An elliptic regularization approach to the Stefan problem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866916070790529024 |
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| author | Paiano, Filippo Velichkov, Bozhidar |
| author_facet | Paiano, Filippo Velichkov, Bozhidar |
| contents | In this paper, we develop the theory for the two-phase Stefan problem with finite energy, possibly non-empty mushy region, and space-dependent melting temperature. Specifically, we prove the existence of weak solutions with an elliptic regularization scheme. Our existence theorem provides information about the regularity of the solutions: we prove that the temperature of weak solutions is in $H^1$ for all times, that the enthalpy is well defined and bounded for all times, and that both the enthalpy and the temperature are weakly continuous in time. Finally, we establish a comparison principle for weak solutions on general unbounded domains and use it to show that every weak solution is recovered by the approximation scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_00885 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An elliptic regularization approach to the Stefan problem Paiano, Filippo Velichkov, Bozhidar Analysis of PDEs In this paper, we develop the theory for the two-phase Stefan problem with finite energy, possibly non-empty mushy region, and space-dependent melting temperature. Specifically, we prove the existence of weak solutions with an elliptic regularization scheme. Our existence theorem provides information about the regularity of the solutions: we prove that the temperature of weak solutions is in $H^1$ for all times, that the enthalpy is well defined and bounded for all times, and that both the enthalpy and the temperature are weakly continuous in time. Finally, we establish a comparison principle for weak solutions on general unbounded domains and use it to show that every weak solution is recovered by the approximation scheme. |
| title | An elliptic regularization approach to the Stefan problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2606.00885 |