Global well-posedness of the one-phase Muskat problem with surface tension
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866909024850542592 |
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| author | Dong, Hongjie Kwon, Hyunwoo |
| author_facet | Dong, Hongjie Kwon, Hyunwoo |
| contents | In this paper, we establish the global well-posedness of the one-phase Muskat problem with surface tension for small initial data. This problem describes the motion of the interface separating a wet region from a dry region within a porous medium, a process governed by Darcy's law. Although physically essential, the inclusion of surface tension introduces an additional challenge. We prove that if the initial free boundary is sufficiently small in $H^s$, $s>d/2+1$, then the problem admits a unique global strong solution. Moreover, the solution converges to zero in Lipschitz norm as $t\rightarrow\infty$. To the best of our knowledge, this work constitutes the first global well-posedness result for the one-phase Muskat problem with surface tension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06545 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global well-posedness of the one-phase Muskat problem with surface tension Dong, Hongjie Kwon, Hyunwoo Analysis of PDEs Fluid Dynamics 35R35, 35Q35, 35D35, 76B03 In this paper, we establish the global well-posedness of the one-phase Muskat problem with surface tension for small initial data. This problem describes the motion of the interface separating a wet region from a dry region within a porous medium, a process governed by Darcy's law. Although physically essential, the inclusion of surface tension introduces an additional challenge. We prove that if the initial free boundary is sufficiently small in $H^s$, $s>d/2+1$, then the problem admits a unique global strong solution. Moreover, the solution converges to zero in Lipschitz norm as $t\rightarrow\infty$. To the best of our knowledge, this work constitutes the first global well-posedness result for the one-phase Muskat problem with surface tension. |
| title | Global well-posedness of the one-phase Muskat problem with surface tension |
| topic | Analysis of PDEs Fluid Dynamics 35R35, 35Q35, 35D35, 76B03 |
| url | https://arxiv.org/abs/2604.06545 |