Weak solutions and sharp interface limit of the anisotropic Cahn-Hilliard equation with disparate mobility and inhomogeneous potential

TL;DR

This paper investigates the existence of weak solutions for an anisotropic Cahn-Hilliard equation with degenerate mobility and inhomogeneous potential, analyzing the sharp interface limit to a weighted anisotropic Hele-Shaw flow, supported by numerical simulations.

Contribution

It establishes the convergence of weak solutions to BV solutions of a weighted anisotropic Hele-Shaw flow under energy convergence assumptions, including numerical analysis of anisotropic effects.

Findings

Weak solutions exist for the anisotropic Cahn-Hilliard equation.

Solutions converge to BV solutions of a weighted anisotropic Hele-Shaw flow.

Numerical simulations illustrate the impact of anisotropy.

Abstract

We study the existence of weak solutions and the corresponding sharp interface limit of an anisotropic Cahn-Hilliard equation with disparate mobility, i.e., the mobility is degenerate in one of the two pure phases, making the diffusion in that phase vanish. The double-well potential is polynomial and is weighted by a spatially inhomogeneous coefficient. In the limit when the parameter of the interface width tends to zero, and under an energy convergence assumption, we prove that the weak solutions converge to BV solutions of a weighted anisotropic Hele-Shaw flow. We also add some numerical simulations to analyze the effects of anisotropy on the Cahn-Hilliard equation.