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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.18741 |
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| _version_ | 1866914449655332864 |
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| author | Munoz, Sebastian |
| author_facet | Munoz, Sebastian |
| contents | We study the regularity and well-posedness of physical solutions to the supercooled Stefan problem. Assuming only that the initial temperature is integrable, we prove that the free boundary, known to have jump discontinuities as a function of the time variable, is $C^1$ as a function of the space variable, and is $C^{\infty}$ outside of a closed, countable set, which we describe explicitly. We also prove that, as conjectured in arXiv:1902.05174, the set of positive times when a jump occurs cannot have accumulation points. In addition, we prove that short-time uniqueness of physical solutions implies global uniqueness, which allows us to obtain uniqueness for very general initial data that fall outside the scope of the current well-posedness regime. In particular, we answer two questions left open in arXiv:1811.12356, arXiv:2302.13097, regarding the global uniqueness of solutions. We proceed by deriving a weighted obstacle problem satisfied by the solutions, which we exploit to establish regularity and non-degeneracy estimates and to classify the free boundary points. We also establish a backward propagation of oscillation property, which allows us to control the occurrence of future jumps in terms of the past oscillation of the solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18741 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free boundary regularity and well-posedness of physical solutions to the supercooled Stefan problem Munoz, Sebastian Analysis of PDEs Probability 80A22, 35B44, 35R35, 60H30, 35B05 We study the regularity and well-posedness of physical solutions to the supercooled Stefan problem. Assuming only that the initial temperature is integrable, we prove that the free boundary, known to have jump discontinuities as a function of the time variable, is $C^1$ as a function of the space variable, and is $C^{\infty}$ outside of a closed, countable set, which we describe explicitly. We also prove that, as conjectured in arXiv:1902.05174, the set of positive times when a jump occurs cannot have accumulation points. In addition, we prove that short-time uniqueness of physical solutions implies global uniqueness, which allows us to obtain uniqueness for very general initial data that fall outside the scope of the current well-posedness regime. In particular, we answer two questions left open in arXiv:1811.12356, arXiv:2302.13097, regarding the global uniqueness of solutions. We proceed by deriving a weighted obstacle problem satisfied by the solutions, which we exploit to establish regularity and non-degeneracy estimates and to classify the free boundary points. We also establish a backward propagation of oscillation property, which allows us to control the occurrence of future jumps in terms of the past oscillation of the solution. |
| title | Free boundary regularity and well-posedness of physical solutions to the supercooled Stefan problem |
| topic | Analysis of PDEs Probability 80A22, 35B44, 35R35, 60H30, 35B05 |
| url | https://arxiv.org/abs/2506.18741 |