LARGE TIME BEHAVIOR OF A CONSERVED PHASE-FIELD SYSTEM

. We investigate the large time behavior of a conserved phase-ﬁeld system that describes the phase separation in a material with viscosity eﬀects. We prove a well-posedness result, the existence of the global attractor and its upper semicontinuity, when the heat capacity tends to zero. Then we prove the existence of inertial manifolds in one space dimension, and for the case of a rectangular domain in two space dimension. We also construct robust families of exponential attractors that converge in the sense of upper and lower semicontinuity to those of the viscous Cahn-Hilliard equation. Continuity properties of the intersection of the inertial manifolds with bounded absorbing sets are also proven. This work extends and improves some recent results proven by A. Bonfoh for both the conserved and non-conserved phase-ﬁeld systems.


1.
Introduction. The conserved phase-field system that takes into account the effects of viscosity in a material occupying either a bounded domain Ω of R d , with smooth boundary, or Ω = Π d i=1 (0, L i ), L i > 0, for d ≤ 3, reads (cf. [7,13,17]) τ φ t − ∆(δφ t − ∆φ + g(φ) − u) = 0, where τ > 0 is a relaxation time, δ ≥ 0 is the viscosity parameter, ≥ 0 is the heat capacity, φ is the order parameter, u is the absolute temperature and g = G with G a double-well potential. These systems of equations describe phase transition processes such as melting or solidification. If = 0, then System (1) reduces to the viscous Cahn-Hilliard equation: which can be written in an equation for the single unknown, namely (cf. [20]; cf. also [10,12]). In [13], the author proved a well-posedness result for a problem in 3d including Problem (1), with irregular potentials such as logarithmic functions, and subject to dynamic boundary conditions. Global and exponential attractors and also some stability results were proven in [16,17] for a 3d conserved phase-field system with viscosity and memory terms and subject to Neumann boundary conditions. System (1) with δ = 0 were considered in [1,5,6,14]. In [5,6], the authors considered the problem, subject to Neumann boundary conditions, and with an arbitrary polynomial g(φ) when d = 1, 2, and g(φ) = 1 4 (φ 3 − φ) when d = 3. Introducing the change of variable v = u + φ, they showed the existence of the global and exponential attractors for the problem in the variable (φ, v). In [14], A. Miranville considered the problem subject to Dirichlet boundary conditions, for d = 2 or 3 and for a large class of functions G(φ) including polynomials of any arbitrary odd degree with a strictly positive leading coefficient. He proved the existence of exponential attractors. More recently, in [1], A. Bonfoh proved the existence of the global and exponential attractors and also inertial manifolds. He also proved some convergence properties of the dynamics to the one of the Cahn-Hilliard equation as goes to zero. Note that the exponential attractors attract bounded sets of a closed subspace of the space H 1 (Ω) × L 2 (Ω) in [1] while this space is H 2 0 (Ω) ∩ H 1 0 (Ω) × H 1 0 (Ω) in [14] and H 2 (Ω) × H 2 (Ω) in [5,6]. Our aim in this paper is to extend and improve the analysis carried out in [1] to System (1) but with δ > 0 and subject to the boundary conditions either of Neumann or periodic type. More precisely, we prove the existence of the global and exponential attractors and also inertial manifolds. Then we compare the dynamics of (1) with the one of the viscous Cahn-Hilliard equation. Let us now mention an earlier work of A. Bonfoh [2] where a similar study was done on a non-conserved phase-field system having the viscous Cahn-Hilliard equation as singular limit. The present paper also aims to improve some methods and results of [2]. Here, we give a direct proof of the existence of inertial manifolds that differs from the method used in [2] (inspired by [8]) that was based on introducing the change of variable w = 2 −1/2 φ + √ u and an auxiliary problem in the variable (φ, w). Also, the continuity properties do not require smoothness of exponential attractors and absorbing sets as previously needed in [1,2]. This paper is organized as follows. In Section 2, we set the problem. In Sections 3 and 4 we derive a priori estimates and we demonstrate the well-posedness of the problem and the existence of the global attractor which is upper semicontinuous at = 0. Then, in Section 5, we prove the existence of inertial manifolds in one space dimension, and for the case of a rectangular domain in two space dimension. In the final Section 6, we construct exponential attractors that are continuous at = 0 in a metric independent of . Continuity properties of intersection of the inertial manifolds with bounded sets are also examined.
