Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866911851859673088 |
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| author | Abels, Helmut Matioc, Bogdan-Vasile |
| author_facet | Abels, Helmut Matioc, Bogdan-Vasile |
| contents | We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an $L_p$-setting with $p\in(1,\infty)$. The Sobolev space $W^s_p(\mathbb{R})$ with $s=1+1/p$ is a critical space for this problem. We prove, for $s\in (1+1/p,2),$ that the Rayleigh-Taylor condition identifies an open subset of $W^s_p(\mathbb{R})$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_07656 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces Abels, Helmut Matioc, Bogdan-Vasile Analysis of PDEs 35R37, 35K55, 35Q35, 42B20 We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an $L_p$-setting with $p\in(1,\infty)$. The Sobolev space $W^s_p(\mathbb{R})$ with $s=1+1/p$ is a critical space for this problem. We prove, for $s\in (1+1/p,2),$ that the Rayleigh-Taylor condition identifies an open subset of $W^s_p(\mathbb{R})$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property. |
| title | Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces |
| topic | Analysis of PDEs 35R37, 35K55, 35Q35, 42B20 |
| url | https://arxiv.org/abs/2003.07656 |