Unbounded solutions for the Muskat problem
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910289640816640 |
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| author | Sánchez, Omar |
| author_facet | Sánchez, Omar |
| contents | We prove the local existence of solutions of the form $x^2+ct+g,$ with $g\in H^s(\mathbb R)$ and $s\geq 3,$ for the Muskat problem in the stable regime. We use energy methods to obtain a bound of $g$ in Sobolev spaces. In the proof we deal with the loss of the Rayleigh-Taylor condition at infinity and a new structure of the kernels in the equation. Remarkably, these solutions grow quadratically at infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08347 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unbounded solutions for the Muskat problem Sánchez, Omar Analysis of PDEs 35Q35, 76S05, 47J35 We prove the local existence of solutions of the form $x^2+ct+g,$ with $g\in H^s(\mathbb R)$ and $s\geq 3,$ for the Muskat problem in the stable regime. We use energy methods to obtain a bound of $g$ in Sobolev spaces. In the proof we deal with the loss of the Rayleigh-Taylor condition at infinity and a new structure of the kernels in the equation. Remarkably, these solutions grow quadratically at infinity. |
| title | Unbounded solutions for the Muskat problem |
| topic | Analysis of PDEs 35Q35, 76S05, 47J35 |
| url | https://arxiv.org/abs/2312.08347 |