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arXiv · 2409.20421

The supercooled Stefan problem with transport noise: weak solutions and blow-up

Sean Ledger·Andreas Sojmark
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Abstract

We derive two weak formulations for the supercooled Stefan problem with transport noise on a half-line: one captures a continuously evolving system, while the other resolves blow-ups by allowing for jump discontinuities in the evolution of the temperature profile and the freezing front. For the first formulation, we establish a probabilistic representation in terms of a conditional McKean--Vlasov problem, and we then show that there is finite time blow-up with positive probability when part of the initial temperature profile is supercooled below a critical value. On the other hand, the system is shown to evolve continuously when the initial profile is everywhere above this value. In the presence of blow-ups, we show that the conditional McKean--Vlasov problem provides global solutions of the second weak formulation. Finally, we identify a solution of minimal temperature increase over time and we show that its discontinuities are characterised by a natural resolution of emerging instabilities with respect to an infinitesimal external heat transfer.

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Sean Ledger, Andreas Sojmark. 2026-03-09. The supercooled Stefan problem with transport noise: weak solutions and blow-up. https://arxiv.org/abs/2409.20421

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