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

## TL;DR

This paper studies the supercooled Stefan problem in one dimension, providing a probabilistic framework for global solutions that handle blow-ups, describing regularity transitions, and proving uniqueness of solutions with singular growth behavior.

## Contribution

It introduces a probabilistic reformulation for global solutions to the supercooled Stefan problem, including cases with blow-ups, and establishes the first uniqueness result for such growth processes.

## Key findings

- Complete description of solution regularity transitions
- Rediscovery of square root growth behavior
- Proof of uniqueness for solutions with blow-ups

## 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 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) Holder 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 paper [Ste89] from 1889 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.