Regularity of the free boundary for the supercooled Stefan problem in arbitrary dimensions

TL;DR

This paper establishes a comprehensive regularity theory for the free boundary in the supercooled Stefan problem across arbitrary dimensions, revealing its decomposition into regular, singular, and jump parts, and characterizing the nature of singularities.

Contribution

It provides the first detailed regularity analysis of the free boundary in the supercooled Stefan problem in any dimension, including decomposition and characterization of singularities.

Findings

Free boundary decomposes into regular, singular, and jump parts.

The free boundary is a continuously differentiable graph in time.

Singularities occur with infinite speed and are characterized by the critical set of the freezing time.

Abstract

We study the free boundary in the supercooled Stefan problem, a classical model for the solidification of water below its freezing temperature. In contrast with the melting problem, physical experiments and heuristics indicate that the water--ice interface in the supercooled problem may exhibit fractal freezing sets, infinite-speed propagation of the frozen front, and nucleation (the spontaneous appearance of ice). Despite this, we show that the free boundary has a robust structure. We decompose the free boundary into three parts: (1) a regular part that advances with finite speed in time; (2) a singular part consisting of points where the front attains infinite speed or nucleates, but with controlled space-time (i.e., $\leq d-1$ parabolic) dimension; and (3) a jump component, which can have large dimension in a time slice, but which is contained in a space-time smooth graph and occurs only at a zero-dimensional set of times. Examples show that each of these parts can be nonempty. Furthermore, we prove that the free boundary is the graph $t=s(x)$ of a continuously differentiable freezing time $s$, and the singular set coincides with the critical set of $s$, proving that singularities in supercooled freezing always occur with infinite speed. These results provide the first free boundary regularity theory for the supercooled Stefan problem in arbitrary dimensions.