CONVERGENCE OF EXPONENTIAL ATTRACTORS FOR A TIME SPLITTING APPROXIMATION OF THE CAGINALP PHASE-FIELD SYSTEM

. We consider a time semi-discretization of the Caginalp phase-ﬁeld model based on an operator splitting method. For every time-step parameter τ , we build an exponential attractor M τ of the discrete-in-time dynamical system. We prove that M τ converges to an exponential attractor M 0 of the continuous-in-time dynamical system for the symmetric Hausdorﬀ distance as τ tends to 0. We also provide an explicit estimate of this distance and we prove that the fractal dimension of M τ is bounded by a constant independent of τ .


1.
Introduction. The Caginalp phase-field system, which has been proposed in [9] to model phase transition phenomena such as melting-solidification, has a great importance in materials science. Other models of phase transition can also be derived from it as singular limits (e.g. the Allen-Cahn, the Cahn-Hilliard and the Stefan problems). We refer the reader to [1,3,7,10,8,16,31,40] and the references therein for more details.
In this paper, we will consider the Caginalp system where all physical parameters are set equal to one. It reads where ϕ is the order parameter, u is the relative temperature, and g is the derivative of a double-well potential G.
The asymptotic behaviour of this model has been extensively studied. Convergence to a stationary state, existence of global and exponential attractors has been proved for regular and singular potentials, and for various types of boundary conditions, see [4,6,12,13,14,24,26,27,29,30,43].
In this paper, we consider a time semi-discretization of (1)-(2) based on an operator splitting approach. For every time-step parameter τ > 0 small enough, the scheme defines a discrete dynamical system; the case τ = 0 corresponds to the continuous dynamical system associated to (1)- (2). Our purpose is to build a family M τ of exponential attractors associated to these dynamical systems and which is continuous at τ = 0 for the symmetric Hausdorff distance.
An exponential attractor is a compact positively invariant set which contains the global attractor, has finite fractal dimension and attracts exponentially the trajectories. In comparison with the global attractor, an exponential attractor is expected to be more robust to perturbations: global attractors are generally upper semi-continuous with respect to perturbations, but the lower semi-continuity can be proved only in some particular cases (see e.g. [2,28,33,35,37,38,39]). We can also note that an exponential attractor is not necessarily unique.
In the initial construction proposed by Eden et al. [17], based on a "squeezing property", the continuity of exponential attractors was shown for classical Galerkin approximations, but only up to a time shift (see also [23,25]). In [20], Efendiev, Miranville and Zelik proposed a construction of exponential attractors based on a "smoothing property" and an appropriate error estimate, where the continuity holds without time shift. Their construction has been adapted to many situations, including singular perturbations. We refer the reader to the review [28] and the references therein for more details.
In [32], the second author was able to use the construction in [20] in order to build a robust family of exponential attractors for a time semi-discretization of a generalized Allen-Cahn equation. An abstract result was first derived, and then applied to the backward Euler scheme. The robustness result in [20,32] also includes an upper bound on the fractal dimension of the family of exponential attractors (and therefore of the global attractors). This bound is independent of the time step and, although crude, it is explicit in terms of the physical parameters of the model. In contrast, only a few authors have obtained upper bounds independent of the discretization parameters for the dimensions of attractors, by using more specific methods [41,42,44].
Our proof in this paper uses the abstract result from [32], but here, the time discretization is a first order operator splitting method, which consists in decoupling the equations (1) and (2): at every time iteration, a semi-linear elliptic equation is first solved, followed by a linear elliptic equation. In comparison, the backward Euler scheme would require the resolution of a semi-linear elliptic system at every time iteration.
We choose this particular scheme because operator splitting methods, also known as fractional step methods, are ubiquitous in the numerical resolution of evolutionary systems of partial differential equations. They allow a very efficient reduction of the computational complexity. The Chorin-Temam [15,36] scheme for the resolution of the incompressible Navier-Stokes equation, the convex splitting of the energy for Allen-Cahn and Cahn-Hilliard type equations [21,22], or the exponential formula for the sum of maximal monotone operators [5] are some famous (nonexhaustive) examples of these methods.
The paper is structured as follows. The estimates for the continuous problem are first given in Section 2, their discrete counterparts are derived in Section 3, and the error between the discrete solution and the continuous solution is estimated in Section 4. The main result, Theorem 5.3, is stated in the last section.
2.1. The continuous semi-group. We consider the Caginalp system (1)- (2) in Ω × (0, +∞), where Ω is a bounded domain of R I (I = 1, 2 or 3) with smooth boundary ∂Ω. The function g is a polynomial of odd degree with a positive leading coefficient, For sake of simplicity, we assume that p = 2 when I = 3 (no condition on p is required if I = 1 or 2). This guarantees that the continuous imbeddings The potential G is defined by Following [6], we will work with the unknowns and we consider Neumann boundary conditions. The problem ("problem (P)") reads Here, ν is the outward unit normal to the boundary ∂Ω. We set where the norm in L 2 (Ω) is denoted | · | 0 and the scalar product (·, ·) 0 ; the norm in It is therefore convenient to introduce the function spaces where β ∈ R and α > 0; H β is an affine subspace of H whereas H α is a closed convex (unbounded) subset of H. We denote R + the interval [0, +∞).
The following result is proved in [6]: Moreover, the mapping is Lipschitz continuous on H for all t ≥ 0.
2.2. Some useful inequalities. By Poincaré's inequality (see e.g. [6]), there exists a constant c P = c P (Ω) such that for all v ∈ H 1 (Ω), By considering the leading coefficients of the polynomials G and g, it is easily seen (see e.g. [37]) that and for some positive constants c 1 , c 1 , c 2 and c 2 . There is also a nonnegative constant c g such that g (s) ≥ −c g ∀s ∈ R.

