Pullback attractor for a non-linear evolution equation in elasticity

Pullback attractor for a non–linear evolution equation in elasticity Tom´as Caraballoa, Renato Coluccib aDepartamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain bDepartamento de Matem´aticas Pontificia Universidad Javeriana Cra. 7 No. 43-82, Bogot´a, Colombia Abstract We prove the existence of a pullback attractor for a non–autonomous fourth order evolution equation arising in the field of phase transitions and elasticity theory. The existence of several families of bounded absorbing sets is first proved in several spaces, and owing to the compactness of some inclusions between Sobolev spaces, we can then ensure the existence of a family of compact absorbing sets in the pullback sense and, as a consequence, the existence of a pullback attractor. Key words: Pullback attractor, phase transitions, pullback absorbing family, two–parameter semigroup. 2000 MSC: 35B41, 82B26 1. Introduction In [10], [11] and [12] it has been studied the asymptotic behavior of the following evolution equation:    ut=−ε2uxxxx +1 2W00(ux)uxx, u=uxx = 0,on ∂I, u(0, t) = u0, (1.1) Email addresses: [email protected] (Tom´as Caraballo), [email protected] (Renato Colucci) Partly supported by FEDER and Ministerio de Econom´ıa y Competitividad (Spain) under grant MTM2011-22411. Preprint submitted to Elsevier June 6, 2013 where u:I→R, 0 < ε << 1, I= (0,1) and W(p) = (p2−1)2is the so called double well potential. The equation (1.1) represents the L2-gradient dynamics of the non convex energy: Fε(u) = 1 2ε2ZI u2 xxdx +1 2ZI W(ux)dx. (1.2) The equation arises in the field of elasticity theory, phase transition and image processing. The functional (1.2) with ε= 0 is a simple model describing microstructures that arise from solid-solid phase transitions in certain elastic crystals (see [18] and [19]) such as In-Th Cu-Al-Ni Ni-Ti. These materials present a variety of microstructures which are important for technological application in the context of material theory (shape memory effects, pseudoelasticity, etc). From a physical point of view, the functional (1.2)(with ε= 0) describes the elastic energy required for deformation of the crystal, while the function W(·) is the stored energy density functions describing the properties of the material. The add of the regularizing term, depending on the second derivative of uand on the small parameter ε, solve the problem of non–uniqueness of minimizers (and of the ill–posedness of the equation of the associated L2 gradient dynamics) without changing the relevant macroscopic features of the model. It is natural to consider the associated dynamical problem (see [1] and [2]), that is, the study of the gradient flow of the energy (1.2) which can be written in the form of system (1.1). In [3] they studied the global dynamics of (1.1). By numerical experiments they show the existence of three well separated time scales with peculiar dynamical behavior. In a first time scale of order t>Tε=O(ε2) the energy of the initial data is drastically reduced and microstructures (see [18]) appears in the regions where uxfalls in the non convex region of W. In a second time scale of order t > T =O(1) the equation exhibits a heath equation-like behavior in the regions without microstructures, while the dynamic is slow in the regions with microstructures. In the third time scale t > 1 Tε=O(ε−2) the equation shows a finite dimensional character, i.e. the solutions are approximately the union of consecutive segments. In particular the existence of an inertial manifold has been conjectured in [3]. In the papers [10], [11] the finite dimensional reduction of the system has been proved. In particular the authors prove the existence, giving an estimate of the dimension, of the global attractor, exponential attractor and inertial manifold. 