Global Attractor and Omega-Limit Sets Structure for a Phase-Field Model of Thermal Alloys

GLOBAL ATTRACTOR AND OMEGA-LIMIT SETS STRUCTURE FOR A PHASE-FIELD MODEL OF THERMAL ALLOYS DEDICATED TO PROF. JOS ´ E REAL ON HIS 60T H BIRTHDAY PEDRO MAR´ IN-RUBIO AND GABRIELA PLANAS Abstract. In this paper, the existence of weak solutions is established for a phase-field model of thermal alloys supplemented with Dirichlet boundary conditions. After that, the existence of global attractors for the associated multi-valued dynamical systems is proved, and relationship among these sets is established. Finally, we provide a more detailed description of the asymptotic behaviour of solutions via the omega-limit sets. Namely, we obtain a characterization –through the natural stationary system associated to the model– of the elements belonging to the omega-limit sets under suitable assumptions. 1. Introduction In recent years the phase-field methodology has achieved considerable importance in the modelling and numerical simulation of a range of phase transitions and complex growth structures like dendrites occurring during solidification (see, for instance, [24]). There exists a wide literature devoted to phase-field modelling of various phase transition phenomena, from the classical phase-field system of Caginalp [7] to problems with constraints leading to variational inequalities (cf. [15] and the references therein). Phase-field models have also been used to describe the evolution in time of diffusive phase interfaces in solid materials, in which martensitic phase transformations driven by configurational forces take place. In such models, partial differential equations of linear elasticity were coupled with a degenerate parabolic equation of second order for the phase-field (cf. [2]). In addition, we remark that the method has been extended to multi-phase systems by introducing a multiphase approach, see [13] for an isothermal phase-field model of ternary two-phase diffusion couples. In this paper we study an initial boundary value problem of a model describing a binary mixture with thermal properties and a phase transition. Phase-field models for binary alloys were proposed in [9, 8] and its derivation is based on thermodynamic principles. These models include as a special case the phase-field equations for a pure material introduced by Caginalp [7]. Date: November 28, 2011. 2010 Mathematics Subject Classification. 35B40, 35B41, 35K55, 80A22. Key words and phrases. Phase-field models, global attractor, omega-limit sets, asymptotic behaviour. 1 2 PEDRO MAR´ IN-RUBIO AND GABRIELA PLANAS With several simplifications and idealization, it is possible to obtain the “simplest” phase-field-alloy model which retains the key characteristics of an alloy. It involves the following highly nonlinear parabolic system of three partial differential equations with three independent variables: phase-field, solute concentration, and temperature                    αε2φt−ε2∆φ=1 2(φ−φ3) + β(θ−θAc−θB(1 −c)) in Q, CVθt+l 2φt=∇ · [K1(φ)∇θ] in Q, ct=K2(∆c+M∇ · [c(1 −c)∇φ]) in Q, 0≤c≤1 in Q, (1) where Q= Ω ×(0,∞),being Ω an open connected bounded subset of RNwith N= 2 or 3,and with smooth boundary ∂Ω.The order parameter (phase-field) φ is the state variable characterizing the different phases; the function θrepresents the temperature; the concentration c∈[0,1] denotes the fraction of one of the two materials in the mixture. The parameter α > 0 is the relaxation scaling; the parameter β≥0 is given by β=ε[s]/(3σ),where ε > 0 is a measure of the interface width, σthe surface tension and [s] the entropy density difference between phases; CV>0 is the specific heat; the constant l > 0 is the latent heat; θAand θB are the respective melting temperatures of each of the two materials in the alloy; K2>0 is the solute diffusivity; Mis a constant related to the slopes of solidus and liquidus lines; and K1denotes the thermal conductivity. This physical parameter is assumed, as in [18, 5, 20], to be a function depending on the order parameter φ. In the previous paper [5] global existence of weak solutions for such model was proved under Neumann boundary conditions. Several results are available on the asymptotic behaviour in time of phase-field models for