ON ANISOTROPIC CAGINALP PHASE-FIELD TYPE MODELS WITH SINGULAR NONLINEAR TERMS

Our aim in this paper is to study the well-posedness and the existence of the global attractor of anisotropic Caginalp phase-field type models with singular nonlinear terms. The main difficulty is to prove, in one and two space dimensions, that the order parameter remains in the physically relevant range and this is achieved by deriving proper a priori estimates.


Introduction
The Caginalp phase-field system, was proposed in [6] to model phase transition phenomena, such as melting-solidification phenomena. Here, u is the order parameter, T is the relative temperature (defined as T =T − T E , whereT is the absolute temperature and T E is the equilibrium melting temperature) and f is the derivative of a double-well potential F (a typical choice of potential is F (s) = 1 4 (s 2 − 1) 2 , hence the usual cubic nonlinear term f (s) = s 3 − s). Furthermore, here and below, we set all physical parameters equal to one. This system has been much studied; we refer the reader to, e.g., [3-5, 8, 10, 12, 13, 15, 16, 18-20, 25-27, 31].
These equations can be derived as follows. One introduces the (total Ginzburg-Landau) free energy where Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of R n , n = 1 or 2, with boundary Γ), and the enthalpy H = u + T. (1.4) As far as the evolution equation for the order parameter is concerned, one postulates the relaxation dynamics (with relaxation parameter set equal to one) where D Du denotes a variational derivative with respect to u, which yields (1.1). Then, we have the energy equation where q is the heat flux. Assuming finally the usual Fourier law for heat conduction, we obtain (1.2). Now, one essential drawback of the Fourier law is that it predicts that thermal signals propagate at an infinite speed, which violates causality (the so-called paradox of heat conduction). To overcome this drawback, or at least to account for more realistic features, several alternatives to the Fourier law, based, e.g., on the Maxwell-Cattaneo law or recent laws from thermomechanics, have been proposed and studied, in the context of the Caginalp phase-field system, in [20,21].
In the late 1960's, several authors proposed a heat conduction theory based on two temperatures (see [33][34][35]). More precisely, one now considers the conductive temperature T and the thermodynamic temperature θ. In particular, for simple materials, these two temperatures are shown to coincide. However, for non-simple materials, they differ and are related as follows: (1.8) The Caginalp system, based on this two temperatures theory and the usual Fourier law, was studied in [3].
Our aim in this paper is to study a variant of the Caginalp phase-field system based on the type III thermomechanics theory with two temperatures recently proposed in [36].
In that case, the free energy reads, in terms of the (relative) thermodynamic temperature θ, and (1.5) yields, in view of (1.8), the following evolution equation for the order parameter: Furthermore, the enthalpy now reads which yields, owing to (1.6), the energy equation Finally, the heat flux is given, in the type III theory with two temperatures, by (see [36]) is the conductive thermal displacement. In (1.3) (or (1.9)), the term |∇u| 2 models short-ranged interactions. It is however interesting to note that such a term is obtained by truncation of higher-order ones (see [30]); it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions (see [11]).
G. Caginalp and E. Esenturk recently proposed in [7] higher-order phase-field models in order to account for anisotropic interfaces (see also [29] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these authors proposed the following modified (total) free energy and, for β = (0, 0, 0), . Noting that T = ∂α ∂t , this then yields the following evolution equation for the order parameter u: (1. 16) In particular, for k = 1 (anisotropic Caginalp phase-field system), we have an equation of the form and, for k = 2 (fourth-order anisotropic Caginalp phase-field system), we have an equation of the form It is however important to note that, in phase transition, regular nonlinear terms actually are approximations of thermodynamically relevant logarithmic ones of the form f (s) = −λ 1 s + λ2 2 ln 1+s 1−s , s ∈ (−1, 1), 0 < λ 2 < λ 1 , which follow from a meanfield model (see [12]; in particular, the logarithmic terms correspond to the entropy of mixing).
Our aim in this paper is to consider the second-order anisotropic equation (1.17) and the energy equation with general singular nonlinear terms (containing the physically relevant logarithmic ones). In particular, we prove, in one and two space dimensions, the existence and uniqueness of classical solutions, as well as the existence of the global attractor of the associated dynamical system. Here, we do not address the higher-order models. Indeed, when k 2, we are not able to prove the existence of classical solutions and have to deal with a different notion of a solution, based on a variational inequality (see [9,22]; see also [18]).

