TY - JOUR

T1 - Stefan problem with surface tension

T2 - global existence of physical solutions under radial symmetry

AU - Nadtochiy, Sergey

AU - Shkolnikov, Mykhaylo

N1 - Funding Information:
S. Nadtochiy is partially supported by the NSF CAREER Grant DMS-1651294. M. Shkolnikov is partially supported by the NSF grant DMS-2108680.
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/10

Y1 - 2023/10

N2 - We consider the Stefan problem with surface tension, also known as the Stefan–Gibbs–Thomson problem, in an ambient space of arbitrary dimension. Assuming the radial symmetry of the initial data we introduce a novel “probabilistic” notion of solution, which can accommodate the discontinuities in time (of the radius) of the evolving aggregate. Our main result establishes the global existence of a probabilistic solution satisfying the natural upper bound on the sizes of the discontinuities. Moreover, we prove that the upper bound is sharp in dimensions d⩾ 3 , in the sense that none of the discontinuities in the solution can be decreased in magnitude. The detailed analysis of the discontinuities, via appropriate stochastic representations, differentiates this work from the previous literature on weak solutions to the Stefan problem with surface tension.

AB - We consider the Stefan problem with surface tension, also known as the Stefan–Gibbs–Thomson problem, in an ambient space of arbitrary dimension. Assuming the radial symmetry of the initial data we introduce a novel “probabilistic” notion of solution, which can accommodate the discontinuities in time (of the radius) of the evolving aggregate. Our main result establishes the global existence of a probabilistic solution satisfying the natural upper bound on the sizes of the discontinuities. Moreover, we prove that the upper bound is sharp in dimensions d⩾ 3 , in the sense that none of the discontinuities in the solution can be decreased in magnitude. The detailed analysis of the discontinuities, via appropriate stochastic representations, differentiates this work from the previous literature on weak solutions to the Stefan problem with surface tension.

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U2 - 10.1007/s00440-023-01206-8

DO - 10.1007/s00440-023-01206-8

M3 - Article

AN - SCOPUS:85158167170

SN - 0178-8051

VL - 187

SP - 385

EP - 422

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 1-2

ER -