Cahn-Hilliard equations on an evolving surface

TL;DR

This paper develops a mathematical framework for analyzing the Cahn-Hilliard equation on surfaces that evolve over time, establishing well-posedness and conditions for global solutions with various potentials.

Contribution

It introduces a functional analysis framework for the Cahn-Hilliard equation on evolving surfaces, including results for regular and singular potentials, and explores conditions for global existence.

Findings

Established well-posedness for regular potentials.

Proved global existence conditions for singular potentials.

Described alternative models preserving weighted integrals.

Abstract

We describe a functional framework suitable to the analysis of the Cahn-Hilliard equation on an evolving surface whose evolution is assumed to be given \textit{a priori}. The model is derived from balance laws for an order parameter with an associated Cahn-Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular nonlinearities -- the thermodynamically relevant logarithmic potential and a double obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution, and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.