The singular set in the Stefan problem

2021/03/24 by Figalli, Alessio, Ros-Oton, Xavier, Serra, Joaquim · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2103.13379

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^∞-expansion at all singular points, up to a set of parabolic Hausdorff dimension at most n-2. - In \mathbb R³, the free boundary is smooth for almost every time t, and the set of singular times \mathcal S⊂ \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.

Cited by

• On the existence of solutions to some singular parabolic free boundary problems
• Regularity of the free boundary for the supercooled Stefan problem in arbitrary dimensions
• Free boundary regularity and well-posedness of physical solutions to the supercooled Stefan problem

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