We denote by W the dual space of W .

ON A CONSERVED PHASE-FIELD SYSTEM 1079
System (1) is subject to the boundary conditions either of Neumann or periodic type (the symbol ∂ n denotes the outward normal derivative) if Ω is a bounded domain of R d , with smooth boundary ∂Ω, or for φ and the derivatives of φ of order ≤ 3, if Ω = Π d i=1 (0, L i ). Let us define the linear unbounded operator, with domain D(N ), (4), which is self-adjoint and nonnegative. If N is restricted to D(N ) ∩L 2 (Ω), then it turns to be positive with compact inverse N −1 . Moreover, one can define the powers N r of N for r ∈ R (cf. [23] at page 57). The spaces V r = D(N r/2 ) are Hilbert spaces. In particular, V −1 = (H 1 (Ω)) or (H 1 per (Ω)) , V 0 = L 2 (Ω), V 1 = H 1 (Ω) or H 1 per (Ω). The injection V r1 → V r2 is compact whenever r 1 > r 2 . We denote by . and (., .) the usual norm and scalar product in L 2 (Ω) (and also in L 2 (Ω) d ). When r is positive, V r is a subspace of H r (Ω) and ϕ r = N r/2 ϕ 2 + |m(ϕ)| 2 1/2 is a norm on V r which is equivalent to the usual H r (Ω)−norm; we endow V r with the norm Note that, for k ∈ N and any ϕ ∈ V k , we have N k/2 ϕ = ∆ k/2 ϕ when k ≥ 2 is even, and We consider the problem where ∈ (0, 1] and δ > 0. We denote the function G(s) = s 0 g(ς)dς and we assume that g ∈ C 2 (R) and the following conditions hold (cf., e.g., [4]): (where C 2 , C 3 are bounded when γ is bounded) where p > 0 is arbitrary when d = 1, 2 and p ∈ [0, 3] when d = 3. For instance, g(s) = s 3 − s satisfies (7)- (10). However, we note that, in one space dimension, no growth assumption on g is needed.
The space X Y denotes the closure of a metric space X ⊂ Y in the topology of the complete metric space Y . Furthermore, there exist two positive constants C 6 , C 7 such that For every r ≥ 0, we endow the Hilbert space Note that (I + N ) r/2 . 2 is a norm on V r which is equivalent to . r . Sometimes, we will use the equivalent norm (ϕ, ψ) Ur, = (I + N ) r/2 ϕ 2 + (I + N ) (r−1)/2 ψ The Hausdorff semi-distance with respect to the metric of E is defined as: whereas the symmetric Hausdorff distance between A and B is for some α ≥ 0 and σ ≥ 0.

4.
Well-posedness and the global attractor. We start this section by proving a well-posedness result.
Theorem 4.1. We assume that (7)-(10) hold. If (φ 0 , u 0 ) ∈ U 1 , then (6) possesses a unique solution (φ, u) such that Proof. (i) Existence: The existence follows from standard arguments, using Galerkin approximations and then passing to the limit (see for instance [23]). If (φ 0 , u 0 ) ∈ U 1 , then the approximate solutions (φ m , u m ) are bounded independently of m (cf. (15)), and using weak compactness, we find a subsequence still denoted by (φ m , u m ) and a pair of Since g is continuous, we can pass to the limit as m → ∞ in the approximate problem, and (φ, u) is solution to (6). From classical compactness theorems, it follows that φ is weakly continuous from [0, T ] into V 1 , and u is weakly continuous from [0, T ] into L 2 (Ω). Using (11), we can see that the real function t → u 2 + ∇φ 2 is continuous on [0, T ]. We can conclude that φ is strongly continuous from [0, T ] into V 1 , and u is strongly continuous from [0, T ] into L 2 (Ω). If (φ 0 , u 0 ) ∈ U 2 , we can proceed like in part (i) to show the existence of a pair of functions (φ, u) solution to (6) (ii) Uniqueness: Let (φ 1 , u 1 ) and (φ 2 , u 2 ) be two solutions of (6).
