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arXiv · 2512.10138

The supercooled Stefan problem: fractal freezing and the fine structure of maximal solutions

Raymond Chu·Inwon Kim·Sebastian Munoz
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Abstract

We study the supercooled Stefan problem in arbitrary dimensions. First, we study general solutions and their irregularities, showing generic fractal freezing and nucleation, based on a novel Markovian gluing principle. In contrast, we then establish regularity properties of maximal solutions, which are obtained by maximizing a suitable notion of "average" freezing time. Unexpectedly, we show that maximal solutions have a transition zone that is open modulo a low-dimensional set: this allows us to apply obstacle problem theory for a finer regularity analysis. We further show that maximal solutions are in general non-universal, and we obtain sharp stability results under perturbation of each maximal solution. Lastly, we study maximal solutions in both the radial and the one-dimensional setting. We show that in these cases the maximal solution is universal and minimizes nucleation, in agreement with phenomena observed in the physics literature.

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Raymond Chu, Inwon Kim, Sebastian Munoz. 2025-12-10. The supercooled Stefan problem: fractal freezing and the fine structure of maximal solutions. https://arxiv.org/abs/2512.10138

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