2.3.
A priori estimates for the solution. In this section, we collect some results from [6]. Proposition 2.3 (Absorbing set in H α ). For any α > 0, there exist a constant R 0 = R 0 (α) > 0 and a monotonic function T 0 (·) such that for all (ϕ 0 , v 0 ) ∈ H α , Let r denote a positive constant. We have: Proposition 2.4 (Absorbing set in H α ∩V ). For any α > 0, there exists a constant These results imply the existence of a compact and connected global attractor For the existence of an exponential attractor, we need a few additional estimates. The following result (Theorem 2.2 in [6]) will prove useful: There exists a positive constant D depending only on g such that in the remainder of this section. We have (see [6, (2.15)]): Next, we prove: In particular, for all 0 ≤ t 1 , t 2 ≤ T , we have Proof. First, we multiply (3) by ∂ϕ ∂t and we integrate over Ω. We find We integrate this relation over [0, t], we use (11), the Sobolev imbedding H 1 (Ω) ⊂ L 2p (Ω), and we find for some positive constant C( ϕ 0 1 ) = C( ϕ 0 1 , Ω, p, c 1 , c 1 ). Next, we multiply (4) by ∂v ∂t and we integrate over Ω. After some standard computations, we obtain We add (16) and (17) and we conclude, using Proposition 2.6, that (15) holds. The conclusion follows by a standard calculation.

2.4.
Estimates for the difference of two solutions. The purpose here is to obtain a "smoothing property" (cf. [28]), which is the key in our construction of the exponential attractor for the continuous dynamical system. In contrast, a "squeezing property" was used in [6] in the construction of an exponential attractor for the same problem. We note that in the Hilbert setting, the "smoothing property" implies the "squeezing property" (see [18, Remark (i)]).
Proof. We first multiply (18) by ψ and integrate over Ω. We obtain where we have used (14). Thus, Similarly, on multiplying (19) by w, we obtain Now, we add (22) to (23) and we use Gronwall's lemma. The conclusion follows with D g = max{2c g , 1}.
The following result is an H-V "smoothing property".
We note that the constant C 3 (R 1 , T ) blows up as T → 0 + .

