Global-in-time estimates for the 2D one-phase Muskat problem with contact points
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908946380357632 |
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| author | Bocchi, Edoardo Castro, Ángel Gancedo, Francisco |
| author_facet | Bocchi, Edoardo Castro, Ángel Gancedo, Francisco |
| contents | In this paper, we study the dynamics of a two-dimensional viscous fluid evolving through a porous medium or a Hele-Shaw cell, driven by gravity and surface tension. A key feature of this study is that the fluid is confined within a vessel with vertical walls and below a dry region. Consequently, the dynamics of the contact points between the vessel, the fluid and the dry region are inherently coupled with the surface evolution. A similar contact scenario was recently analyzed for more regular viscous flows, modeled by the Stokes [GuoTice2018] and Navier-Stokes [GuoTice2024] equations. Here, we adopt the same framework but use the more singular Darcy's law for modeling the flow. We prove global-in-time a priori estimates for solutions initially close to equilibrium. Taking advantage of the Neumann problem solved by the velocity potential, the analysis is carried out in non-weighted $L^2$-based Sobolev spaces and without imposing restrictions on the contact angles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19286 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global-in-time estimates for the 2D one-phase Muskat problem with contact points Bocchi, Edoardo Castro, Ángel Gancedo, Francisco Analysis of PDEs In this paper, we study the dynamics of a two-dimensional viscous fluid evolving through a porous medium or a Hele-Shaw cell, driven by gravity and surface tension. A key feature of this study is that the fluid is confined within a vessel with vertical walls and below a dry region. Consequently, the dynamics of the contact points between the vessel, the fluid and the dry region are inherently coupled with the surface evolution. A similar contact scenario was recently analyzed for more regular viscous flows, modeled by the Stokes [GuoTice2018] and Navier-Stokes [GuoTice2024] equations. Here, we adopt the same framework but use the more singular Darcy's law for modeling the flow. We prove global-in-time a priori estimates for solutions initially close to equilibrium. Taking advantage of the Neumann problem solved by the velocity potential, the analysis is carried out in non-weighted $L^2$-based Sobolev spaces and without imposing restrictions on the contact angles. |
| title | Global-in-time estimates for the 2D one-phase Muskat problem with contact points |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.19286 |