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

TL;DR

This paper develops weak formulations for the supercooled Stefan problem with transport noise, analyzing conditions for blow-up and continuous evolution, and providing probabilistic representations and solutions with jump discontinuities.

Contribution

It introduces two weak formulations for the stochastic supercooled Stefan problem, linking probabilistic McKean--Vlasov representations to blow-up phenomena and discontinuous solutions.

Findings

Finite time blow-up occurs with positive probability for supercooled initial profiles.

The system evolves continuously if the initial temperature is above the critical value.

A minimal temperature solution with jump discontinuities is characterized and resolved.

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.