pure materials; concerning with Neumann boundary conditions, we can cite [6, 10, 17, 18, 25] among others. In particular, Lauren¸cot [18] was able to describe the omega-limit sets of singletons in a precise way. More exactly, they also are singletons, namely, stationary points of the problem. However, there exists obvious differences between his model and that in [5], that do not allow the same treatment; in particular, an energy decay result was known explicitly in [18], but not for the alloy model in [5]. The asymptotic behaviour of the dynamical systems related to problem (1) fulfilled with Neumann boundary conditions was investigated in [20] in the framework of the theory of attractors. The Neumann boundary conditions require Poincar´e- Wirtinger inequality in estimates and lead to a result of existence of attractor in a level-set formulation, similarly to the results in [6]. Besides the above comments concerning to phase-field models with Neumann boundary conditions, we must cite that there exist many references in the literature involving phase-field models with Dirichlet boundary conditions, see for instance [4, 1, 22, 14]. OMEGA-LIMIT SETS STRUCTURE FOR A PHASE-FIELD MODEL 3 However, as commented before, in the literature there exist more involved studies on the structure of omega-limit sets, and even for single trajectories, for this kind of problems. Most of the results on stability and asymptotic convergence towards stationary solutions uses in an essential way the fact that there exists a natural energy functional that plays the role of a Lyapunov functional for the solutions. Nevertheless, it is not clear wether system (1) possesses a global Lyapunov functional (as in [18]) or, more specifically, if it satisfies a Lojasiewicz-Simon property (as in [1]). Our goal in this paper is two-fold: on the one hand, we aim to study existence of solutions to problem (1) fulfilled with Dirichlet boundary conditions. On the other hand, we analyze deeply the long-time behaviour of the problem. The choice of Dirichlet boundary conditions will show to provide a more dissipative structure, in contrast with the results in [20]. Firstly, we prove the existence of global attractors for the two natural associated dynamical systems (depending on the regularity of the initial data) and the relationship among them. As far as uniqueness of solution for the considered system is unknown, we must use multi-valued dynamical systems for our approach. Secondly, and following some ideas from [18], we describe more precisely the elements belonging to the omega-limit sets. Due to the troublesome nonlinearities on the system, we are able to perform the analysis of the omega-limit sets under suitable assumptions. It is convenient to rewrite the system by performing a change of variables in (1). Namely, we introduce the enthalpy variable u=CV(θ−θB) + l 2φ. We fulfill the system with homogeneous Dirichlet boundary conditions in the three unknowns φ, u, and c, and complete it with initial conditions, leading to the following problem                                  αε2φt−ε2∆φ=1 2(φ−φ3) + β CV (u−l 2φ) + β(θB−θA)cin Q, CVut=∇ · [K1(φ)∇u]−l 2∇ · [K1(φ)∇φ] in Q, ct=K2(∆c+M∇ · [c(1 −c)∇φ]) in Q, 0≤c≤1 in Q, φ= 0, u = 0, c = 0 on ∂Ω×(0,∞), φ(0) = φ0, u(0) = u0, c(0) = c0. (2) Observe that in this way we are assuming that the original variable θ, which is the controlled temperature on ∂Ω,has the constant value θB. We assume that K1is a (globally) Lipschitz continuous function and there exist positive constants k1, k1such that 0< k1≤K1(r)≤k1∀r∈R.(3) This condition will be assumed all through this paper. The paper is organized as follows. In Section 2 we establish existence and regularity properties of weak solutions to problem (2). In Section 3 we recall briefly 4 PEDRO MAR´ IN-RUBIO AND GABRIELA PLANAS some abstract results concerning multi-valued dynamical systems and the study of their long-time behaviour, in particular, on existence of global attractors. These results will be used later, in Section 4, for the above problem. Namely, absorbing, compact, and continuity properties of the associated semiflows are established. In particular, we deduce that the global attractors in the two natural phase-spaces of the problem, which are (L2)3and H1 0×(L2)2,coincide thanks to the regularizing property of the model. Some other results related to omega-limits are also explored in this paragraph. Section 5 is devoted to a more involved study of the structure of the omega-limits. Actually, we improve some estimates that lead to identify the limiting problem (which is the natural stationary system associated to the model) satisfied by any element in an omega-limit. 