Degenerate Cahn-Hilliard equation: From nonlocal to local

TL;DR

This paper introduces a new method to rigorously derive the local degenerate Cahn-Hilliard equation from nonlocal models, extending previous work to include degenerate mobilities relevant in biological tissue modeling.

Contribution

It presents a novel approach using nonlocal inequalities to prove convergence from nonlocal to local degenerate Cahn-Hilliard equations, addressing a gap in existing literature.

Findings

Established convergence of nonlocal to local degenerate Cahn-Hilliard equations

Developed new nonlocal Poincaré and compactness inequalities

Extended the theoretical framework to models with degenerate mobilities

Abstract

There has been recently an important interest in deriving rigorously the Cahn-Hilliard equation from the nonlocal equation, also called aggregation equation. So far, only non-degenerate mobilities were treated. Since we are motivated by models for the biomechanics of living tissues, it is useful to include degenerate motilities. In this framework, we present a new method to show the convergence of the nonlocal to the local degenerate Cahn-Hilliard equation. The method includes the use of nonlocal Poincar\'e and compactness inequalities.