The singular set in the Stefan problem

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.