2. Existence of solutions Let us firstly introduce some notation which will be used hereafter all through the paper. We will denote by (·,·) and |·|the inner product and its associated norm in L2(Ω) or in (L2(Ω))N. Otherwise, the norm in other spaces will be fully specified. The duality product between H1 0(Ω) and H−1(Ω) will be denoted by h·,·i. The first eigenvalue of −∆ with Dirichlet boundary conditions will be denoted by λ1,so that we have the Poincar´e inequality |f|2≤λ−1 1|∇f|2for any f∈H1 0(Ω). We define the concept of weak solution to system (2). Definition 1. We say that the triplet (φ, u, c)is a weak solution to system (2) in (0, T)for T > 0,if (i) φ∈L2 (0, T;H1 0(Ω))∩C([0, T]; L2(Ω))∩L4 (0, T;L4(Ω)), φt∈L2 (0, T;H−1(Ω)) +L4/3(0, T;L4/3(Ω)), φ(0) = φ0, (ii) u∈L2(0, T;H1 0(Ω)) ∩C([0, T]; L2(Ω)), ut∈L2(0, T ;H−1(Ω)), u(0) = u0, (iii) c∈L2(0, T;H1 0(Ω)) ∩C([0, T]; L2(Ω)), ct∈L2(0, T ;H−1(Ω)), c(0) = c0, 0≤c≤1a.e. in Ω×(0, T), and satisfies the equations αε2ZT 0 hφt(t), η(t)idt +ε2ZT 0 (∇φ(t),∇η(t))dt =1 2ZT 0 (φ(t)−φ3(t), η(t))dt +β CVZT 0 (u(t)−l 2φ(t), η(t))dt +β(θB−θA)ZT 0 (c(t), η(t))dt, for any η∈L2(0, T;H1 0(Ω)) ∩L4(0, T;L4(Ω)), CVZT 0 hut(t), η(t)idt +ZT 0 (K1(φ(t))∇u(t),∇η(t))dt =l 2ZT 0 (K1(φ(t))∇φ(t),∇η(t))dt, OMEGA-LIMIT SETS STRUCTURE FOR A PHASE-FIELD MODEL 5 for any η∈L2(0, T;H1 0(Ω)),and ZT 0 hct(t), η(t)idt +K2ZT 0 (∇c(t),∇η(t))dt +K2MZT 0 (c(t)(1 −c(t))∇φ(t),∇η(t))dt = 0, for any η∈L2(0, T;H1 0(Ω)). Additionally, we will say that (φ, u, c) : Q→R3is a global solution to system (2) if its restriction to (0, T)is a weak solution for any T > 0. We state now a result on existence of weak solutions to problem (2). Theorem 2. Let be given (φ0, u0)∈(L2(Ω))2,and c0∈L2(Ω; [0,1]),i.e., c0∈ L2(Ω) such that 0≤c0(x)≤1a.e. in Ω. Then there exists at least one weak solution (φ, u, c)to system (2) in (0, T)for any T > 0. If in addition φ0∈H1 0(Ω), then, for any solution (φ, u, c)in (0, T ),one has that φ∈L2(0, T;H2(Ω)) ∩C([0, T ]; H1 0(Ω)), φt∈L2(0, T;L2(Ω)), and φsatisfies the first equation in (2) a.e. in Ω×(0, T). Proof. Let us fix T > 0.The existence of weak solution to (2) can be related to an auxiliary problem. To this end, let Π be the function Π(r) =      0 if r < 0, rif 0 ≤r≤1, 1 if r > 1, and consider the following problem                          αε2φt−ε2∆φ=1 2(φ−φ3) + β CV (u−l 2φ) + β(θB−θA)cin Ω ×(0, T ), CVut=∇ · [K1(φ)∇u]−l 2∇ · [K1(φ)∇φ] in Ω ×(0, T ), ct=K2(∆c+M∇ · [Π(c)(1 −Π(c))∇φ]) in Ω ×(0, T ), φ(0) = φ0u(0) = u0, c(0) = c0in Ω ×(0, T ), φ= 0, u = 0, c = 0 on ∂Ω×(0, T ). (4) A weak solution to this auxiliary problem is that given in Definition 1, but with the natural modification, according to the new third equation in (4). Namely, cmust satisfy now ZT 0 hct(t), η(t)idt +K2ZT 0 (∇c(t),∇η(t))dt +K2MZT 0 (Π(c(t))(1 −Π(c(t)))∇φ(t),∇η(t))dt = 0, for any η∈L2(0, T;H1 0(Ω)). Let us split the proof into three steps. 6 PEDRO MAR´ IN-RUBIO AND GABRIELA PLANAS Step 1. Assume that (φ, u, c) is a weak solution of (4), and that 0 ≤c0≤1 a.e. in Ω.Then, 0 ≤c≤1,therefore Π(c) = c, and so this triplet is indeed a weak solution for (2). First, we prove that if c0≤1 a.e. in Ω,then c(t)≤1 for all t∈[0, T] and a.e. in Ω. Let us consider the positive part of (c−1),namely (c−1)+= max(c−1,0). According to [11], we have that ∇(c−1)+=(∇cif c−1≥0, 0 otherwise. Multiplying the third equation in (4) by (c−1)+(s)χ(0,t)(s) and integrating in (0, t), we have 1 2|(c−1)+(t)|2+K2Zt 0 |∇(c−1)+(s)|2ds =1 2|(c0−1)+|2−K2MZt 0ZΩ Π(c(s))(1 −Π(c(s)))∇φ(s)∇(c−1)+(s)dxds. Since c0≤1,one has that |(c0−1)+|= 0. Moreover, if c < 1,it follows that ∇(c−1)+= 0 and, if c≥1,we have that Π(c) = 1 and thus, 1 −Π(c)=0.So the last integral vanishes and we can conclude that |(c−1)+(t)|2≤0∀t∈[0, T]. Therefore, (c−1)+(t) = 0 for all 0 ≤t≤Tand a.e. in Ω, which implies that c(t)≤1 for all 0 ≤t≤Tand a.e. in