A new reformulation of the Muskat problem with surface tension
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910421960622080 |
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| author | Matioc, Anca--Voichita Matioc, Bogdan--Vasile |
| author_facet | Matioc, Anca--Voichita Matioc, Bogdan--Vasile |
| contents | Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density $\vartheta$ to the $L_2$-adjoints of these operators evaluated at the density $\vartheta'$ are used to recast the Muskat problem with surface tension and general viscosities as a system of equations with nonlinearities expressed in terms of the $L_2$-adjoints of these operators. An advantage of this formulation is that the nonlinearities appear now as a derivative. This aspect and abstract quasilinear parabolic theory are then exploited to establish a local well-posedness result in all subcritical Sobolev spaces $W^s_p(\mathbb{R})$ with $p\in(1,\infty)$ and $s\in (1+1/p,2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_13567 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A new reformulation of the Muskat problem with surface tension Matioc, Anca--Voichita Matioc, Bogdan--Vasile Analysis of PDEs 35R37, 35K59, 35K93, 35Q35, 42B20 Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density $\vartheta$ to the $L_2$-adjoints of these operators evaluated at the density $\vartheta'$ are used to recast the Muskat problem with surface tension and general viscosities as a system of equations with nonlinearities expressed in terms of the $L_2$-adjoints of these operators. An advantage of this formulation is that the nonlinearities appear now as a derivative. This aspect and abstract quasilinear parabolic theory are then exploited to establish a local well-posedness result in all subcritical Sobolev spaces $W^s_p(\mathbb{R})$ with $p\in(1,\infty)$ and $s\in (1+1/p,2)$. |
| title | A new reformulation of the Muskat problem with surface tension |
| topic | Analysis of PDEs 35R37, 35K59, 35K93, 35Q35, 42B20 |
| url | https://arxiv.org/abs/2203.13567 |