Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions

Our aim in this paper is to prove the existence and uniqueness of solutions to Cahn-Hilliard and Allen-Cahn type equations based on a modification of the Ginzburg-Landau free energy proposed in [12] (see also [16]) which takes into account strong anisotropy effects. In particular, the free energy contains a regularization term, called Willmore regularization.

1. Introduction. In [6] and [18], a modified Ginzburg-Landau free energy which takes into account strong anisotropy effects arising during the growth and coarsening of thin films is considered, namely, Here, u is the order parameter, n = ∇u |∇u| , Ω is the domain occupied by the material, and f (u) = F (u) = u 3 − u.
(1.3) Furthermore, γ(n) is a bounded function which describes the anisotropy effects, G(u) = ω 2 , ω = f (u) − ∆u, is called nonlinear Willmore regularization and β is a small positive regularization parameter. Such a regularization is relevant, e.g., in determining the equilibrium shape of a crystal in its own liquid matrix, when anisotropy effects are strong. Indeed, in that case, the equilibrium interface may not be a smooth curve and may present facets and corners with slopes of discontinuities (see, e.g., [19]).
The author in [15] proved the well-posedness for a one-dimensional Allen-Cahn system based on (1.1). The analysis in [15] consists in regularizing γ. Unfortunately, the estimates obtained there are not uniform with respect to the regularization parameter, so that one is not able to pass to the limit.

AHMAD MAKKI AND ALAIN MIRANVILLE
We recall that the original Ginzburg-Landau free energy, plays a fundamental role in phase separation and transition, see, e.g., [4] and [5]. Actually, in [12], the author proposed another modification of the Ginzburg-Landau free energy which takes into account anisotropy effects, namely, This model describes dendritic pattern formation for one component melt growth and plays an important role in crystal growth. Compared with (1.1), we will not have to regularize γ when studying the corresponding models. Indeed, obtaining a variational derivative (see below) of (1.1) is not clear without such a regularization. We considered in [16] the Cahn-Hilliard and Allen-Cahn systems based on (1.5) in one space dimension and proved the existence and uniqueness of solutions. We can note that, in this particular case, n only takes the values ±1, which makes the analysis easier than in higher space dimensions.
In this article, we now deal with the two-dimensional case and (1.5) can be rewritten in the form and Actually, what follows can also be easily extended to the three-dimensional case (see Remark 1, b) below).
We can write, formally, for a small variation and assuming proper boundary conditions (see Section 3 and Remark 2 below), and the variational derivative of Ψ AM GL with respect to u reads Our aim in this article is to prove the existence and uniqueness of solutions for the Cahn-Hilliard and Allen-Cahn systems associated with the Ginzburg-Landau free energy (1.6), i.e., with (1.9), in higher dimensions. In particular, (1.6) and (1.9) lead to a sixth-order Cahn-Hilliard equation and a fourth-order Allen-Cahn equation.