Ω. Next, we prove that if c0≥0 a.e. in Ω,then c(t)≥0 for all t∈[0, T] and a.e. in Ω. For this we consider the negative part of c, namely c−= max(−c, 0).Observe that now we have that (see [11]) ∇c−=(0 if c≥0, −∇cotherwise. Multiplying the third equation in (4) by −c−(s)χ(0,t)(s),after integration in (0, t) we obtain 1 2|c−(t)|2+K2Zt 0 |∇c−(s)|2ds =1 2|c− 0|2+K2MZt 0ZΩ Π(c(s))(1 −Π(c(s)))∇φ(s)∇c−(s)dxds. Similarly as before, since c0≥0 we have that |c− 0|= 0. Moreover, if c≥0 the last integral vanishes and, if c < 0,we have that Π(c)=0,thus, the last integral also vanishes. Therefore, we deduce |c−(t)|2≤0∀t∈[0, T]. Hence, c−(t) = 0 for all 0 ≤t≤Tand a.e. in Ω, which implies that c(t)≥0 for all 0≤t≤Tand a.e. in Ω. This proves the claim in Step 1. So, in order to obtain the existence of solutions, we only have to check the following claim. OMEGA-LIMIT SETS STRUCTURE FOR A PHASE-FIELD MODEL 7 Step 2. There exists at least one weak solution to (4). We apply the classical Faedo-Galerkin method. Let us sketch the main ideas. First, we introduce the Galerkin approximations. Let {wn}n∈Nbe a Hilbert basis of L2(Ω) composed by eigenfunctions of the operator −∆ with homogeneous Dirichlet boundary conditions, Vk=span{w1, .., wk},and Pkbe the orthogonal projection from L2(Ω) onto Vk. For each m∈N, we consider the approximate problem of finding φm(t) = m X j=1 φmj(t)wj, um(t) = m X j=1 umj(t)wj, cm(t) = m X j=1 cmj(t)wj satisfying                    αε2(φm t, wj) + ε2(∇φm,∇wj) =1 2φm−(φm)3, wj+β CVum−l 2φm, wj+β(θB−θA)cm, wj, CV(um t, wj) + K1(φm)∇um,∇wj=l 2K1(φm)∇φm,∇wj, (cm t, wj) + K2(∇cm,∇wj) + K2MΠ(cm)(1 −Π(cm))∇φm,∇wj= 0, (5) for 1 ≤j≤m, and φm(0) = Pmφ0, um(0) = Pmu0, cm(0) = Pmc0.(6) Problem (5)-(6) is in fact an initial value problem for a system of 3mordinary differential equations. Since the non-linear terms are locally Lipschitz continuous functions, the above problem has a unique solution (φm, um, cm) defined on some maximal time interval [0, Tm) with 0 < T m≤T. Next, we derive some estimates which show in particular that all solutions are well-defined in [0, T ]. Multiplying the first equation in (5) by φmj(t) and summing from j= 1 to m, we deduce that for all t∈(0, Tm) αε2d dt|φm|2+ 2ε2|∇φm|2 =|φm|2− kφmk4 L4(Ω) +2β CV (um, φm)−βl CV |φm|2+ 2β(θB−θA)(cm, φm) ≤ −kφmk4 L4(Ω) +C1|φm|2+|um|2+|cm|2,(7) where C1is a constant independent of m. Multiplying the second equation in (5) by umj(t) and summing from j= 1 to m, we obtain 1 2 d dt|um|2+1 CV (K1(φm)∇um,∇um) = l 2CV (K1(φm)∇φm,∇um) ≤k1 2CV |∇um|2+k2 1l2 8CVk1 |∇φm|2, and using again (3) we have that 1 2 d dt|um|2+k1 2CV |∇um|2≤k2 1l2 8CVk1 |∇φm|2.(8) 8 PEDRO MAR´ IN-RUBIO AND GABRIELA PLANAS Multiplying the third equation in (5) by cmj(t) and summing from j= 1 to m, gives 1 2 d dt|cm|2+K2|∇cm|2=−K2MΠ(cm)(1 −Π(cm))∇φm,∇cm ≤K2 2|∇cm|2+K2M2 2|∇φm|2,(9) where we have used that 0 ≤Π(cm)(1 −Π(cm)) ≤1. Now, adding (7), (8) multiplied by 4CVε2k1 k2 1l2,and (9) multiplied by ε2 K2M2,we arrive at d dtαε2|φm|2+2CVε2k1 k2 1l2|um|2+ε2 2K2M2|cm|2 +kφmk4 L4(Ω) +ε2|∇φm|2+2ε2k2 1 k2 1l2|∇um|2+ε2 2M2|∇cm|2 ≤C1|φm|2+|um|2+|cm|2, for any m≥1. It follows from Gronwall Lemma that |φm(t)|2+|um(t)|2+|cm(t)|2≤C2(T)∀t∈[0, Tm),∀m≥1. So we can take Tm=Tand we infer from the above estimates, after integration in time, that ZT 0kφm(s)k4 L4(Ω) +|∇φm(s)|2+|∇um(s)|2+|∇cm(s)|2ds ≤C3|φ0|2+|u0|2+|c0|2+C2(T), where C3>0 is independent of m. From the uniform estimates obtained above, we deduce that there exist functions ξ∈L4/3(0, T;L4/3(Ω)) and (φ, u, c)∈L2(0, T ; (H1 0(Ω))3)∩L∞(0, T; (L2(Ω))3) with φ∈L4(0, T;L4(Ω)),and that from the sequence {(φm, um, cm)}mwe can extract a subsequence (cf. e.g., [19]), relabelled the same, such that (φm, um, cm)*(φ, u, c) weakly in L2(0, T; (H1 0(Ω))3), (φm, um, cm)∗ *(φ, u, c) weakly-star in L∞(0, T; (L2(Ω))3), φm* φ weakly in L4(0, T;L4(Ω)), (φm)3* ξ weakly in L4/3(0, T;L4/3(Ω)). From the first equation in (5), {φm t}mis bounded in L2(0, T ;H−1(Ω))+L4/3(0, T; L4/3(Ω)); and from the second and third equations in (5) it follows that the sequences {um t}mand {cm t}mare bounded in L2(0, T;H−1(Ω)). Then, by taking into account that H1 0(Ω) is compactly embedded into L2(Ω),we conclude (up to a subsequence) that (φm, um, cm)→(φ, u, c) strongly in L2(0, T; (L2(Ω))3) and