Nonlocal-to-local convergence of the Cahn-Hilliard equation with degenerate mobility and the Flory-Huggins potential

TL;DR

This paper investigates the convergence of the nonlocal to local Cahn-Hilliard equation with degenerate mobility and Flory-Huggins potential, extending previous results to more complex mobility functions and singular kernels.

Contribution

It adapts existing analytical tools to handle the specific mobility m(u)=u(1-u), including boundedness and nonlinear term management, for the first time in this context.

Findings

Established convergence results for the degenerate mobility case.

Extended analysis to singular kernels and cell adhesion models.

Provided new mathematical techniques for handling nonlinearities.

Abstract

The Cahn-Hilliard equation is a fundamental model for phase separation phenomena. Its rigorous derivation from the nonlocal aggregation equation, motivated by the desire to link interacting particle systems and continuous descriptions, has received much attention in recent years. In the recent article, we showed how to treat the case of degenerate mobility for the first time. Here, we discuss how to adapt the exploited tools to the case of the mobility $m(u)=u\,(1-u)$ as in the original works of Giacomin-Lebowitz and Elliott-Garcke. The main additional information is the boundedness of $u$, implied by the form of mobility, which allows handling the nonlinear terms. We also discuss the case of (mildly) singular kernels and a model of cell-cell adhesion with the same mobility.