The anisotropic Cahn--Hilliard equation with degenerate mobility: Existence of weak solutions
Harald Garcke, Patrik Knopf, Andrea Signori
TL;DR
This work proves the existence of weak solutions for the anisotropic Cahn--Hilliard equation with concentration-dependent degenerate mobility under periodic boundaries. The authors regularize the degenerate mobility, potential, and initial data to obtain well-posed approximate problems, derive uniform energy-entropy bounds, and then pass to the limit using compactness and the strong monotonicity of the anisotropy operator. The analysis yields a solution that stays in the physically admissible range $|\varphi|\le 1$ and satisfies a weak energy dissipation law, extending prior isotropic results to general anisotropy with degeneracy. The results contribute to rigorous foundations for anisotropic surface diffusion and solid-state dewetting models in thin films.
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.