Regularisation and separation for evolving surface Cahn-Hilliard equations

TL;DR

This paper investigates the regularisation and separation properties of solutions to the evolving surface Cahn-Hilliard equation with logarithmic potential, establishing higher regularity and validating the double-well approximation on two-dimensional surfaces.

Contribution

It extends well-posedness and regularity results for the Cahn-Hilliard equation to evolving surfaces, proving separation properties and higher-order regularity.

Findings

Solutions stay uniformly away from pure phases after positive time

Regularisation properties of weak solutions are established

Validation of the double-well approximation on evolving surfaces

Abstract

We consider the Cahn-Hilliard equation with constant mobility and logarithmic potential on a two-dimensional evolving closed surface embedded in $\mathbb R^3$, as well as a related weighted model. The well-posedness of weak solutions for the corresponding initial value problems on a given time interval $[0,T]$ have already been established by the first two authors. Here we first prove some regularisation properties of weak solutions in finite time. Then, we show the validity of the strict separation property for both the problems. This means that the solutions stay uniformly away from the pure phases $\pm1$ from any positive time on. This property plays an essential role to achieve higher-order regularity for the solutions. Also, it is a rigorous validation of the standard double-well approximation. The present results are a twofold extension of the well-known ones for the classical equation in planar domains.