The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

TL;DR

This paper proves that in two dimensions, area-preserving mean curvature and Mullins-Sekerka flows starting from any bounded set of finite perimeter asymptotically converge exponentially to a union of equally sized disks.

Contribution

It establishes the first general asymptotic convergence results for area-preserving mean curvature and Mullins-Sekerka flows in two dimensions.

Findings

Solutions converge exponentially to unions of disks

Results apply to both mean curvature and Mullins-Sekerka flows

First such general asymptotic convergence proof in 2D

Abstract

We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with exponential rate to a finite union of equally sized disjoint disks. A similar result is established also for the periodic two-phase Mullins-Sekerka flow.