We multiply (20) by N −1 φ t , and (21) by u and we integrate over Ω, respectively, and we obtain 1 2 and 2 d dt When d = 1, we have When d = 2, we have (22) and (23), we deduce where Applying the Gronwall's lemma to (24), we deduce that Thanks to Theorem 4.1, we can define the semigroup is the solution to (6) at time t. The semigroup S (t) is strongly continuous. We apply the Gronwall's lemma to (15) and we deduce the existence of an absorbing set for S (t) on K α,σ of the form where r 1 is independent of . Note that, if (φ, u) ∈ K α,σ , then the constant c 0 in (15) is bounded from below by a strictly positive constant that does not depend on m(φ 0 ) and m(u 0 ), the other constants are also independent of m(φ 0 ) and m(u 0 ).
We also deduce from (15) Applying the uniform Gronwall lemma to (19), we deduce the existence of an absorbing set for S (t) on K α,σ of the form where r 2 is independent of . The semigroup S (t) restricted to K α,σ is then uniformly compact. We apply [23, Chap. 1, Theorem 1.1], and we state the following result.
Theorem 4.2. For every ∈ (0, 1], the semigroup S (t) has the global attractor A α,σ in K α,σ , that is, The semigroup S(t) generated by the unperturbed problem (for the variable φ alone) possesses the global attractor A α on K α (see [4]). We now wish to compare the global attractor A α,σ with A α for small values of . Observe that, a solution of the unperturbed problem for both variables φ and u (at time t) is given by We define We now prove the following stability property.
Theorem 4.3. The global attractor A α,σ is upper semicontinuous at = 0, that is, Proof. Let (φ 0 , u 0 ) ∈ A α,σ (we have A α,σ ⊂ B 2 ; cf. Sect. 6). Thus, owing to the definition of the global attractor A α and also (97), If η > 0, then we can show, on account of (70) and (27), that there exist (φ, ζ) belonging to (A α ) σ , t η > 0 and η (all depending only on η) such that The upper semicontinuity (26) follows from (28) and the invariance property 5. Inertial manifolds. In this section only, we take d = 1 or 2, and we assume Ω = Π d i=1 (0, L i ) and L 1 /L 2 is a rational number. In order to prove the existence of an inertial manifold for Problem (6)-(10), we introduce the "prepared problem": where and θ : R + → [0, 1] is a C ∞ function such that θ(s) is equal to 1 when 0 ≤ s ≤ 1, and is equal to 0 when s > 2, and |θ (s)| ≤ 2, ∀s ≥ 0. Then we write (29) in the following form: where U = (φ, u), The operator A : U 3 → U 1 is non self-adjoint, positive and has a discrete spectrum for k = 0, 1, 2, ... and corresponding eigenfunctions {λ k } are the eigenvalues of N ordered in an increasing sequence and {e k } are the corresponding eigenfunctions which is an orthogonal basis of L 2 (Ω). These eigenvalues and eigenfunctions have the form: There exists n such that λ n ≥ 1 and Indeed, if d = 1, then this is immediate; and if d = 2, then the result is due to I. Richards (see [21]), since L 1 /L 2 is rational. Of course, the latter estimate implies that Let us now prove the following results.
Lemma 5.1. Provided that n is large enough for (32) to hold, there exists (n) suitably small such that the following inequalities are satisfied: Similarly the term D n+1 − √ D n and D n+1 − D n have the same sign, and and we have f 2 n < x − ≤ x + , hence (iii). (iv) A computation shows that and Summing (35) with (36), we deduce that (iv) is equivalent to which holds whenever (32) is satisfied (refer to the proof of (i)).
We have the following result.
