GLOBAL SOLUTIONS TO THE SUPERCOOLED STEFAN PROBLEM WITH BLOW-UPS: REGULARITY AND UNIQUENESS

François Delarue, Sergey Nadtochiy, Mykhaylo Shkolnikov

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

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 languageEnglish (US)
Pages (from-to)171-213
Number of pages43
JournalProbability and Mathematical Physics
Volume3
Issue number1
DOIs
StatePublished - 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

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