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

Raymond Chu, Inwon Kim, Sebastian Munoz

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.