Proof. It is clear that g − k is positive for every k. We have Thanks to (i) and (ii) of Lemma 5.1, the inequality We set ∆ n = A − B, with so that, (38) is exactly the following We observe that Now, on account of (v) of Lemma 5.1, inequality (39), in turn, is equivalent to , which can be rewritten as The inequality (40), in turn, can be rewritten as Noting that j − n = g + n g − n , we compute that (j − n ) 2 + (g − n ) 4 − 2j + n (g − n ) 2 = 0, and we find that (41) is equivalent to On account of (i), (iii) and (iv) of Lemma 5.1, the inequality (42) holds for every ∈ (0, 3 (n)], for some 3 (n) > 0. Finally, it results from (42), (ii) and (v) of Lemma 5.1 that (37) holds for every ∈ (0,˜ (n)], where˜ (n) = min{ (n), 3 (n)}.
We now prove the existence of an inertial manifold for Problem (6).

U1
, We introduce the scalar product ., . in U 1 (inspired by [24]) defined by where P Xn and P Yn are, respectively, the projections from U 1 onto X n and Y n and the functions Ψ 1 : X n × X n → R and Ψ 2 : Y n × Y n → R are defined by with U = (u, v), V = (y, z) in X n and Y n , respectively. Indeed, we have Thus, for U − l ∈ X n1 and U + l ∈ X n2 , noting that (e l , e k ) = δ lk , we have that As a consequence, X n1 is orthogonal to X n2 and to Y n , and the decomposition K α,σ = X n1 ⊕ X n2 ⊕ Y n is orthogonal with respect to the scalar product ., . and we set U 1 1 = X n1 and U 1⊥ 1 = X n2 ⊕ Y n . Let P and Q be the unique orthogonal projections onto U 1 1 and U 1⊥ 1 , respectively. We now define the norm |||U ||| = U, U 1/2 .
From (45) and (46), we can deduce that there exists c independent of such that Now, we note that g, g and g are bounded continuous functions on K α ∩ D(Λ), and there exists a constant c > 0 such that hence, there exists c > 0 such that, where Γ = I, if d = 1, Moreover, there exist C 1 , C 2 > 0, independent of , such that for every ∈ (0, 1]. It follows from the existence theorem of inertial manifolds [23, Chap. 9, Theorem 2.1] (cf. also [22]) that the semigroupS (t) generated by Equation (31) admits an inertial manifold M α,σ in K α,σ ∩ U d . More precisely, there exists a Lipschitz mapping Φ α,σ : K α,σ ∩ PU d → QU d such that the graph of Φ α,σ defines an inertial manifold forS (t) (or Equation (31)) of dimension n, with respect to the metric induced by the norm |||Γ.|||. Since Problems (31) and (6) are identical inside the absorbing set B d , it follows that the restriction ofS (t) to M α,σ coincides with that of S (t) on M α,σ . Hence, M α,σ is also the desired inertial manifold for S (t), that is, M α,σ = M α,σ .
6.1. Estimates of the difference of two solutions. Firstly, we estimate the difference of two solutions of (6).
There exists t * > 0 such that S (t)B k ⊂ B k for all t ≥ t * . From now on, we set and we will always assume that t * ≥ 1. Then we have that Note that B k is a bounded absorbing set for S (t)| Kα,σ as well. Now, we prove the following Proposition 3. There exists c > 0, independent of , such that for any z ∈ B 2 and any ∈ (0, 1]. Proof. Multiplying (6) 2 by N u and integrating over Ω, we obtain From (6) 1 , we deduce that Substituting (66) into (65), we find We have since φ(t) 2 ≤ c, ∀t ≥ 0, and we deduce from (67) that We first multiply (68) by e ct/ , then we integrate between s and t + 1, for any s ≤ t + 1. This yields Integrating now (69) between t and t + 1 with respect to s, we deduce u(t) 2 1 ≤ c, ∀t ≥ 1, since |m(u 0 )| ≤ σ, hence the result.
From now on, we will assume that B = {ϕ ∈ D(N k/2 ), (I + N ) k/2 ϕ ≤ r k } is an absorbing set of S(t) on K α , where r k > 0 are the same as in B k , k = 1,2.
