Relaxation of the Cahn-Hilliard equation with singular single-well potential and degenerate mobility

TL;DR

This paper introduces a relaxation system for the degenerate Cahn-Hilliard equation with singular potential and degenerate mobility, enabling easier numerical simulation while preserving key energy and entropy properties.

Contribution

It proposes a new relaxation approach with two second order equations that are compatible with the original equation's energy and entropy structures.

Findings

Proves global existence of the relaxation system

Shows convergence to the original degenerate Cahn-Hilliard equation

Analyzes long-time asymptotic behavior and steady states

Abstract

The degenerate Cahn-Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn-Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system; global existence and convergence of the relaxed system to the degenerate Cahn-Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.