Global stability of steady states in the classical Stefan problem

TL;DR

This paper proves the global stability of steady states in the classical Stefan problem for arbitrary smooth initial shapes, extending previous results limited to nearly spherical geometries.

Contribution

It establishes the global-in-time stability of steady states in the Stefan problem without convexity restrictions, broadening the understanding of shape stability.

Findings

Global stability of steady states proven for arbitrary smooth domains

No convexity restrictions required on initial shapes

Extension of previous nearly spherical shape results

Abstract

The classical one-phase Stefan problem (without surface tension) allows for a continuum of steady state solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove global-in-time stability of such steady states, assuming a sufficient degree of smoothness on the initial domain, but without any a priori restriction on the convexity properties of the initial shape. This is an extension of our previous result [28] in which we studied nearly spherical shapes.