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

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.

💡 Research Summary

The paper studies a stochastic version of the one‑dimensional supercooled Stefan problem on the half‑line, where a transport noise term of the form θ∂ₓv dWₜ is added to the heat equation. The deterministic counterpart (θ = 0) is a classical free‑boundary problem describing the evolution of temperature v(t,x) in a liquid that is initially frozen on