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

Helmut Abels, Christoph Hurm, Maximilian Moser

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.

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

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 -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 -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 -estimates for the nonlocal solutions.

Paper Structure

This paper contains 10 sections, 8 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Assumptions
  5. Inequalities
  6. Results for the Local Allen-Cahn Equation
  7. Properties of Solutions to the Nonlocal Allen-Cahn Equation
  8. Existence, Uniqueness and Maximum Principle
  9. Uniform $H^3$-Bounds
  10. Proof of Theorem \ref{['th_conv']} (Convergence)

Key Result

Theorem 1.1

Let $n\in\{2,3\}$, $T_0>0$ and $(\Gamma_t)_{t\in[0,T_0]}$ with $\Gamma_t\subset(-\pi,\pi)^n$ for all $t\in[0,T_0]$ be a smoothly evolving compact closed hypersurface in $\mathbb{T}^n$ satisfying mean curvature flow, i.e. $V_{\Gamma_t}=H_{\Gamma_t}$ for all $t\in[0,T_0]$, where $V_{\Gamma_t}$ is the

Theorems & Definitions (18)

  • Theorem 1.1: Convergence
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4: Nonlocal Ehrling Inequality
  • proof
  • Theorem 2.5: Approximate Solution for Local Allen-Cahn Equation
  • proof
  • ...and 8 more