Global Stability and Decay for the Classical Stefan Problem

Abstract

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.

Global stability of steady states in the classical Stefan problem for general boundary shapes

“…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 (Hadžić & Shkoller 2014 Commun. Pure Appl.…”

“…Remark 1.7. An analogous theorem was stated in [11], for perturbations of steady surfaces initially close to a sphere. Therefore, this work generalizes that result.…”

“…We introduce the following new variables set on the fixed domain Ω: The classical Stefan problem on the fixed domain Ω is written as [7,11] …”

“…For an overview, we refer the reader to [13,22,23] as well as the introduction to [11]. First, weak solutions were defined in [18,24,25].…”

“…[8][9][10]). Our hybrid approach, employing nonlinear Pucci operators, appears to be new and is a natural extension of our previous work [11], which necessitated perturbations of spherical initial domains.…”