{
  "id": "2409.20421",
  "version": "v2",
  "published": "2024-09-30T15:44:05.000Z",
  "updated": "2024-11-07T13:27:49.000Z",
  "title": "The supercooled Stefan problem with transport noise: weak solutions and blow-up",
  "authors": [
    "Sean Ledger",
    "Andreas Sojmark"
  ],
  "comment": "37 pages, 3 figures",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "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 exceeds a critical value. On the other hand, the system is shown to evolve continuously when the initial profile is everywhere below 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 characterized by a natural resolution of emerging instabilities with respect to an infinitesimal external heat transfer.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-11-07T13:27:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H30",
      "80A22",
      "35B44"
    ],
    "keywords": [
      "supercooled stefan problem",
      "transport noise",
      "weak solutions",
      "conditional mckean-vlasov problem",
      "infinitesimal external heat transfer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}