A potential theory approach to the capillarity-driven Hele-Shaw problem
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910237693313024 |
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| author | Matioc, Bogdan-Vasile Walker, Christoph |
| author_facet | Matioc, Bogdan-Vasile Walker, Christoph |
| contents | In this paper, we demonstrate that potential theory provides a powerful framework for analyzing quasistationary fluid flows in bounded geometries, where the bulk dynamics are governed by elliptic equations with constant coefficients. This approach is illustrated by the two-dimensional Hele-Shaw problem with surface tension, for which we derive local well-posedness and parabolic smoothing in (almost) optimal function spaces. In addition, we establish a generalized principle of linearized stability for a particular class of abstract quasilinear parabolic problems, which enables us to show that the stationary solutions to the Hele-Shaw problem are exponentially stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15491 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A potential theory approach to the capillarity-driven Hele-Shaw problem Matioc, Bogdan-Vasile Walker, Christoph Analysis of PDEs In this paper, we demonstrate that potential theory provides a powerful framework for analyzing quasistationary fluid flows in bounded geometries, where the bulk dynamics are governed by elliptic equations with constant coefficients. This approach is illustrated by the two-dimensional Hele-Shaw problem with surface tension, for which we derive local well-posedness and parabolic smoothing in (almost) optimal function spaces. In addition, we establish a generalized principle of linearized stability for a particular class of abstract quasilinear parabolic problems, which enables us to show that the stationary solutions to the Hele-Shaw problem are exponentially stable. |
| title | A potential theory approach to the capillarity-driven Hele-Shaw problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.15491 |