The Cahn-Hilliard equation on an evolving surface

TL;DR

This paper investigates the asymptotic behavior of the Cahn-Hilliard equation on moving surfaces, extending asymptotic methods to evolving domains and deriving related interface problems with numerical validation.

Contribution

It extends matched asymptotic expansions to evolving surfaces and explores various mobility and potential scalings, deriving new interface problems influenced by domain evolution.

Findings

Derivation of Mullins-Sekerka and surface diffusion problems on evolving surfaces.

Extension of asymptotic methods to account for domain movement.

Numerical simulations supporting the asymptotic analysis.

Abstract

We study the asymptotic limit of the Cahn-Hilliard equation on an evolving surface with prescribed velocity. The method of formally matched asymptotic expansions is extended to account for the movement of the domain. We consider various forms for the mobility and potential functions, in particular, with regards to the scaling of the mobility with the interface thickness parameter. Mullins-Sekerka, but also surface diffusion type problems, can be derived featuring additional terms which are due to the domain evolution. The asymptotic behaviour is supported and further explored with some numerical simulations.