The supercooled Stefan problem with transport noise: weak solutions and blow-up
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| Format: | Preprint |
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2024
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| _version_ | 1866910045414883328 |
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| author | Ledger, Sean Sojmark, Andreas |
| author_facet | Ledger, Sean Sojmark, Andreas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20421 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The supercooled Stefan problem with transport noise: weak solutions and blow-up Ledger, Sean Sojmark, Andreas Probability Analysis of PDEs 60H15, 60H30, 80A22, 35B44 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. |
| title | The supercooled Stefan problem with transport noise: weak solutions and blow-up |
| topic | Probability Analysis of PDEs 60H15, 60H30, 80A22, 35B44 |
| url | https://arxiv.org/abs/2409.20421 |