Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow

TL;DR

This paper proves that the nonlocal Allen-Cahn equation converges to mean curvature flow as the kernel parameter diminishes, extending the analysis to $W^{1,1}$-kernels in periodic settings for 2D and 3D.

Contribution

It establishes convergence of the nonlocal Allen-Cahn equation to mean curvature flow for $W^{1,1}$-kernels, using spectral estimates and a nonlocal Ehrling-type inequality.

Findings

Convergence proven in the sharp interface limit.

Extension to $W^{1,1}$-kernels in periodic domains.

Uniform $H^3$-estimates for nonlocal solutions.

Abstract

We prove convergence of the nonlocal Allen-Cahn equation to mean curvature flow in the sharp interface limit, in the situation when the parameter corresponding to the kernel goes to zero fast enough with respect to the diffuse interface thickness. The analysis is done in the case of a $W^{1,1}$-kernel, under periodic boundary conditions and in both two and three space dimensions. We use the approximate solution and spectral estimate from the local case, and combine the latter with an $L^2$-estimate for the difference of the nonlocal operator and the negative Laplacian from Abels, Hurm arXiv:2307.02264. To this end, we prove a nonlocal Ehrling-type inequality to show uniform $H^3$-estimates for the nonlocal solutions.