We now show the following estimate.
We have due to (72), and, therefore, We multiply (92) by N −1 v and integrate over Ω, and we deduce and then v(t) 2 ≤ ce c t , ∀t ≥ 0.
Hence the result.

6.2.
A robust family of exponential attractors. We now give sufficient conditions ensuring the existence of uniform exponential attractors that are continuous with respect to (cf. [3,Theorem 5.1]; cf. also [2,11,15]). More precisely, we have the following theorem.
and endow them with the following norms 1], with the convention that E 0 = E 1 , V 0 = V 1 , and W 0 = W 1 . Let B (r) denote a closed ball in W of radius r > 0 and centered at zero. Consider a one-parameter family of strongly continuous semigroups {S (t)} acting on the phase-space E , for each ∈ [0, 1]. Then assume that there exist α, β, γ, ϑ ∈ (0, 1], κ ∈ (0, 1 2 ), Υ j ≥ 0, and > 0 (all independent of ) such that, setting B = B ( ), the following conditions hold: 1. There exists a map L : B 0 → V 2 which is Hölder continuous of exponent α. Here B 0 is endowed with the metric topology of E 1 . 2. There exists t > 0, independent of , such that and B is uniformly bounded (with respect to ) in the E 1 −norm. Moreover, setting S (t ) = S , the map S satisfies, for every z 1 , z 2 ∈ B , 3. For any z ∈ B , there hold Here the "lifting" map L : B 0 → E is defined by (i) E attracts B with an exponential rate which is uniform with respect to , that is, for some M 1 > 0 and some ω > 0. (ii) The fractal dimension of E is uniformly bounded with respect to , that is, (iii) The family E is Hölder continuous with respect to , that is, there exist a positive constant M 3 and τ ∈ (0, 1 2 for all 0 < ≤ 1. In addition, there exist a positive constant M 4 and σ ∈ (0, 1 2 for all 0 < ≤ 1, and Here ω, τ , σ and M j are independent of , and they can be computed explicitly.
We prove the following result.
Proof. We first observe that the semigroup S(t) has an exponential attractor E α on K α (see [4]). Let us now prove the theorem. On account of Theorem 6.1, we let E = U 1 , V = W = U 2 , B = B 2 and we check all the assumptions 1-5. To verify Assumption 1, we show that there exists a constant c such that for any φ 1 and φ 2 in B. Assumption 2 is satisfied by Propositions 2 and 3. Assumption 3 is given by Proposition 4. We choose t * such that (70) is satisfied. Thus, we are left to check Assumptions 4 and 5. Indeed, defining with z 0i = (φ 0i , u 0i ) ∈ B 2 , i = 1, 2, and t ∈ [t * , 2t * ], we obtain On the one hand, we have Therefore, On the other hand, it follows from (51) that Hence, we conclude with This shows the existence of exponential attractors on B 2 U1 that satisfy (95) and (96). Then, like in [12], we can extend the basin of attraction to the whole phase-space U 1 by using the transitivity property of the exponential attraction.
To prove the remaining estimate on u t , we multiply (6) 2 by u t , and we integrate over Ω, and we find Substituting (66) into (102), we find Using (99) and (94), we deduce from (103) that Integrating (104) between 0 and t, and using (71) and (101), the result follows.
6.3. Continuity of inertial manifolds. We now want to prove some stability properties of the inertial manifolds. Firstly, we recall that the semigroup S(t) possesses an inertial manifold M α on K α (see [10,19]). More precisely, there exists a Lipschitz mapping Φ α : P K α ∩ D(Λ) → QD(Λ) such that the graph of Φ α defines an inertial manifold for the unperturbed "prepared problem": where g is defined by (30). Here P is the unique orthogonal projection in D(Λ) onto the space spanned by {e 0 (x), e 1 (x), ..., e n (x)} (cf. Sect. 5), and Q = I − P . For any arbitrary R > 0, we define the following bounded sets which are intersection of the inertial manifolds M α,σ proven in Theorem 5.2 and M α with bounded sets (we recall that L β is defined by (25)).
We prove the following results.