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Singular limit of an Allen–Cahn equation with nonlinear diffusion

Perla El Kettani, Tadahisa Funaki, Danielle Hilhorst, Hyunjoon Park and Sunder Sethuraman

Vol. 4 (2022), No. 4, 719–754
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.

Keywords
Allen–Cahn equation, mean curvature flow, singular limit, nonlinear diffusion, interface, surface tension
Mathematical Subject Classification
Primary: 35B25, 35B40, 35B51, 35K57, 35K59
Milestones
Received: 26 December 2021
Revised: 22 June 2022
Accepted: 14 July 2022
Published: 15 January 2023
Authors
Perla El Kettani
University of Toulon
Toulon
France
Tadahisa Funaki
Graduate School of Mathematical Sciences
Department of Mathematics
University of Tokyo
Tokyo
Japan
Department of Mathematics
Waseda University
Tokyo
Japan
Danielle Hilhorst
CNRS and Laboratoire de Mathématiques
Paris-Saclay University
Orsay
France
Hyunjoon Park
Department of Mathematical Sciences
Korean Advanced Institute of Science and Technology
Daejeon
South Korea
Sunder Sethuraman
Department of Mathematics
University of Arizona
Tucson, AZ
United States