2 Nonlinear dynamical systems are subjected to aleatoric influences, that is the reason why in the present paper we consider a small non–autonomous perturbation of equation (1.1) and consider the effects on the dynamics and in particular on the third time scale. The theory of global attractors has been generalized in the case of non– autonomous systems by introducing the concept of uniform attractor (see, for instance Chepyzhov and Vishik [9]). However this concept of uniform attractor lacks of the property of invariance which may be a serious inconvenience in the analysis of the long time behavior of the system. For this reason, an alternate possibility has been developed within the framework of pullback attractors theory (see Kloeden and Rassmussen [17], Carvalho et al. [8] and some other papers included in their bibliography sections). One advantage of the latter is that it is not necessary to impose very restrictive assumptions on the time dependent terms in the equation. Another advantage of this theory comes from the fact that, originally, the concept of pullback attractor was introduced in the field of random dynamical systems (see Schmalfuss [22], Crauel and Flandoli [14]), and for this reason it is a suitable concept to treat random and non-autonomous features in the models. The study of non–autonomous or stochastic perturbation for fourth order evolution equation is quite recent. An important fourth order PDE is the Cahn-Hilliard equation that describes phase transitions in a binary metal alloy. In [21] the author studied the existence of an exponential attractor for the viscous Cahn-Hilliard equation: ∂ ∂t[v+ε(−4v)] + 42v− 4W(v) = m(t), where the non–autonomous perturbation satisfies ZI m(t)dx = 0. Stochastic perturbation have also been studied. For instance, in [7] it is considered a stochastic perturbation of the type: vt+4(4v−W(u)) = σ(u)˙ Bt, proving the existence and uniqueness of solutions. Btrepresents a Wiener process and σis a bounded Lipschitz function. In the article [15] they showed 3 the existence of a random attractor for a stochastic Cahn-Hilliard equation with dynamics and stochastic boundary conditions, while the stochastic three-dimensional Lagrangian averaged Navier–Stokes equations are analyzed in [6]. These arguments suggest the importance of considering perturbation of equation (1.1), in particular there is a deep relation between Cahn Hilliard equation and equation (1.1), in fact if vis the solution of the Cahn-Hilliard equation: vt+4[ε24v−W0(v)] = 0, with Neumann boundary condition ∂ ∂nv=∂ ∂n4v= 0, x ∈ {0,1}, then u(x) = Zx 0 v(s)ds, is the solution of equation (1.1) with boudary conditions u=uxx = 0, x ∈ {0,1}. The rest of the paper is organized as follows: in section 2 we present some useful preliminary estimates and remarks, while section 3 contains the main results, that is, the existence of several absorbing sets and the existence of a pullback attractor in L2(I). 2. Preliminary estimates and definitions We consider a non–autonomous perturbation of the problem (1.1)    ut=−ε2uxxxx +1 2W00(ux)uxx +f(x, t), t > τ, x ∈I, u=uxx = 0,on ∂I, u(τ, x) = u0(x), x ∈I, (2.1) where f∈L2 loc(R;L2(I)) satisfies Zt −∞ eσs(kf(s)k2+kf0(s)k2)ds < ∞,∀t∈R,(2.2) 4 for some σ∈0,ε2 2,u0∈L2(I), and k · k is the norm of the space L2(I), where by f0(t) we denote ∂f ∂t (t, x). We will carry on our analysis in the following spaces H=L2(I), V =H2(I)∩H1 0(I), and we will denote by u(·;τ, u0) the solution of (2.1) corresponding to the initial datum u0∈Hat time τ. Since |I|= 1 we have that for all u∈V: kuk≤kuxk ≤ kuxxk.(2.3) Although, for simplicity, we will work with the interval I= [0,1], we remark that our analysis is also valid, with suitable modifications, for any bounded interval I= [a, b]. For this reason, we will keep denoting by |I|the length of this interval instead of writing 1 in what follows. Using a slight modification of the classical results in the autonomous framework, mainly of the Faedo-Galerkin method (see [24]), we can establish the following result (see [23] or Theorems 3.1 and 3.3 in [24]; see also [10]): Theorem 2.1. Suppose f∈L2 loc(R;H). The problem (2.1) admits a unique solution usuch that, for any T > τ, it follows: if u0=u(τ)∈H⇒u∈C0([τ, T], H)∩L2(τ, T;V); if u0=u(τ)∈V⇒u∈C0([τ, T], V )∩L2(τ, T;D(A)); where D(A) = {u∈H4(I) : u=uxx = 0 in ∂I}is the domain of the linear operator A=∂4 ∂x4. Remark 2.2. As in the autonomous case (see [10]) it is possible to prove that the solution is Lipschitz continuous with respect to initial data. We anticipate here the calculations that uses the estimates of the L2norms of u,uxand uxx (see next section for details). Indeed, let w(·) = u1(·)−u2(·)be the difference of two solutions associated to the initial data ui(τ) = uτ,i,i= 1,2, i.e. ui(·) = u(·;τ, uτ,i). Consider the equation fulfilled by w, multiply it by w, and integrate over I: 1 2 d dtkwk2+ε2kwxxk2+1 2ZI [W0(u1x)−W0(u2x)]wxdx = 0, 5 from which 1 2 d dtkwk2+ε2kwxxk2≤2kwxk2−2ZI w2 x¯vdx ≤2(1 + K(t))kwkkwxxk ≤(1 + K(t)) ε2kwk2+ε2kwxxk2. Then, integrating