Free boundary regularity and well-posedness of physical solutions to the supercooled Stefan problem

TL;DR

This paper investigates the regularity and uniqueness of solutions to the supercooled Stefan problem, establishing new smoothness results for the free boundary and resolving open questions on solution uniqueness.

Contribution

It proves the free boundary is smooth in space, describes jump discontinuities explicitly, and establishes global uniqueness of solutions under broad initial conditions.

Findings

Free boundary is $C^1$ in space and $C^{}$ outside a countable set.

Positive jump times cannot accumulate, confirming a conjecture.

Short-time uniqueness implies global uniqueness for general initial data.

Abstract

We study the regularity and well-posedness of physical solutions to the supercooled Stefan problem. Assuming only that the initial temperature is integrable, we prove that the free boundary, known to have jump discontinuities as a function of the time variable, is $C^1$ as a function of the space variable, and is $C^{\infty}$ outside of a closed, countable set, which we describe explicitly. We also prove that, as conjectured in arXiv:1902.05174, the set of positive times when a jump occurs cannot have accumulation points. In addition, we prove that short-time uniqueness of physical solutions implies global uniqueness, which allows us to obtain uniqueness for very general initial data that fall outside the scope of the current well-posedness regime. In particular, we answer two questions left open in arXiv:1811.12356, arXiv:2302.13097, regarding the global uniqueness of solutions. We proceed by deriving a weighted obstacle problem satisfied by the solutions, which we exploit to establish regularity and non-degeneracy estimates and to classify the free boundary points. We also establish a backward propagation of oscillation property, which allows us to control the occurrence of future jumps in terms of the past oscillation of the solution.