The singular set in the Stefan problem

TL;DR

This paper investigates the structure of singularities in the Stefan problem, establishing bounds on their size and smoothness, and revealing that in three dimensions, the free boundary is mostly smooth over time.

Contribution

The paper provides new bounds on the Hausdorff dimension of the singular set and shows the smoothness of the free boundary at almost all times in three dimensions.

Findings

Singular set has parabolic Hausdorff dimension at most n-1

Solution admits a smooth expansion at singular points except on a smaller set

In R^3, the free boundary is smooth for almost every time, with singular times having Hausdorff dimension at most 1/2

Abstract

In this paper we analyze the singular set in the Stefan problem and prove the following results: - The singular set has parabolic Hausdorff dimension at most $n-1$. - The solution admits a $C^\infty$-expansion at all singular points, up to a set of parabolic Hausdorff dimension at most $n-2$. - In $\mathbb R^3$, the free boundary is smooth for almost every time $t$, and the set of singular times $\mathcal S\subset \mathbb R$ has Hausdorff dimension at most $1/2$. These results provide us with a refined understanding of the Stefan problem's singularities and answer some long-standing open questions in the field.