a.e. in Ω ×(0, T), and, by using [19, Lem.1.3, p.12], one obtains that ξ=φ3. Hence we can pass to limit in (5)-(6), noting that K1and Π are bounded Lipschitz continuous functions (cf. [20] for similar arguments) and find out that there OMEGA-LIMIT SETS STRUCTURE FOR A PHASE-FIELD MODEL 9 exists at least one weak solution for (4) in (0, T),which was the claim of Step 2. Step 3. Regularity of the solution. Finally, if φ0∈H1 0(Ω) we observe that the sequence {φm 0=Pmφ0}mconverges to φ0in H1 0(Ω).This allows us to obtain another estimate for {φm}m.Multiplying the first equation in (5) by λjφmj,where λjis the eigenvalue associated to wj,and summing from j= 1 to m, we deduce that αε2 2 d dt|∇φm|2+ε2|∆φm|2+3 2ZΩ (φm)2|∇φm|2dx +βl 2CV |∇φm|2 =−1 2(φm,∆φm)−β CV (um,∆φm)−β(θB−θA)(cm,∆φm) ≤1 4ε2|φm|2+β2 C2 Vε2|um|2+β2(θB−θA)2 ε2|cm|2+3ε2 4|∆φm|2in (0, T). From (2) and the above inequality, we deduce that the sequence {φm}mis bounded in L2(0, T;H2(Ω)) ∩L∞(0, T ;H1 0(Ω)).In the limit we obtain the desired regularity for φ. The proof is then complete.  Remark 3. From the above result, it is clear that by a recursive procedure of concatenation on intervals, for instance, of the form (T, 2T),(2T, 3T),etc, there exist global solutions to (2) for any initial datum (φ0, u0, c0)∈(L2(Ω))2×L2(Ω; [0,1]). As a consequence of the previous result we have a regularizing effect in the problem. Proposition 4. Any global solution (φ, u, c)of (2) with initial datum (φ0, u0, c0)∈ (L2(Ω))2×L2(Ω; [0,1]) satisfies φ∈C((0,∞); H1 0(Ω)) ∩L2(, T;H2(Ω)) ∀, T such that 0<  < T, and moreover, (φ, u, c)satisfies the first equation in (2) a.e. in Q. Proof. Since the first component, φ, of any solution (φ, u, c) to (2), satisfies φ∈ L2(0, T;H1 0(Ω)),we may consider a positive time τ < T (almost sure) such that φ(τ)∈H1 0(Ω),whence the solution in [τ, T] becomes more regular. The uniqueness of solution for the first equation, with uand cfixed, concludes the proof.  3. Abstract results for multi-valued dynamical systems In this section we summarize some basic results on the existence of global attractors for multi-valued dynamical systems. In order to do that, firstly we recall some basic definitions on multi-valued dynamical systems and the study of their long-time behaviour (we refer to [21] for a more detailed exposition on this topic). For a given metric space (X, d),we will denote by P(X), B(X), C(X),and K(X) the classes of nonempty subsets of X,nonempty and bounded subsets of X, nonempty and closed subsets of X,and nonempty and compact subsets of Xrespectively. We will denote by distX(·,·) the Hausdorff semidistance between elements of P(X), given by distX(A, B) = supa∈Ainfb∈Bd(a, b). 16 PEDRO MAR´ IN-RUBIO AND GABRIELA PLANAS Corollary 19. For any bounded sets B1∈B((L2(Ω))2×L2(Ω; [0,1])) and B2∈ B(H1 0(Ω) ×L2(Ω) ×L2(Ω; [0,1])),the following equalities hold ω1(B1) = ω2(G(t, B1)) ∀t > 0, ω1(B2) = ω2(B2). Proof. All the omega-limit sets in the statement are well-defined by Theorem 18 and Remark 7 (b)(iii). Observe that by Proposition 13, and since ω1(B1) = ω1(G(t, B1)) for any t > 0, then the first equality is a consequence of the second. So, we only have to prove that ω1(B2) = ω2(B2). In order to do that, firstly observe that the inclusion to the right is trivial by the continuous injection of H1 0(Ω) into L2(Ω). For the opposite, consider sequences {bn}n⊂B2,{tn}n⊂(0,∞) with tn↑ ∞,and {(φ(tn), u(tn), c(tn))}n,with (φ(tn), u(tn), c(tn)) ∈G(tn, bn) such that (φ(tn), u(tn), c(tn)) →(φ∞, u∞, c∞) in (L2(Ω))3. Secondly, observe that G(tn, B2) = G(tn−t1, G(t1, B2)) for all n≥2, G(t1, B2) is bounded in H1 0(Ω) ×L2(Ω) ×L2(Ω; [0,1]) by Proposition 13, and so, by Corollary 16, there exists a converging subsequence {(φ(tn0), u(tn0), c(tn0))}to some element (˜ φ∞,˜u∞,˜c∞)∈ω2(B2).By uniqueness of the limit, one deduces that (φ∞, u∞, c∞)=(˜ φ∞,˜u∞,˜c∞) and so the inclusion ω1(B2)⊂ω2(B2) holds.  Remark 20. An alternative proof of the equality of the attractors in Theorem 18 follows from Remark 14, Remark 7 (iv), and the above result. 