Local well-posedness and Global stability of the Two-Phase Stefan problem

TL;DR

This paper proves local well-posedness and global stability for the two-phase Stefan problem, extending previous work on the one-phase case and introducing new energy methods for smooth domains and small initial temperatures.

Contribution

It extends the analysis of Stefan problems from one-phase to two-phase, providing a simplified proof and establishing stability for smooth domains and small initial data.

Findings

Established local-in-time well-posedness.

Proved global-in-time stability.

Developed a higher-order energy method with natural weights.

Abstract

The two-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, composed of two regions separated by an a priori unknown moving boundary which is transported by the difference (or jump) of the normal derivatives of the temperature in each phase. We establish local-in-time well-posedness and a global-in-time stability result for arbitrary sufficiently smooth domains and small initial temperatures. To this end, we develop a higher-order energy with natural weights adapted to the problem and combine it with Hopf-type inequalities. This extends the previous work by Hadzic and Shkoller [31,32] on the one-phase Stefan problem to the setting of two-phase problems, and simplifies the proof significantly.