Singular limit of an Allen-Cahn equation with nonlinear diffusion

TL;DR

This paper studies the singular limit of an Allen-Cahn equation with nonlinear diffusion, revealing how interfaces form and evolve according to a modified mean curvature flow influenced by nonlinearity.

Contribution

It introduces a new analysis of the Allen-Cahn equation with nonlinear diffusion, deriving a homogenized speed for interface evolution from the nonlinearity.

Findings

Generation and propagation of interfaces in the singular limit

Interface evolution governed by a modified mean curvature flow

Emergence of a homogenized speed related to surface tension and mobility

Abstract

We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scaling limit of certain interacting particle systems. We investigate its singular limit and show the generation and propagation of an interface in the limit. The evolution of this limit interface is governed by mean curvature flow with a novel, homogenized speed in terms of a surface tension-mobility parameter emerging from the nonlinearity in our equation.