An asymptotic limit of a class of Cahn–Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic type, nonlinear diffusion of Penrose–Fife type, fast diffusion equation and so on. Namely...

The Cahn–Hilliard and viscous Cahn–Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved.

In this paper, we study the relation between the second inner variations of the Allen-Cahn functional and its Gamma-limit, the area functional. Our result implies that the Allen-Cahn functional only approximates well the area functional up to the first order. However, as an application of our result, we prove, assuming the singlemultiplicity property of the limiting energy, that the Morse indic...

We propose a novel second order in time, decoupled and unconditionally stable numerical scheme for solving the Cahn-Hilliard-Darcy (CHD) system which models two-phase flow in porous medium or in a Hele-Shaw cell. The scheme is based on the ideas of second order convex-splitting for the Cahn-Hilliard equation and pressure-correction for the Darcy equation. We show that the scheme is uniquely sol...

Numerical method for Cahn-Hilliard equation has been well-studied, but few can be generalized to fractional Cahn-Hilliard equation. In this project to modified the numerical method proposed by Brian Wetton et al in the paper High accuracy solutions to energy gradient flows from material science models[1]. The method they described in the paper is a pseudo-spectral method suitable for considerin...

A formal asymptotic method is used to derive a differential-algebraic system of equations characterizing the metastable motion of a pattern of n (n ≥ 2) internal layers for the one-dimensional viscous Cahn-Hilliard modeling slow phase separation. Similar slow motion results are obtained for the Cahn-Hilliard equation and the constrained Allen-Cahn equation by introducing a homotopy parameter in...

We study perturbations of the Allen–Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and apply previous results for the Allen–Cahn action functional. Finally we discuss...

Abstract In this article, we present a second-order in time implicit-explicit (IMEX) local discontinuous Galerkin (LDG) method for computing the Cahn-Hilliard equation, which describes the phase separation phenomenon. It is well-known that the Cahn-Hilliard equation has a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. The discretized Cahn-Hilliard...

This thesis analyzes two types of phase transition models, namely the Cahn–Hilliard model and the Becker–Döring model. In the Cahn–Hilliard setting, this thesis establishes a second-order Γ-convergence result for the mass-constrained Cahn–Hilliard energy. This is obtained using a new variant of the Pòlya–Szegő inequality, along with some new regularity results for the isoperimetric function. Fo...

We present an existence result for the Cahn{Hilliard equation with a concentration dependent mobility which allows the mobility to degenerate. Formal asymptotic results relate the Cahn{Hilliard equation with a degenerate mobility to motion by surface diiusion V = ? S. We state a local existence result for this geometric motion and show that circles are asymptotically stable.