Well-posedness of a modified degenerate Cahn-Hilliard model for surface diffusion

TL;DR

This paper establishes the well-posedness of a modified degenerate Cahn-Hilliard model for surface diffusion, ensuring the model's mathematical soundness and correct sharp interface limit.

Contribution

It introduces a notion of weak solutions for the nonlinear model and proves their existence using approximation methods, including a related nonlocal model.

Findings

Existence of weak solutions for the modified degenerate Cahn-Hilliard model

Validation of the model's sharp interface limit

Extension to a nonlocal chemical potential model

Abstract

We study the well-posedness of a modified degenerate Cahn-Hilliard type model for surface diffusion. With degenerate phase-dependent diffusion mobility and additional stabilizing function, this model is able to give the correct sharp interface limit. We introduce a notion of weak solutions for the nonlinear model. The existence result is obtained by approximations of the proposed model with nondegenerate mobilities. We also employ this method to prove existence of weak solutions to a related model where the chemical potential contains a nonlocal term originated from self-climb of dislocations in crystalline materials.