## Abstract

We consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows us to define global solutions, even in the presence of blow-ups of the freezing rate. We provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Hölder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root behavior of the growth process pointed out by Stefan in his seminal 1889 paper for the ordinary Stefan problem. In our second main theorem, we establish the uniqueness of the global solutions, a first result of this kind in the context of growth processes with singular self-excitation when blow-ups are present.

Original language | English (US) |
---|---|

Pages (from-to) | 171-213 |

Number of pages | 43 |

Journal | Probability and Mathematical Physics |

Volume | 3 |

Issue number | 1 |

DOIs | |

State | Published - 2022 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Atomic and Molecular Physics, and Optics
- Statistical and Nonlinear Physics

## Keywords

- blow-ups
- free boundary problem
- heat equation
- interacting particle systems
- mean-field interaction
- physical solutions
- probabilistic reformulation
- self-excitation
- supercooled Stefan problem
- zero set