Accepted Paper
Inserted: 9 apr 2021
Last Updated: 9 feb 2023
Journal: J. Amer. Math. Soc
Year: 2023
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 R 3 , the free boundary is smooth for almost every time t, and the set of singular times S ⊂ R has Hausdorff dimension at most 12. These results provide us with a refined understanding of the Stefan problem’s singularities and answer some long-standing open questions in the field.
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