Setting of the problem
We consider the following initial and boundary value problem: We assume that a i > 0, i ∈ {1, 2, 3}, (2.5) and we introduce the elliptic operator A defined by where H −1 (Ω) is the topological dual of H 1 0 (Ω). Furthermore, ((.,.)) denotes the usual L 2 -scalar product, with associated norm . ; more generally, we denote by . X the norm on the Banach space X. We can note that is bilinear, symmetric, continuous and coercive, so that is indeed well defined. It then follows from elliptic regularity results for linear elliptic operators of order 2 (see [1,2]) that A is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain where, for v ∈ D(A), We further note that D(A We finally note that (see, e.g., [28]) A. (resp., Remark 2.1. Note that similar properties hold for the operator −∆, with obvious changes. Having this, we rewrite (2.1) as As far as the nonlinear term f is concerned, we assume that In particular, it follows from (2.9)-(2.10) that Here, the only difficulty is to prove that F (s) f (s)s + c, c 0, s ∈ (−1, 1). To do so, it suffices to study the variations of the function s → f (s)s − F (s) + c0 2 s 2 , whose derivate has, owing to (2.10), the sign of s. We further assume that and that the following compatibility conditions hold: (2.14) Throughout the paper, the same letters c, c and c denote (generally positive) constants which may vary from line to line. Similary, the same letter Q denotes (positive) monotone increasing (with respect to each argument) and continuous functions which may vary from line to line.

A priori estimates
The estimates derived in this section are formal, but they can easily be justified within a Galerkin scheme.
We first assume that u is regular and a priori satisfies where T > 0 is an arbitrary final time.

Separation property
Our aim now is to prove that u a priori satisfies where δ ∈ (0, 1) depends only on the initial data and the final time T . In one space dimension, we have the Proof. In one space dimension, we have, owing to the embedding H 1 (Ω) ⊂ L ∞ (Ω), It is then not difficult to prove the separation property (4.1) for solutions to the parabolic equation Indeed, let δ ∈ (0, 1) be such that We multiply (4.3) by v + = max(v, 0) and obtain, owing to (2.10) and (4.2), which yields, owing to Growall's lemma and noting that v Proceeding as above and noting that f is odd, we also prove that We now consider the case dim Ω = 2. In that case, we do not have the embedding H 1 (Ω) ⊂ L ∞ (Ω) and, consequently, we are not able to obtain estimate (4.1) for all the nonlinearities satisfying (2.9). Nevertheless, using the embedding of H 1 (Ω) into an appropriate Orlicz space, we obtain this result for a wide class of nonlinearities, which includes the thermodynamically relevant logarithmic nonlinearities. Theorem 4.1. We assume that dim Ω = 2 and that the nonlinearity f satisfies assumptions (2.9) and the following additional condition: with some positive constants c and c . Then every solution u(t) of equation (2.7) satisfies estimate (4.1).
The proof of this result is based on the the following lemma.
Lemma 4.1. We assume that dim Ω = 2 and that the nonlinearity f satisfies assumptions (2.9). Then, for every L > 0, the following estimate holds: where c = c(L) depends only on the initial data and the final time T .
To conclude, we use the following Orlicz's embedding inequality ( [32]) : where c depends only on Ω and N . It then follows from (4.9), (4.17) and (4.18) that where c depends only on the initial data, T and L. Noting finally that where c depends on T and L, (4.19) yields the desired inequality (4.7) .
We finally rewrite (2.2) in the (functional) form where B denotes the minus Laplace operator with Dirichlet boundary conditions.
Taking the scalar product (in L 2 (Ω)) of (4.27) by B 3 dα dt , we have and we deduce from (4.26) and (4.28) that Rewriting again (2.7) in the form we have, owing to the above estimates, and the separation property follows as in the one-dimensional case.
Remark 4.1. In three space dimensions, one can also prove the strict separation property, but under growth assumptions on the singular nonlinear term which are not satisfied by the thermodynamically relevant ones, see, e.g., [17,22].

