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

Sean Ledger, Andreas Sojmark

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.

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

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.
Paper Structure (14 sections, 18 theorems, 49 equations, 3 figures)

This paper contains 14 sections, 18 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Weak formulations
  3. Allowing for temperature discontinuities
  4. Càdlàg weak formulation
  5. Instability under an external heat transfer
  6. Main results
  7. The associated McKean--Vlasov problem
  8. Proofs of Theorem \ref{['thm:prob_rep_cont']} and Theorem \ref{['thm:existence_jump']}
  9. Proof of Theorem \ref{['thm:prob_rep_cont']}
  10. Proof of Theorem \ref{['thm:existence_jump']}
  11. Proofs of Theorem \ref{['thm:jump_nonzero_prob']} and Theorem \ref{['prop:initial_global_cont']}
  12. Initial considerations
  13. Proof of Theorem \ref{['thm:jump_nonzero_prob']}
  14. Proof of Theorem \ref{['prop:initial_global_cont']}

Key Result

Theorem 2.2

If $(v, s)$ is a (global) continuous weak solution in the sense of Definition def:weak_cont, then $(v,s)$ is characterised by for all $t\geq0$, almost surely, where for a Brownian motion $B$ independent of $W$. If $(v,s)$ is a local continuous weak solution on $[0,\tau)$, for some $\mathcal{F}_t$-stopping time $\tau$, then the above holds on $[0,\tau)$.

Figures (3)

  • Figure 1.1: Domain of the free boundary problems \ref{['eq:classical_soln']} and \ref{['eq:classical_noise']}. One should have in mind a continuously differentiable boundary for \ref{['eq:classical_soln']} versus something akin to Minkowski's question-mark function for \ref{['eq:classical_noise']}.
  • Figure 2.1: Illustration of a jump discontinuity in the temperature profile at time $t$, along with the corresponding instantaneous advance of the freezing front according to \ref{['eq:simplified_energy_balance']}. In the upper picture, $s(r)\uparrow s(t-)$ as $r\uparrow t$.
  • Figure 2.2: Domain of the supercooled Stefan problem for a single jump discontinuity at time $t^\prime$ with $s(t^\prime)=x^\prime$. In fact, $s(t)$ is only differentiable a.e., but the smooth picture is meant to illustrate the divergence in the rate of increase towards $t^\prime$. We also note that $t^\prime$ could instead be an accumulation point of small subsequent jumps.

Theorems & Definitions (25)

  • Definition 2.1: Continuous weak solutions
  • Theorem 2.2: Probabilistic McKean--Vlasov representation
  • Definition 2.3: Càdlàg weak solutions
  • Remark 2.4: Physical jumps
  • Theorem 3.1: Probabilistic càdlàg solutions
  • Theorem 3.2: Temperature discontinuities
  • Theorem 3.3: Minimal temperature increase
  • Theorem 3.4: Pathwise characterisation of \ref{['eq:equality_jumps']}
  • Theorem 3.5: Unique continuous solution
  • Theorem 3.6: Initial and global continuity
  • ...and 15 more