Preliminaries.
We assume in what follows that the function γ in (1.5) is of class C 2 .
Lemma 2.1. The function g defined in (1.7) is of class C 1 .
Proof. We first prove that g is a continuous function. We note that, for all s = (s 1 , which implies that We then have, for (s 1 , s 2 ) = (0, 0), Noting that s 2 i |s| ≤ |s i | ≤ |s| and s 1 s 2 |s| ≤ 1 2 |s|, this yields that ∂g ∂s 1 and ∂g ∂s 2 are continuous and, consequently, g is of class C 1 , Proof. We compute, for instance, ∂ 2 g ∂s 1 ∂s 2 for (s 1 , s 2 ) = (0, 0) (the proof is similar for the other second-order derivatives) and have Noting that |s| 4 ≤ 1, we finally deduce that Proof. Let u, v ∈ R 2 . We set ϕ = ∂g ∂s i , i = 1, 2.
and note that δ(s) = 0, ∀s ∈ [0, 1]. In that case, we have It thus follows from Lemma 2.2 that • If 0 ∈]u, v[, we can assume that (up to a rotation and a translation) We then take δ(s) = −a cos(πs) a sin(πs) , which gives and have It then again follows from Lemma 2.2 that • If u = 0 or v = 0, taking, for instance, v = 0, then (see the proof of Lemma In conclusion, we have, in all cases, and ϕ is Lipschitz continuous.
Remark 2.4. a) It follows from Lemma 2.3 that g is Lipschitz continuous, where g denotes the differential of g. b) We can proceed in a similar way in three space dimensions. In particular, in the second case, we can assume, without loss of generality, that u and v belong to the plane (O, being an orthonormal frame) and proceed as in the proof of Lemma 2.3.
We set, whenever it makes sense, · = 1 Vol(Ω) Ω · dx, being understood that, for , and we note that is a norm on H −1 (Ω) which is equivalent to the usual one. Here, Ω = (0, Throughout the article, the same letter c (and, sometimes, c or c M when accounting for the dependence on a parameter M ) denotes constants which may vary from line to line. Similarly, the same letter Q denotes monotone increasing (with respect to each argument) functions which may vary from line to line.
As far as the nonlinear term f is concerned, we assume more generally that f is of class C 2 and that Note that these assumptions are satisfied by the cubic nonlinear term (1.3). We can also note that (2.1)-(2.4) are satisfied by polynomials of the form f (s) = 2p+1 i=1 a i s i , a 2p+1 > 0. Assumption (2.5) is, on the contrary, much more restrictive and is needed to obtain dissipative estimates (see below). It is however reasonable, since it is satisfied by the cubic function f (s) = s 3 − s which is usually considered in the Cahn-Hilliard and Allen-Cahn theories (we can further note that this cubic function actually is an approximation of the logarithmic function (see [4]), which also satisfies (2.5)). As far as the function g is concerned, we assume that 3. Cahn-Hilliard system. The Cahn-Hilliard equation is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary alloy spontaneously separate and form a fined-grained structure in which each of the two components appears more or less alternatively. This equation was initially derived as a model for spinodal decomposition in solid materials [4] and has been extended to many other physical systems.
We also note that there is currently a strong interest in the study of sixth-order Cahn-Hilliard equations; these equations also arise in situations such as atomistic models of crystal growth (see [2], [3] and [8]), the description of growing crystalline surfaces with small slopes which undergo faceting (see [17]), oil-water-surfactant mixtures (see [9] and [10]) and mixtures of polymer molecules (see [7]). We refer the reader to [6], [11], [13], [14] and [18] for the mathematical and numerical analysis of such models.
Setting of the problem. Writing the mass conservation, i.e., ∂u ∂t = − div h, where h is the mass flux which is related to the chemical potential µ by the constitutive relation h = −∇µ, and that the chemical potential is the variational derivative of (1.6) with respect to u, we end up with the following sixth-order Cahn-Hilliard system, taking, for simplicity, β = 1, 3) together with periodic boundary conditions, u, µ, ω are Ω − periodic, (3.4) and the initial condition Remark 3.1. Actually, the Cahn-Hilliard equation usually is endowed with Neumann boundary conditions. In our case, these conditions read where, here, Ω is a regular and bounded domain of R 2 with boundary Γ and ν denotes the unit outer normal. We also note that g (∇u) · ν = γ 2 ∇u |∇u| ∂u ∂ν In particular, ∂u ∂ν = 0 on Γ does not necessarily imply g (∇u) · ν = 0 on Γ, contrary to what we had in the one-dimensional case [16] (we need the condition ∂u ∂ν = 0 on Γ for the existence scheme, based on Galerkin approximations, see below). We can note that this holds provided that we have the compatibility relation Unfortunately, (3.7) yields, considering polar coordinates, that γ(s) = ϕ(|s|), i.e., γ(n) is constant. One idea, to handle such boundary conditions could be to consider a different approximation scheme (e.g., a finite differences scheme). This will be addressed elsewhere.
Our main aim in this section is to prove the 3.1. A priori estimates. We first note that, integrating (formally) (3.1) over Ω, we obtain the conservation of mass, namely, We then multiply (3.2) by ∂u ∂t and integrate over Ω to find We note that it follows from (3.3) that We finally deduce from (3.9)-(3.12) that In particular, (3.13) yields that the free energy decreases along the trajectories, as expected.
Let v 0 , v 1 , · · · be an orthonormal (in L 2 (Ω)) and orthogonal (in H 1 per (Ω)) family associated with the eigenvalues 0 = λ 0 < λ 1 ≤ · · · of the operator −∆ associated with Neumann boundary conditions (note that v 0 is a constant). We set where u 0,m = P m u 0 , P m being the orthogonal projector from L 2 (Ω) onto V m . The existence of a local (in time) solution to (3.58)-(3.61) is standard. Indeed, we have to solve a Lipschitz continuous finite-dimensional system of ODE's to find u m , which yields ω m and then µ m .
As far as the passage to the limit is concerned, the most delicate part is to prove that for ϕ regular enough. We have, say, for The passage to the limit in the first integral in the right-hand side of (3.62) is straightforward, while the passage to the limit in the second one follows from the above convergences which yield, in particular, the inequality Finally, recalling that g is Lipschitz continuous, we have Assuming the relaxation dynamics ∂u ∂t = − DΨ AM GL Du , we obtain the following Allen-Cahn system, taking again β = 1, ∂u ∂t − div(g (∇u)) + f (u) + ωf (u) − ∆ω = 0, (4.1) 2) together with Dirichlet boundary conditions, where Γ is the boundary of Ω (we assume that Ω is a bounded and regular domain of R 2 ), and the initial condition Remark 4.1. We have considered, for simplicity, Dirichlet boundary conditions; actually, the Allen-Cahn equation usually is endowed with such boundary conditions. However, periodic boundary conditions, can also be considered (as far as Neumann boundary conditions are concerned, the situation is similar to that in Section 3). In that case, we can obtain the same results by replacing (2. 2) by Note that, in this section, we do not need (2.4).