our system (VSMA). We conclude this subsection by recalling two technical results which will be useful in the course of our analysis. We start with the so-called Uniform Gronwall Lemma (for the proof see [32, Lemma III.1.1]). Lemma 2.4. Let y, a, b ∈ L 1 loc (0, +∞) be three non negative functions such that y ′ ∈ L 1 loc (0, +∞) and y ′ (t) ≤ a(t)y(t) + b(t) for a.e. t > 0, and let a 1 , a 2 , a 3 be three non negative constants such that Then, we have y(t + 1) ≤ (a 2 + a 3 ) exp a1 for all t > 0.
Next, we prepare a Gronwall-type lemma prompted by [25].
Proof. The existence, uniqueness and continuous dependence result is essentially proved in [26]. Here, we give a proof of the energy equality (2.38) that will be of fundamental importance in proving the uniform asymptotic compactness of the system. To show (2.38) we rely on an approximation argument similar to the one devised in [13,Appendix]. If u ∈ H 2 loc (0, +∞, V ′ ) ∩ C 1 ([0, +∞), H) ∩ C 0 ([0, +∞); V ) solves the hyperbolic equation (2.21), for any ε > 0 we let u ε be the unique solution of u ε , v + ε 2 a(u ε , v) = u, v , a.e. in (0, +∞), ∀v ∈ V . (2.39) The behaviour of u ε as ε ց 0 is well known (cf., e.g., [22]). In particular, for all T ∈ [0, +∞) we have that as well as other convergences that can be inferred from the regularity of u and (2.39). Now, putting v = u ε t + (c/2)u ε in (2.21) and using also a Green formula, we obtain for a.e. t ∈ (0, +∞). Next, we multiply (2.44) by e c(t−M) , with M > 0, and integrate between 0 and M . We infer that The goal is plainly to prove that all of these J ε i (M )-terms tend to 0 as ε ց 0. Indeed, from (2.40)-(2.43) and the regularity of u it is clear that the left-hand side and the first three terms in the right-hand side of (2.45) converge to their respective limits in (2.38). On the other hand, the convergence to 0 of the residual can be shown using the methods employed in [13,Appendix], to which we refer for getting the right hints on how to manage things. Just for helping the reader a bit, let us develop the computation for and note that the last line tends to 0 as ε ց 0 because of (2.41)-(2.42) and (2.4).
Observe however that we cannot immediately conclude from (2.36) that the the mapping (z 0 , σ) → U σ (t, τ )z 0 is continuous from X × Σ to X . Indeed, the metric on X (cf. (2.13)) involves the gradients of the phase variable ( χ 1 , χ 2 ) and the continuous dependence estimate (2.36) entails no pointwise control in time for the gradients of ( χ 1 , χ 2 ). Nonetheless, in the next theorem we will actually check such continuity property.
(2.56) Standard energy estimates (see [26] for details) entail the boundedness of the sequences (cf. the regularity in (2.15) and (2.17)) uniformly with respect to n, for all t ≥ τ ≥ 0. Thus, by well-known compactness arguments, the related weak or weak star convergences to χ 1 , χ 2 , u, u t hold. Note that the whole sequences converge since the limits are perfectly identified and, in particular, let us point out the following convergence χ jn ⇀ χ j in H 1 (τ, t; H) for all t ≥ τ ≥ 0 and for j = 1, 2. (2.57) Now, in order to show (2.56), we exploit the following semicontinuity comparison argument. Formally test (2.19) at level n by the vector of components χ 1nt , χ 2nt and then integrate over (τ, t), with t ≥ τ, τ ≥ 0. We get ) Ω (s)ds. (2.58) Observe that the same identity follows rigorously from [6, Théorème 3.6, pp. 72-73].
Taking the lim sup as n ր +∞ of both sides of (2.58), our aim is clearly to verify that the terms on the right-hand side actually pass to the limit. This is the case. First, note that α(ϑ n ) div u n − α(ϑ) div u = α(ϑ n )(div u n − div u) + (α(ϑ n ) − α(ϑ)) div u tends to 0 strongly in L 2 (τ, t; H) due to (2.10), (2.53), α(ϑ n ) → α(ϑ) a.e. in Ω×(τ, t) for a subsequence, and the Lebesgue dominated convergence theorem. Then, thanks to (2.57) and the lower semicontinuity of norms with respect to weak convergence, from (2.53) and the convergence of ϑ n to ϑ it follows that (2.59)

PIERLUIGI COLLI AND ANTONIO SEGATTI
Then, by recovering the identity analogous to (2.58) for the limiting pair ( χ 1 , χ 2 ), a comparison with (2.58) yields lim sup The converse lim inf inequality clearly follows from the weak lower semicontinuity of norms and from the fact that χ jn (t) → χ j (t), j = 1, 2, strongly in H and weakly in V for all t ≥ τ ≥ 0, due to (2.55) and the boundedness of { χ jn }, j = 1, 2, in L ∞ (τ, T ; V ) for all T > 0. Thus, we have that ∇ χ jn (t) 2 H → ∇ χ j (t) 2 H for j = 1, 2 and any t ≥ τ . This convergence, combined with the abovementioned weak convergence, plainly leads to the strong convergence of χ jn (t) to χ j (t) in V for j = 1, 2 and any t ≥ τ ≥ 0. Then, recalling again (2.53)-(2.54), it turns out that the theorem is completely proved.
3. Uniform attractor for our system. In this section, we prove that the system (2.19)-(2.21) possesses the compact uniform attractor A. We advise the reader that in the sequel we will make often use of some formal estimates, which can be rigorously justified by adopting some, by now standard, approximation argument. The occurrence of these formal estimates will be however pointed out to the reader. To simplify the notation, from now on we denote by C (or C i , i = 1, 2, . . .) some possibly different constants depending on the data of the problem. Moreover, we let c = 1 in (2.21).
We start with the proof of the dissipativity of the system and state the following result. such that the X -ball with radius D turns out to be a uniform absorbing set for the family U (G,ϑ) (t, τ ), (G, ϑ, ϑ t ) ∈ F . Moreover, for any R > 0 there is a constant Proof. The notation of Definition 2.7 being in force, we test (2.19) by Then, we obtain and let z 0n = (( χ 0 1n , χ 0 2n ), u 0n , v 0n ) denote a sequence in X that weakly converges to z 0 := (( χ 0 (3.10) Proof. As in the proof of Theorem 2.8, we agree to set (( χ 1n (t), χ 2n (t)), u n (t), u nt (t)) = U (Gn,ϑn) (t, τ )z 0n .
where the product α ′ (ϑ n )∇ϑ n has to be understood properly. Then, summing the resulting inequalities and so on, with the help of (2.3), (2.5), (2.10) and the Gronwall lemma it is not difficult to obtain the following bound where the constant C depends on data and on t, but is independent of n due to the convergences z 0n ⇀ z 0 and (G n , ϑ n ) → (G, ϑ) which we have assumed.