Convergence of a mass conserving Allen-Cahn equation whose Lagrange multiplier is nonlocal and local

TL;DR

This paper analyzes a mass conserving Allen-Cahn equation with a mixed nonlocal and local Lagrange multiplier, proving its solutions converge to volume-preserving mean curvature flow as the diffuse layer thickness approaches zero.

Contribution

It provides a rigorous convergence proof for the Allen-Cahn equation with a novel mixed Lagrange multiplier to volume-preserving mean curvature flow.

Findings

Formal asymptotic expansions as the diffuse layer thickness tends to zero.

Rigorous proof of convergence to volume-preserving mean curvature flow.

Analysis of the error between actual and approximate Lagrange multipliers.

Abstract

We consider the mass conserving Allen-Cahn equation proposed in \cite{Bra-Bre}: the Lagrange multiplier which ensures the conservation of the mass contains not only nonlocal but also local effects (in contrast with \cite{Che-Hil-Log}). As a parameter related to the thickness of a diffuse internal layer tends to zero, we perform formal asymptotic expansions of the solutions. Then, equipped with these approximate solutions, we rigorously prove the convergence to the volume preserving mean curvature flow, under the assumption that classical solutions of the latter exist. This requires a precise analysis of the error between the actual and the approximate Lagrange multipliers.