NARCISSE BATANGOUNA AND MORGAN PIERRE
3. The time semi-discrete problem. For the time semi-discretization, we apply a splitting scheme to the system (18)- (19) in the variables (ϕ, v). Let τ > 0 denote the time step. The scheme reads: let (ϕ 0 , v 0 ) ∈ H β and for n = 0, 1, 2, . . . , let This scheme is nonlinear in the ϕ n+1 variable and linear in the v n+1 variable. By induction, we see that Thus, an important feature of the continuous problem (see (8)) is preserved on the discrete level.
3.1. The discrete semi-group. The following result shows that the discrete semi- Moreover, the mapping S τ : We note that the Lipschitz constant blows up as τ → 0 + .
In the remainder of the paper, we assume that the time step τ of the discrete system satisfies at least 0 < τ ≤ 1/(1 + c g ). We also denote · the floor integer function.
We note that (42) implies a n ≤ a 0 + C/γ for all n ≥ 0. Thus, we have: For the absorbing set in V , we will need the following lemma from [34]. for all k 0 ≥ n 0 , with r = τ N > 0. Then In the next result, r ≥ 2 is a fixed positive constant (cf. Proposition 2.4).
Proof. We multiply (28) by −∆ϕ n+1 and (29) by v n+1 , and we add the two resulting equations. Using Young's inequality and (38), we find Thus, by Proposition 3.5, for nτ ≥ T 0 ( (ϕ 0 , v 0 ) H ), we have On the other hand, estimate (41) shows that We are in position to apply the discrete uniform Gronwall lemma (Lemma 3.4) with g n = 0, a 1 = 0, and h n = C g R 2 0 (α) (a constant). We find . This shows the assertion.
Let us now derive a useful estimate.
3.3. Estimates for the difference of solutions, uniform in τ . In this section, (ϕ n , v n ) and (φ n ,v n ) are two sequences generated by the scheme (28)- (30). We denote ψ n = ϕ n −φ n and w n = v n −v n their difference, which satisfies We first have (compare with Lemma 2.8): Lemma 3.7. Assume that τ ≤ 1/(4c g ). Then for all n ≥ 0 we have where D g is a constant which depends only on c g .
The smoothing property reads as follows (compare with Lemma 2.9).

5.
Convergence of exponential attractors. 5.1. Some (standard) definitions. Before stating our main result, we recall some definitions (see e.g. [20,32,37]). The Hilbert space H is defined as previously by (7) and K denotes a closed subset of H. A continuous-in-time semi-group {S(t), t ∈ R + } on K is a family of (nonlinear) operators such that S(t) is a continuous operator from K into itself, for all t ∈ R + , with S(0) = Id (identity in K) and S(t + s) = S(t) • S(s), ∀s, t ∈ R + .
A discrete-in-time semi-group {S(t), t ∈ N} on K is a family of (nonlinear) operators which satisfy these properties with R + replaced by N. A discrete-in-time semigroup is usually denoted {S n , n ∈ N}, where S(= S(1)) is a continuous (nonlinear) operator from K into itself.
A (continuous or discrete) semi-group {S(t), t ≥ 0} defines a (continuous or discrete) dynamical system: if u 0 is the state of the dynamical system at time 0, then u(t) = S(t)u 0 is the state at time t ≥ 0. The term "dynamical system" will sometimes be used instead of "semi-group".  Proof. We apply Theorem 2.5 in [32] with the spaces H and V defined by (7), and the set B = {(ψ, w) ∈ H α : (ψ, w) V ≤ max{R 1 (α), R 1 (α)}} . We note that V is compactly imbedded in H, and that B is absorbing in H α , uniformly with respect to τ ∈ [0, τ 0 ]. The estimates of Sections 2-4 show that assumptions (H1)-(H9) of Theorem 2.5 in [32] are satisfied. Thus, the conclusions of Theorem 5.3 hold for τ ∈ [0, τ 0 ], for some τ 0 ∈ (0, τ 0 ] small enough (we note that Theorem 2.5 is stated for a family of semi-groups which act on the whole space H, but with a minor modification of its proof, it can be applied to our situation).
A standard argument (see e.g. [32]) shows that M τ is an exponential attractor for S τ on H α , with a fractal dimension bounded by a constant independent of τ ∈ [τ 0 , τ 0 ], and which attracts the bounded sets of H α , uniformly with τ ∈ [τ 0 , τ 0 ]. This concludes the proof (the continuity holds only at τ = 0).
As in [32, Corollary 6.2], we have: Corollary 5.4. For every τ ∈ [0, τ 0 ], the semi-group {S τ (t), t ≥ 0} possesses a global attractor A τ in H α which is bounded in V , compact and connected in H. Moreover, dist H (A τ , A 0 ) → 0 as τ → 0 + , and the fractal dimension of A τ is bounded by a constant independent of τ .