Zero kinetic undercooling limit in the supercooled Stefan problem

Graeme Baker, Mykhaylo Shkolnikov

Research output: Contribution to journalArticlepeer-review

Abstract

We study the solutions of the one-phase supercooled Stefan problem with kinetic undercooling, which describes the freezing of a supercooled liquid, in one spatial dimension. Assuming that the initial temperature lies between the equilibrium freezing point and the characteristic invariant temperature throughout the liquid our main theorem shows that, as the kinetic undercooling parameter tends to zero, the free boundary converges to the (possibly irregular) free boundary in the supercooled Stefan problem without kinetic undercooling, whose uniqueness has been recently established in (Delarue, Nadtochiy and Shkolnikov (2019), Ledger and Søjmark (2018)). The key tools in the proof are a Feynman-Kac formula, which expresses the free boundary in the problem with kinetic undercooling through a local time of a reflected process, and a resulting comparison principle for the free boundaries with different kinetic undercooling parameters.

Original languageEnglish (US)
Pages (from-to)861-871
Number of pages11
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume58
Issue number2
DOIs
StatePublished - May 2022

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Feynman-Kac formula
  • Free boundary problems
  • Kinetic undercooling
  • Local time
  • Reflected processes
  • Supercooled Stefan problem

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