Regularity of the free boundary for the supercooled Stefan problem in arbitrary dimensions

Abstract: We study the free boundary in the supercooled Stefan problem, a classical model for the solidification of water below its freezing temperature. In contrast with the melting problem, physical experiments and heuristics indicate that the water--ice interface in the supercooled problem may exhibit fractal freezing sets, infinite-speed propagation of the frozen front, and nucleation (the spontaneous appearance of ice). Despite this, we show that the free boundary has a robust structure. We decompose the free boundary into three parts: (1) a regular part that advances with finite speed in time; (2) a singular part consisting of points where the front attains infinite speed or nucleates, but with controlled space-time (i.e., $\leq d-1$ parabolic) dimension; and (3) a jump component, which can have large dimension in a time slice, but which is contained in a space-time smooth graph and occurs only at a zero-dimensional set of times. Examples show that each of these parts can be nonempty. Furthermore, we prove that the free boundary is the graph $t=s(x)$ of a continuously differentiable freezing time $s$, and the singular set coincides with the critical set of $s$, proving that singularities in supercooled freezing always occur with infinite speed. These results provide the first free boundary regularity theory for the supercooled Stefan problem in arbitrary dimensions.