Quantitative convergence of the nonlocal Allen--Cahn equation to volume-preserving mean curvature flow

TL;DR

This paper establishes a quantitative convergence rate of the nonlocal Allen--Cahn equation to volume-preserving mean curvature flow using novel gradient flow calibrations and the relative entropy method.

Contribution

It introduces a new notion of gradient flow calibrations with a tangential component, enabling optimal convergence rate without analyzing Lagrange-multiplier closeness.

Findings

Proves convergence of the nonlocal Allen--Cahn equation to volume-preserving mean curvature flow.

Develops a new gradient flow calibration method with a tangential velocity component.

Achieves optimal convergence rate using the relative entropy approach.

Abstract

We prove a quantitative convergence result of the nonlocal Allen--Cahn equation to volume-preserving mean curvature flow. The proof uses gradient flow calibrations and the relative entropy method, which has been used in the recent literature to prove weak-strong uniqueness results for mean curvature flow and convergence of the Allen--Cahn equation. A crucial difference in this work is a new notion of gradient flow calibrations. We add a tangential component to the velocity field in order to prove the Gronwall estimate for the relative energy. This allows us to derive the optimal convergence rate without having to show the closeness of the Lagrange-multipliers.