Weak solutions of the Cahn-Hilliard system with dynamic boundary conditions: A gradient flow approach

TL;DR

This paper proves the existence and uniqueness of weak solutions for a Cahn-Hilliard system with dynamic boundary conditions by employing a gradient flow framework, improving previous geometric assumptions.

Contribution

It introduces a gradient flow approach to establish weak solutions for new models with dynamic boundary conditions, relaxing earlier geometric constraints.

Findings

Existence of weak solutions is proven.

Weak solutions are shown to be unique.

The approach improves upon previous geometric assumptions.

Abstract

The Cahn-Hilliard equation is the most common model to describe phase separation processes of a mixture of two components. For a better description of short-range interactions of the material with the solid wall, various dynamic boundary conditions have been considered in recent times. New models with dynamic boundary conditions have been proposed recently by C. Liu and H. Wu [arXiv:1710.08318]. We prove the existence of weak solutions to these new models by interpreting the problem as a suitable gradient flow of a total free energy which contains volume as well as surface contributions. The formulation involves an inner product which couples bulk and surface quantities in an appropriate way. We use an implicit time discretization and show that the obtained approximate solutions converge to a weak solution of the Cahn-Hilliard system. This allows us to substantially improve earlier results which needed strong assumptions on the geometry of the domain. Furthermore, we prove that this weak solution is unique.