5. Structure of the omega-limit sets As mentioned in the Introduction, because of the strong coupling of the model, the convergence of solutions towards the equilibria is difficult to determine using standard methods. However, we show that under suitable assumptions any weak solution converges to an equilibrium. More precisely, in this section we obtain a characterization of the elements belonging to the omega-limit sets of both semiflows given in Proposition 10. Actually, by the regularizing effect of the problem, we have seen (cf. Corollary 19 above) that they are related. In order to achieve a more complete description of the omega-limit sets we will use one technique employed in [16] (see also [18]). To this end, we improve some estimates obtained previously for the solutions. Proposition 21. Assume that the following relation among the coefficients in the model holds, ε2λ1> β2k2 1l2 2C2 Vε2k2 1λ1 +2β2M2(θB−θA)2 ε2λ1 + 1 −βl CV!+ ,(13) where (·)+denotes the positive part, i.e., (·)+= max{·,0}. Then, there exists a constant Csuch that the following inequality holds for any global solution (φ, u, c)of (2) with initial datum (φ0, u0, c0)∈H1 0(Ω) ×L2(Ω) × OMEGA-LIMIT SETS STRUCTURE FOR A PHASE-FIELD MODEL 17 L2(Ω; [0,1]) : |φ(t)|2+|u(t)|2+|c(t)|2+|∇φ(t)|2 +Zt 0 (|∇φ(s)|2+|φt(s)|2+|∇u(s)|2+|∇c(s)|2)ds ≤C(φ0, u0, c0)∀t≥0. The constant Conly depends on the norms of (φ0, u0, c0),but is independent of t. Proof. We will obtain now several inequalities using different test functions in the system (2). We begin by taking φas test function in the first equation in (2) and applying the Young inequality with arbitrary positive constants c1and c2to be fixed later. So we obtain αε2 2 d dt|φ|2+ε2|∇φ|2+1 2ZΩ (φ4−φ2)dx +βl 2CV |φ|2 ≤β2 2c1C2 V |u|2+c1 2|φ|2+β2(θB−θA)2 2c2 |c|2+c2 2|φ|2a.e. t > 0.(14) By multiplying the first equation in (2) by φtand applying again the Young inequality, we also have αε2|φt|2+ε2 2 d dt|∇φ|2+1 2 d dt ZΩφ4 4−φ2 2dx+βl 4CV d dt|φ|2 ≤β2 αC2 Vε2|u|2+β2(θB−θA)2 αε2|c|2+αε2 2|φt|2a.e. t > 0.(15) By multiplying the third equation in (2) by c, and since c∈[0,1],one deduces similarly as above that 1 2 d dt|c|2+K2|∇c|2≤K2M2 2|∇φ|2+K2 2|∇c|2a.e. t > 0. So, arranging terms and multiplying by ε2 2K2M2,we arrive at ε2 4K2M2 d dt|c|2+ε2 4M2|∇c|2≤ε2 4|∇φ|2a.e. t > 0.(16) Recall that taking uas test function in the second equation in (2), using the boundedness of K1,arranging terms and multiplying by 2k1ε2 k2 1l2we have previously obtained inequality (11). 18 PEDRO MAR´ IN-RUBIO AND GABRIELA PLANAS By multiplying (15) by a constant c3>0 to be fixed later on, and adding to (14), (11), and (16), we conclude d dt αε2 2+lc3β 4CV|φ|2+CVε2k1 k2 1l2|u|2+ε2 4K2M2|c|2+c3ε2 2|∇φ|2! +c3 2 d dt ZΩφ4 4−φ2 2dx+ε2 2|∇φ|2+βl 2CV |φ|2+1 2ZΩ φ4dx +αc3ε2 2|φt|2+ε2k2 1 k2 1l2|∇u|2+ε2 4M2|∇c|2 ≤β2 2c1C2 V +β2c3 αC2 Vε2|u|2+β2(θB−θA)2 2c2 +β2c3(θB−θA)2 αε2|c|2 +(c1+c2+ 1) 2|φ|2a.e. t > 0.(17) Now, we rearrange coefficients using the Poincar´e inequality suitably, in such a manner that positive terms and positive coefficients remain in the left hand side and the right hand side be zero. But for that, before to use the Poincar´e inequality, we must care about two possible cases for the coefficients of |φ|2and, if necessary, of |∇φ|2. These two possibilities can be abbreviated into one expression using the positive part function (·)+,which appears below. Thus, denoting for short by di>0, i = 1,...,4,to the coefficients in the first four derivatives in (17), depending on c3,and by the above arguments, we arrive at d dt(d1|φ|2+d2|u|2+d3|c|2+d4|∇φ|2) + c3 2 d dt ZΩφ4 4−φ2 2dx +αc3ε2 2|φt|2+D1|∇u|2+D2|∇c|2+D3|∇φ|2≤0a.e. t > 0,(18) where D1=ε2k2 1 k2 1l2−β2 2c1C2 V +β2c3 αC2 Vε2λ−1 1, D2=ε2 4M2−β2(θB−θA)2 2c2 +β2c3(θB−θA)2 αε2λ−1 1, D3=ε2 2−c1+c2+ 1 2−βl 2CV+ λ−1 1.(19) Now, from (13) it is possible to choose ci>0, i = 1,...,3,such that Dj>0 for j= 1,...,3.For clarity, the proof of this fact is postponed to an auxiliary result (cf. Lemma 22 below). So, once we have fixed such ci>0,and denoting d5= min{αc3ε2/2, D1, D2, D3}>0, OMEGA-LIMIT SETS STRUCTURE FOR A PHASE-FIELD MODEL 19 from (18), integrating in time and using the inequality x4−2x2≥x2−9/4 for all x∈R,we conclude d1+c3 8|φ|2+d2|u|2+d3|c|2+d4|∇φ|2 +d5Zt 0n|∇φ|2+|φt|2+|∇u|2+|∇c|2ods ≤d1|φ0|2+c3 8kφ0k4 L4(Ω) +d2|u0|2+d3|c0|2+d4|∇φ0|2+9 32c3|Ω| ∀t≥0, whence the claim follows.  