Global stability and decay for the classical Stefan problem

TL;DR

This paper proves the global stability and decay of solutions for the classical Stefan problem in nearly spherical geometries with small initial temperatures, using a new hybrid analytical approach.

Contribution

It introduces a novel hybrid methodology combining energy, decay, and Hopf inequalities to analyze the Stefan problem's stability.

Findings

Global-in-time stability for nearly spherical geometries

Decay estimates for temperature distribution

Effective control of free boundary evolution

Abstract

The classical one-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition, such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free-boundary. We establish a global-in-time stability result for nearly spherical geometries and small temperatures, using a novel hybrid methodology, which combines energy estimates, decay estimates, and Hopf-type inequalities.