The anisotropic Cahn--Hilliard equation with degenerate mobility: Existence of weak solutions

TL;DR

This paper proves the existence of weak solutions for an anisotropic, degenerate mobility Cahn--Hilliard equation, modeling surface diffusion and solid-state dewetting, with complex nonlinear energy estimates.

Contribution

It establishes the existence of weak solutions for a highly nonlinear, anisotropic, degenerate Cahn--Hilliard model using energy and entropy estimates.

Findings

Existence of weak solutions under degenerate mobility conditions

Handling of anisotropic and nonlinear energy terms

Relevance to modeling surface diffusion and dewetting phenomena

Abstract

This paper presents an existence result for the anisotropic Cahn--Hilliard equation characterized by a potentially concentration-dependent degenerate mobility taking into account an anisotropic energy. The model allows for the degeneracy of the mobility at specific concentration values, demonstrating that the solution remains within physically relevant bounds. The introduction of anisotropy leads to highly nonlinear terms making energy and entropy estimates rather involved. As the mobility degenerates in the pure phases, the degenerate Cahn--Hilliard equation describes surface diffusion and is an important model to model solid-state dewetting (SSD) of thin films. We show existence of weak solutions for the anisotropic degenerate Cahn--Hilliard equation by using suitable energy and entropy type estimates.