Existence and uniqueness of solutions
We restrict ourselves to the one-and two-dimensional cases (note however that one can prove an existence result also in three space dimensions, but without the strict separation property, see, e.g., [14]). We have the following result.

Proof. a)Existence:
The proof of existence is standard, once we have the separation property (4.1), since the problem then reduces to one with a regular nonlinearity.
Indeed, we define the approximated function f δ by where δ is the same constant as in (4.1) (note that f δ is not C 1 , but we can make a further regularization, see [10]). We choose δ such that f (δ) δ (resp., f (−δ) −δ).
We then consider the following approximated problem (P δ ) defined by We know that (P δ ) possesses one solution (u δ , α δ , ∂α δ ∂t ) if the functions f δ and F δ satisfie the same properties as those satisfied by f and F (see [17]). We have the The functions f δ and F δ possess the following properties: where c 0 and c 1 are the same constants as those in (2.10) and (2.11).
Proof. We only detail the case: The other are very similar and are omitted. It follows from the definition of f δ that Moreover, since f satisfie (2.11) and f (δ) > δ, we obtain Then, since f (δ) = f δ (s) − s + δ for s > δ, (2.11) also leads to Consequently, we know that problem P δ possesses at least one solution (u δ , α δ , ∂α δ ∂t ) such that the estimates derived above are valid here. In particular, Hence, (u δ , α δ , ∂α δ ∂t ) is also a solution to the original problem.

b)Uniqueness:
We actually prove a more general result, namely, the uniqueness of solutions such that |u(t, x)| < 1 almost everywhere in (0, T ) × Ω and which do not necessarily satisfy the separation property (4.5) (when this property is satisfied, the proof of uniqueness is straightforward).

Existence of global attractor
We saw in Section 4 that the key estimate, in view of the well-posedness, is a separation property of the form u L ∞ ((0,T )×Ω) δ(< 1). (6.1) However, here, δ depends on u 0 ; more precisely, one has δ u 0 L ∞ , meaning, in particular, that (6.1) cannot be a dissipative estimate, i.e., independent of the initial data for large times and for initial data belonging to some bounded set (thus, (6.1) cannot be used to prove the existence of a bounded absorbing set). Actually, in order to have a dissipative estimate, one has to be more accurate. To do so, we fix R > 0 and assume that the initial data satisty Then, one can prove that there exists t 0 = t 0 (R) such that where c is independent of R, hence, for dim Ω = 1, owing to the continuous embedding H 1 (Ω) ⊂ L ∞ (Ω), where c 0 is independent of R. Furthemore, one has where c 1 = c 1 (R) (we can assume, without loss of generality, that c 0 c 1 ). We now choose δ 0 (independent of R) and t 1 t 0 such that f (δ 0 ) c 0 + 1, δ 0 ∈ [γ, 1), where γ > 0 is such that f 0 on [γ, 1), λ(= λ(R)) = 1−δ0 t1 is such that 0 < λ 1, f (1 − λt 0 ) c 1 + 1. Note in particular that δ 0 y + (t) < 1, ∀t > 0, y + (0) = 1.
It follows from the above assumptions that hence G(t) 0, ∀t > 0, t = t 1 .
Noting that y + (t) = δ 0 for t t 1 , where δ 0 is independent of R, and proceeding similarly for a lower bound (note that f is odd), we obtain u(t) L ∞ δ 0 , ∀t t 1 , hence a dissipative estimate. Remark 6.1. It is now not difficult to prove, in view of the strict separation property of u, that A has finite dimension (in the sens of the Hausdorff or the fractal dimension, see, e.g., [23,28]); to do so, we essentially proceed as in the case of regular potentials (see, e.g., [25]). Remark 6.2. When dim Ω = 2, the situation is more involved and will be studied elsewhere.