Lemma 22. Condition (13) is equivalent to the possibility of choosing positive values {ci}3 i=1,such that {Di}3 i=1,given in (19), are positive. Proof. First at all, observe that c3appears in D1and D2in a proportional way, which means that we can take c3>0 as small as desired, and we have the result provided that c1and c2are chosen such that R1=ε2k2 1 k2 1l2−β2 2c1C2 Vλ1 , R2=ε2 2M2−β2(θB−θA)2 c2λ1 , R3=ε2λ1−c1+c2+ 1 −βl CV+ are positive. Assume that (13) holds. We must prove that it is possible to choose c1, c2>0 such that Ri>0 for i= 1,...,3. Observe that R1and R2are strictly increasing functions in relation to c1and c2 respectively. Moreover, there exist positive values c∗ 1and c∗ 2such that Ri= 0,and therefore Ri>0 if and only if ci> c∗ i,for i= 1,2. Then, for any choice ci> c∗ i,since the positive part function is non-decreasing, we have that c∗ 1+c∗ 2+ 1 −βl CV+ ≤c1+c2+ 1 −βl CV+ . From this inequality and condition (13), it holds that for sufficiently close values ci> c∗ i, i = 1,2,then R3is positive, as we desired to prove. Let us prove the opposite implication. Assume that there exist values c1, c2>0 such that Ri>0 for i= 1,...,3.We must prove that (13) holds. Using again the monotonicity of the positive part function (·)+,the proof is analogous to the above, but going back into the steps.  A direct consequence of the Proposition 21 is the following. Corollary 23. Under the assumptions of Proposition 21, there exists a constant C > 0,only depending on the norms of φ0, u0,and c0,such that any global solution 20 PEDRO MAR´ IN-RUBIO AND GABRIELA PLANAS (φ, u, c)with such initial datum satisfies kφtkL2(0,∞;L2(Ω)) +kφkL∞(0,∞;H1 0(Ω)) +kukL2(0,∞;H1 0(Ω)) +kukL∞(0,∞;L2(Ω)) +kckL2(0,∞;H1 0(Ω)) ≤C, kutkL2(0,∞;H−1(Ω)) +kctkL2(0,∞;H−1(Ω)) ≤C. Proof. First estimate is an immediate consequence of Proposition 21. Second estimate follows from the above and second and third equations in (2).  We have the following description of the omega-limit set of trajectories which contains the solutions of the steady state system naturally associated to the evolution problem. Theorem 24. Assume that (13) is satisfied. Then, for any B∈B((L2(Ω))2× L2(Ω; [0,1])) and any (φ∞, u∞, c∞)∈ω1(B),it holds that φ∞∈H4(Ω) ∩H1 0(Ω), u∞∈H2(Ω) ∩H1 0(Ω), c∞∈H3(Ω) ∩H1 0(Ω), and they satisfy                    −ε2∆φ∞=1 2(φ∞−φ3 ∞) + β CV (u∞−l 2φ∞) + β(θB−θA)c∞in Ω, ∇ · [K1(φ∞)∇u∞] = l 2∇ · [K1(φ∞)∇φ∞]in Ω, −K2(∆c∞+M∇ · [c∞(1 −c∞)∇φ∞]) = 0 in Ω, 0≤c∞≤1in Ω. (20) Proof. By Corollary 19, it is clear that we may assume that B∈B(H1 0(Ω)×L2(Ω)× L2(Ω; [0,1])).Let (φ∞, u∞, c∞)∈ω2(B) and tn→ ∞ such that (φ(tn), u(tn), c(tn)) →(φ∞, u∞, c∞) in H1 0(Ω) ×(L2(Ω))2.(21) For n≥1 and t∈(0,1) we define φn(t) = φ(tn+t), un(t) = u(tn+t), cn(t) = c(tn+t). From Corollary 23 we infer some uniform estimates for the sequences {φn}n, {un}n,and {cn}n.More precisely, there exists a constant C > 0 such that for all n≥1 : kφntkL2(0,1;L2(Ω)) +kφnkL∞(0,1;H1 0(Ω)) +kunkL2(0,1;H1 0(Ω)) +kunkL∞(0,1;L2(Ω)) +kcnkL2(0,1;H1 0(Ω)) +kuntkL2(0,1;H−1(Ω)) +kcntkL2(0,1;H−1(Ω)) ≤C. (22) Next, we identify the limit of φn, un,and cnas ngoes to ∞.We observe that for each t∈[0,1],by H¨older inequality, |φn(t)−φ(tn)|=Ztn+t tn φt(s)ds ≤Z∞ tn |φt(s)|2ds1/2, and the right hand side converges to zero as n→ ∞ by Corollary 23. Similarly, we have that un(t) and cn(t) converge to u(tn) and c(tn),respectively, in H−1(Ω) uniformly in t∈[0,1]. Now, we write |φn(t)−φ∞|≤|φn(t)−φ(tn)|+|φ(tn)−φ∞|, OMEGA-LIMIT SETS STRUCTURE FOR A PHASE-FIELD MODEL 21 and using (21) we conclude that φn→φ∞in C([0,1]; L2(Ω)). In a similar way, we prove that (un, cn)→(u∞, c∞) in C([0,1]; (H−1(Ω))2). We also have the following bounds Z1 0 |φnt(s)|2ds =Ztn+1 tn |φt(s)|2ds ≤Z∞ tn |φt(s)|2ds, Z1 0 kunt(s)k2 H−1(Ω)ds ≤Z∞ tn kut(s)k2 H−1(Ω)ds, and Z1 0 kcnt(s)k2 H−1(Ω)ds ≤Z∞ tn kct(s)k2 H−1(Ω)ds, so that φnt→0 in L2(0,1; L2(Ω)), (unt, cnt)→0 in L2(0,1; (H−1(Ω))2). Moreover, from (22) and the Aubin-Lions lemma (e.g., cf. [19]), there exist subsequences of {φn}n,{un}n,and {cn}n(which we still denote the same) satisfying φn ∗ * φ∞weakly-star in L∞(0,1; H1 0(Ω)), φn→φ∞a.e. (0,1) ×Ω, (un, cn)*(u∞, c∞) weakly in L2(0,1; (H1 0(Ω))2), (un, cn)∗ *(u∞, c∞) weakly-star in L∞(0,1; (L2(Ω))2), (un, cn)→(u∞, c∞) strongly in L2(0,1; (L2(Ω))2). Finally, we prove that φ∞, u∞and c∞satisfy (20). To this end, consider %∈ D(0,1) with Z1 0 %(s)ds = 1 and ζ∈ D(Ω). Multiplying