Laurence Cherfils

A numerical analysis of the Cahn---Hilliard equation with non-permeable walls

In this article we consider the numerical analysis of the Cahn–Hilliard equation in a bounded domain with non-permeable walls, endowed with dynamic-type boundary conditions. The dynamic-type boundary conditions that we consider here have been recently proposed in Ruiz Goldstein et al. (Phys D 240(8):754–766, 2011) in order to describe the interactions of a binary material with the wall. The equation is semi-discretized using a finite element method for the space variables and error estimates between the exact and the approximate solution are obtained. We also prove the stability of a fully discrete scheme based on the backward Euler scheme for the time discretization. Numerical simulations sustaining the theoretical results are presented.

On the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard Equation with Logarithmic Nonlinear Terms

Our aim in this paper is to study the existence of local (in time) solutions for the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation with logarithmic nonlinear terms. This equation was proposed in view of applications to binary image inpainting. We also give some numerical simulations which show the efficiency of the model.

Finite-dimensional attractors for a model of Allen-Cahn equation based on a microforce balance

Abstract We are interested in this Note in the long time behavior of a model of AllenCahn equation based on a microforce balance introduced by M. Gurtin in [6]. In particular, we obtain the existence of finite dimensional attractors.

A Cahn---Hilliard System with a Fidelity Term for Color Image Inpainting

In this paper, we propose a model for multi-color image inpainting composed of n colors. In particular, as in the binary model, i.e., the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation (Bertozzi et al. in IEEE Trans Image Proc 16:285–291, 2007, Multiscale Model Simul 6:913–936, 2007), we add a fidelity term to the corresponding Cahn–Hilliard system. We are interested in the study of the asymptotic behavior, in terms of finite-dimensional attractors, of the dynamical system associated with the problem. The main difficulty here is that we no longer have the conservation of mass, i.e., of the spatial average of the order parameter c, as in the Cahn–Hilliard system. Instead, we prove that the spatial average of c is dissipative. We finally give numerical simulations which confirm and extend previous ones on the efficiency of the binary model.

A Complex Version of the Cahn--Hilliard Equation for Grayscale Image Inpainting

Our aim in this article is to propose a generalization of the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation, introduced for binary image inpainting, for grayscale image inpainting. In particular, we consider the solution to the corresponding Cahn--Hilliard inpainting model as a complex valued function. We are interested in the study of the well-posedness and of the asymptotic behavior, in terms of finite-dimensional attractors, of the associated dynamical system. We have to face two major difficulties here. The first one comes from the fact that we no longer have the conservation of mass, i.e., of the spatial average of the order parameter

Higher-Order Allen–Cahn Models with Logarithmic Nonlinear Terms

u

The Cahn-Hilliard Equation with Logarithmic Potentials

, contrary to the classical Cahn--Hilliard equation. The second one is due to the estimates on the nonlinear terms, combined with the fact that the order parameter

On the Caginalp system with dynamic boundary conditions and singular potentials

u

Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials

is complex valued. We finally give numerical simulations which confirm and extend previous ones on the efficiency of the binary model.

A Variational Approach to a Cahn–Hilliard Model in a Domain with Nonpermeable Walls

Our aim in this chapter was to study higher-order (in space) Allen–Cahn models with logarithmic nonlinear terms. In particular, we obtain well-posedness results, as well as the existence of the global attractor.