Sharp Convergence to the Half-Space for Mullins-Sekerka in the Plane

TL;DR

This paper analyzes the Mullins-Sekerka evolution in the plane, establishing sharp convergence rates to a flat interface using an intrinsic distance measure and energy-dissipation framework.

Contribution

It introduces a natural intrinsic distance for the interface and derives the sharp leading order constant for convergence, extending previous algebraic rate results.

Findings

Established sharp convergence rate to the flat interface.

Identified an intrinsic distance measure for interfaces.

Derived the leading order constant for convergence.

Abstract

We revisit the HED Method for the Mullins-Sekerka evolution in the plane. We identify a natural notion of distance, intrinsic to the interface itself. Using this distance, the energy, and the dissipation, we develop natural assumptions on the flow and, assuming existence of a solution satisfying these conditions, establish not just the algebraic rate (previously derived by Chugreeva, Otto, and M. G. Westdickenberg) but also the sharp leading order constant for the convergence to the flat limiting interface.