the first equation in (2) by %(t−tn)ζ(x), and integrating in (0, tn+ 1), αε2Ztn+1 0 hφt(t), ζi%(t−tn)dt +ε2Ztn+1 0 (∇φ(t),∇ζ)%(t−tn)dt =1 2Ztn+1 0 (φ(t)−φ3(t), ζ)%(t−tn)dt +β CVZtn+1 0 (u(t)−l 2φ(t), ζ)%(t−tn)dt +β(θB−θA)Ztn+1 0 (c(t), ζ)%(t−tn)dt. 22 PEDRO MAR´ IN-RUBIO AND GABRIELA PLANAS By using the change of variable s=t−tn, αε2Z1 0 hφnt(s), ζi%(s)ds +ε2Z1 0 (∇φn(s),∇ζ)%(s)ds =1 2Z1 0 (φn(s)−φ3 n(s), ζ)%(s)ds +β CVZ1 0 (un(s)−l 2φn(s), ζ)%(s)ds +β(θB−θA)Z1 0 (cn(s), ζ)%(s)ds. From the above convergences, and by [19, Lem.1.3, p.12], passing to the limit as n→ ∞ we obtain ε2Z1 0 %(s)ds(∇φ∞,∇ζ) =1 2Z1 0 %(s)ds(φ∞−φ3 ∞, ζ) + β CVZ1 0 %(s)ds(u∞−l 2φ∞, ζ) +β(θB−θA)Z1 0 %(s)ds(c∞, ζ). Thus, first equation in (20) is satisfied. Similarly, multiplying by the same function in the second equation in (2) and integrating, it yields CVZ1 0 hunt(s), ζi%(s)ds +Z1 0 (K1(φn(s))∇un(s),∇ζ)%(s)ds =l 2Z1 0 (K1(φn(s))∇φn(s),∇ζ)%(s)ds. Since K1is Lipschitz continuous, we have that K1(φn)→K1(φ∞) strongly in L2(0,1; L2(Ω)), and passing to limit we obtain that second equation in (20) holds. Next, since 0 ≤cn≤1 and cn→c∞in L2(0,1; L2(Ω)),by the Dominated Convergence Theorem, it follows that cn(1−cn) converges to c∞(1−c∞) in Lp(0,1; Lp(Ω)) for any p∈[1,∞),and also 0 ≤c∞≤1.Proceeding analogously as before in the concentration equation in (2), we can pass to the limit in Z1 0 hcnt(s), ζi%(s)ds +K2Z1 0 (∇cn(s),∇ζ)%(s)ds +K2MZ1 0 (cn(s)(1 −cn(s))∇φn(s),∇ζ)%(s)ds = 0, and deduce that third equation in (20) is satisfied. Finally, concerning the regularity of the triplet (φ∞, u∞, c∞),let us observe that they belong not only to H1 0(Ω) ×L2(Ω) ×L2(Ω; [0,1]).By elliptic regularity and bootstrapping in (20), it is not difficult to deduce that φ∞,and later u∞and c∞, gain the regularity claimed in the statement. The proof is then complete.  Remark 25. If the function K1is more regular, the regularity of the elements in the omega-limit can be improved, again by a bootstrapping argument as above. OMEGA-LIMIT SETS STRUCTURE FOR A PHASE-FIELD MODEL 23 Remark 26. The above result identifies the possible cluster points of any single solution, but it does not indicate whether this limit set is properly a singleton. This last question may be answered by the study of the set of stationary states. Indeed, the answer is positive if, for instance, one can show that the number of solutions to (20) is discrete, or the system satisfies some kind of Lojasiewicz-Simon property. None of these properties seem to be easy to obtain here. In fact, they do not hold in several examples (cf. [3, 12]). Acknowledgments. The authors would like to express their warmest congratulations to Prof. Jos´e Real on the occasion of his 60th birthday. Prof. Real was P.M.-R.’s PhD-advisor, and P.M.-R. would like to show his deepest gratitude for the relationship with him for many years, as advisor, colleague, and friend. It has been sincere, warm, and fruitful. The author wishes sincerely to Prof. Real many more years of this relationship, and that he continues enjoying with his clever contributions in research activities. This work has been partially supported by Ministerio de Educaci´on y Ciencia (MEC, Spain), grant MTM2008-00888, Junta de Andaluc´ıa grant P07-FQM-02468, Ministerio de Educaci´on–DGPU (Spain) project PHB2010-0002-PC, and Capes- DGU (Brazil) 238/11. G. Planas also thanks to CNPq - Brazil, grant 303302/2009- 7, FAEPEX - Brazil, grant 964/08, and Fapesp - Brazil, grant 2008/09342-3. References [1] S. Aizicovici, E. Feireisl, and F. Issard-Roch, Long time convergence of solutions to a phasefield system, Math. Methods Appl. Sci. 24 (2001), 277–287. [2] H.-D. Alber and P. Zhu, Solutions to a model with Neumann boundary conditions for phase transitions driven by configurational forces, Nonlinear Anal. Real World Appl. 12 (2011), 1797–1809. [3] B. 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(Pedro Mar´ın-Rubio) Dpto. de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla (Spain) E-mail address, Pedro Mar´ın-Rubio: [email protected] (Gabriela Planas) Departamento de Matem´ atica, Instituto de Matem´ atica, Estat´ ıstica e Computac¸˜ ao Cient´ ıfica, Universidade Estadual de Campinas, 13083-859 Campinas, SP, Brazil E-mail address, G. Planas: [email protected]