in (τ, t)with respect to time, we deduce: ku1−u2k ≤ e1 ε2Rt τ(1+K(s))dskuτ,1−uτ,2k. where ¯v=u1xu2xand we have used (see the next section for the detailed expressions of the functions involved in the computation) k¯vk∞≤ ku1xk∞ku2xk∞≤ ku1xkku2xkku1xxkku2xxk ≤ uε(t)vε(t) := K(t). We recall now some definitions and theoretical results that will be useful in the following analysis. Definition 2.3. A two-parameter semigroup or process S(·,·)on a metric space Xis a family of maps S(t, τ) : X→X, with −∞ < τ ≤t < ∞such that: (a) S(τ, τ) = Id,∀τ∈R, (b) S(t, τ) = S(t, s)S(s, τ)∀τ≤s≤t. Definition 2.4. The family of subsets of X,{B(t)}t∈R, is said to be pullback absorbing with respect to the process Sif for all t∈Rand any bounded subset D⊂X, there exists T(D, t)≤tsuch that S(t, τ)D⊂B(t)for all τ≤T(D, t). Definition 2.5. A family of compact sets A={A(t)}t∈Ris said to be a pullback attractor for the process Sif it satisfies (a) S(t, τ)A(τ) = A(t),∀t≥τ(invariance property), (b) limτ→−∞ dist(S(t, τ)D, A(t)) = 0 for all bounded sets D⊂X(pullback attracting property). We establish the following theorem (see [5]): 6 Theorem 2.6. Let S(t, τ)be a two-parameter semigroup on a complete metric space X, and suppose S(t, τ) : X→Xis continuous for all t≥τ. If there exists a family of compact (pullback) absorbing sets {B(t)}t∈R, then there exists a pullback attractor {A(t)}t∈R, and A(t)⊂B(t)for all t∈R. Furthermore, A(t) = [ D⊂X bounded ΛD(t), where ΛD(t) = \ n∈N[ s≥n S(t, t −s)D, denotes the omega limit set associated to D. Remark 2.7. It is worth emphasizing that it is also possible to choose a more general framework for our analysis in which the attracted sets are time dependent families of sets belonging to certain universe of sets (see, for instance Caraballo et al. [4] for a detailed analysis of this situation). However, for our interest in the present paper, the situation described above is enough. We conclude this section by recalling an important lemma that will be useful later (see [24] pag. 91). Lemma 2.8. (The Uniform Gronwall Lemma). Let y,g,hthree positive locally integrable functions on (t0,+∞)such that y0is locally integrable on (t0,+∞)which satisfy d dty≤gy +h, ∀t≥t0, Zt+r t g(s)ds ≤a1,Zt+r t h(s)ds ≤a2,Zt+r t y(s)ds ≤a3,for t≥t0, where r,a1,a2,a3are positive constants. Then y(t+r)≤a2+a3 rea1,∀t≥t0. 7 3. Existence of Pullback Attractor Thanks to Theorem 2.1 we easily deduce that our problem (2.1) generates a process Sin Hwhich is given by S(t, τ)u0=u(t;τ, u0),for all t≥τ, u0∈H. Moreover, the discussion in Remark 2.2 ensures that S(t, τ) : H→His a continuous mapping for any t≥τ. Consequently, to prove the existence of a pullback attractor for our model, we need to prove that there exists a compact (pullback) absorbing family {B(t)}t∈R⊂H. We will first prove the existence of an absorbing family of bounded sets in H. Afterwards, we will construct another family of bounded absorbing sets in Vand D(A) which, jointly with the compact injections of these two spaces in H, will ensure the existence of the compact absorbing family in H. Proposition 3.1. Suppose fsatisfies (2.2). Then, for any bounded subset D⊂Hand for any t∈R, there exists τH(D, t)≤tsuch that kS(t, τ)u0k2≤1+2e−σt Zt −∞ 1 2ε2eσskf(s)k2+eσt|I|ds := hε(t)2, for any τ≤τH(D, t)and u0∈D. Proof. Let us fix t∈Rand D⊂Hbounded. Denote u(·) = u(·;τ, u0), the solution of our equation (2.1) which takes the initial value u0∈Dat time τ(≤t), i.e. u(r) = S(r, τ)u0for all r∈[τ, t]. Throughout this proof and the rest of the paper, we will omit the time variables for simplicity in the calculations when no confusion is possible. Multiplying equation (2.1) by uwe obtain 1 2 d drkuk2+ε2kuxxk2≤ |I|+1 2ε2kfk2+ε2 2kuk2 ≤ |I|+1 2ε2kfk2+ε2 2kuxxk2,(3.1) from which, using (2.3), we have 1 2 d drkuk2+ε2 2kuk2≤ |I|+1 2ε2kfk2.(3.2) 8 Since σ∈(0, ε2), 1 2 d dr eσrku(r)k2=1 2σeσrkuk2+1 2eσr d drku(r)k2 ≤σ 2−ε2 2eσrku(r)k2+eσr|I|+1 2ε2eσrkf(r)k2, and, integrating the previous inequality over (τ, t), eσtku(t)k2≤eστ ku0k2+ 2 Zt −∞ 1 2ε2eσskf(s)k2+eσt|I|ds, whence kS(t, τ)u0k2≤eσ(τ−t)ku0k2+ 2e−σt Zt −∞ 1 2ε2eσskf(s)k2+eσt|I|ds. As D⊂His bounded, there exists d > 1 such that ku0k ≤ dfor all u0∈D. Thus, kS(t, τ)u0k2≤eσ(τ−t)d2+ 2e−σt Zt −∞ 1 2ε2eσskf(s)k2+eσt|I|ds, for any u0∈D, and we conclude the proof by setting τH(D, t) = t−2 σln d. We now establish an inequality that will be useful later on. 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