Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness

Abstract

We consider the supercooled Stefan problem, which captures the freezing of a\nsupercooled liquid, in one space dimension. A probabilistic reformulation of\nthe problem allows to define global solutions, even in the presence of blow-ups\nof the freezing rate. We provide a complete description of such solutions, by\nrelating the temperature distribution in the liquid to the regularity of the\nice growth process. The latter is shown to transition between (i) continuous\ndifferentiability, (ii) Holder continuity, and (iii) discontinuity. In\nparticular, in the second regime we rediscover the square root behavior of the\ngrowth process pointed out by Stefan in his seminal paper [Ste89] from 1889 for\nthe ordinary Stefan problem. In our second main theorem, we establish the\nuniqueness of the global solutions, a first result of this kind in the context\nof growth processes with singular self-excitation when blow-ups are present.\n