EXISTENCE OF SOLUTIONS FOR A ONE-DIMENSIONAL ALLEN-CAHN EQUATION

Our aim in this paper is to prove the existence and uniqueness of solutions for a one-dimensional Allen-Cahn type equation based on a modification of the Ginzburg-Landau free energy proposed in [10]. In particular, the free energy contains an additional term called Willmore regularization and takes into account anisotropy effects.


Introduction
The Allen-Cahn equation, ∂u ∂t − ∆u + f (u) = 0, (1.1) where u is the order parameter and f (s) = s 3 − s, describes important processes related with phase separation in binary alloys, namely, the ordering of atoms in a lattice (see [1]). This equation is obtained by considering the Ginzburg-Landau free energy, (1.2) where Ω is the domain occupied by the material and F (s) = 1 4 (s 2 − 1) 2 . Assuming a relaxation dynamics, i.e., writing where D Du denotes a variational derivative, we obtain (1.1). In [10] (see also [2]), the authors introduced the following modification of the Ginzburg-Landau free energy: (1.5) where G(u) = 1 2 ω 2 is called nonlinear Willmore regularization, β is a small regularization parameter and the function δ accounts for anisotropy effects. The Willmore 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, but may present facets and corners with slope discontinuities (see, e.g., [8]), which can lead to an ill-posed problem and requires regularization.
The Allen-Cahn equation associated with (1.4) has been studied in [5] in the particular cases δ ≡ 1 (isotropic case) and δ ≡ −1 (in that case, Ψ AGL is also called functionalized Cahn-Hilliard energy in [7]). In particular, well-posedness results have been obtained. The Cahn-Hilliard equation associated with (1.4) (obtained by writing ∂u ∂t = ∆ DΨAGL Du ) has been studied in [4], again, in the isotropic case δ ≡ 1; we also refer the reader to [2] and [11] for numerical studies.
In one space dimension, i.e., taking Ω = (0, L), and setting β equal to one, (1.4) reads We actually consider the following natural regularization of Ψ AGL : In that case, we have, formally, Therefore, (1.8) In this paper, we will consider the simplest case δ(s) = s (note that ∂u ∂x (1.9) Assuming again a relaxation dynamics, we finally obtain the following (regularized) anisotropic Allen-Cahn system: Our aim in this paper is to prove the existence and uniqueness of solutions to (1.10)-(1.11).

A priori estimates
We consider the following initial and boundary value problem: We denote by ((·, ·)) the usual L 2 -scalar product, with associated norm ∥ · ∥, and we denote by ∥ · ∥ X the norm in the Banach space X.
Throughout the paper, the same letter c (and, sometimes, c ′ ) 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.

Existence and uniqueness of solutions
We have the

Proof. a) Existence:
The proof of existence is based on a standard Galerkin scheme and the a priori estimates derived in the previous section.

5)
where u 0,m = P m u, P m being the orthogonal projector from L 2 (0, L) onto V m (for the L 2 -norm). The existence of a local (in time) solution is standard, as we have to solve a (continuous) finite system of ODE's. It then follows from the a priori estimates derived in the previous section that this solution is global.
We then need to pass to the limit in the nonlinear terms. We have Therefore, since a.e. and | ∂um ∂x | ≤ g ∈ L 2 ((0, L) × (0, T )) a.e. (up again to a subsequence which we do not relabel), we deduce from Lebesgue's theorem that in L 2 (0, T ; L 2 (0, L)) (here, we have used the fact that L 2 (0, T ; L 2 (0, L)) is isometric to L 2 ((0, L) × (0, T ))). Similarly, which finishes the proof of the passage to the limit, hence the existence of a solution.

b) Uniqueness:
Let u 1 and u 2 be two solutions to (2.1)-(2.3) (ω 1 and ω 2 being defined as in (2.2)) with initial data u 0,1 and u 0,2 , respectively. Then, setting u = u 1 − u 2 , ω = ω 1 − ω 2 and u 0 = u 0,1 − u 0,2 , we have We multiply (3.7) by u and obtain, owing to (3.8), Noting that it follows that (3.13) Here, we have used the fact, owing to the continuous embedding (3.14) Then, Noting that we find, proceeding as in (3.13), (3.15) Now, (3.16) Finally, (3.17) We thus deduce from (3.11)-(3.17) that which yields, employing the interpolation inequality (2.17), Assuming that δ is of class C 1 and noting that | ∂u ∂x | ≤ 1, we can proceed exactly as above to prove the existence of a solution. Furthermore, assuming that δ is of class C 2 , we can easily adapt the proof of uniqueness and deduce the existence and uniqueness of solutions.

Remark 3.2.
We can note that our estimates are not independent of ϵ, so that we cannot pass to the limit as ϵ goes to 0. This is not surprising, as the problem formally obtained by taking ϵ = 0 cannot correspond to the (Allen-Cahn) problem associated with the free energy (1.6) (see also [2] and [10]). Actually, this is related with a proper functional setting for the limit problem and, more precisely, for the Allen-Cahn system associated with (1.6) and will be studied elsewhere. We can note that anisotropic versions of the Allen-Cahn equation have been studied in [3] and the references therein, based on viscosity solutions. Such an approach is not straightforward here, as there is no maximum/comparison principle for fourth-order in space parabolic equations.

Remark 3.3.
It is also important to study the Cahn-Hilliard system associated with (1.7) (for δ(s) = s), namely, (3.23) We thus have an equation which bears some resemblance with (2.1), except that we have less regularity on ∂u ∂t , which prevents us from proceeding as in the proof of Theorem 3.1. However, if we consider the viscous Cahn-Hilliard equation (introduced in [6] for the usual Cahn-Hilliard equation), (3.24) or, equivalently, then, proceeding as in the proof of Theorem 3.1, we have the existence and uniqueness of solutions.