TY - JOUR

T1 - Zero kinetic undercooling limit in the supercooled Stefan problem

AU - Baker, Graeme

AU - Shkolnikov, Mykhaylo

N1 - Funding Information:
The first author was partially supported by an NSERC PGS-D scholarship and a Princeton SEAS innovation research grant. The second author was partially supported by the NSF grant DMS-1811723 and a Princeton SEAS innovation research grant.
Publisher Copyright:
© 2022 Institute of Mathematical Statistics. All rights reserved.

PY - 2022/5

Y1 - 2022/5

N2 - 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.

AB - 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.

KW - Feynman-Kac formula

KW - Free boundary problems

KW - Kinetic undercooling

KW - Local time

KW - Reflected processes

KW - Supercooled Stefan problem

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U2 - 10.1214/21-AIHP1194

DO - 10.1214/21-AIHP1194

M3 - Article

AN - SCOPUS:85131353026

SN - 0246-0203

VL - 58

SP - 861

EP - 871

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 2

ER -