Vol. 3, No. 1, 2022

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

François Delarue, Sergey Nadtochiy and Mykhaylo Shkolnikov

Vol. 3 (2022), No. 1, 171–213

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.

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blow-ups, free boundary problem, heat equation, interacting particle systems, mean-field interaction, physical solutions, probabilistic reformulation, self-excitation, supercooled Stefan problem, zero set
Mathematical Subject Classification
Primary: 35B05, 35B44, 60H30, 80A22
Received: 1 February 2021
Revised: 1 July 2021
Accepted: 16 July 2021
Published: 11 May 2022
François Delarue
Laboratoire J.A.Dieudonné
Université de Nice Sophia-Antipolis
Sergey Nadtochiy
Department of Applied Mathematics
Illinois Institute of Technology
Rettaliata Engineering Center
Chicago, IL
United States
Mykhaylo Shkolnikov
ORFE Department
Bendheim Center for Finance
Program in Applied and Computational Mathematics
Princeton University
Princeton, NJ
United States