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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2604.26906 |
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| _version_ | 1866913073069031424 |
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| author | Zhang, Yuming Paul |
| author_facet | Zhang, Yuming Paul |
| contents | This paper is a continuation of the work in \cite{kimzhang2024} concerning Hele-Shaw flow with both drift and source terms. We prove that, in a local neighborhood, if the free boundary is Lipschitz continuous with a sufficiently small Lipschitz constant, then the free boundary is $C^{1}$. As a corollary, we also consider the 2D vertical Hele-Shaw (or one-phase Muskat) problem with an advection term. We show that, provided the initial data and the advection term are small and the propagation speed is large, the free boundary becomes uniformly $C^1$ after a finite time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26906 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $C^1$-Regularity of the Free Boundary for Hele-Shaw Flow with Source and Drift Zhang, Yuming Paul Analysis of PDEs 35R35, 35B65, 76D27 This paper is a continuation of the work in \cite{kimzhang2024} concerning Hele-Shaw flow with both drift and source terms. We prove that, in a local neighborhood, if the free boundary is Lipschitz continuous with a sufficiently small Lipschitz constant, then the free boundary is $C^{1}$. As a corollary, we also consider the 2D vertical Hele-Shaw (or one-phase Muskat) problem with an advection term. We show that, provided the initial data and the advection term are small and the propagation speed is large, the free boundary becomes uniformly $C^1$ after a finite time. |
| title | $C^1$-Regularity of the Free Boundary for Hele-Shaw Flow with Source and Drift |
| topic | Analysis of PDEs 35R35, 35B65, 76D27 |
| url | https://arxiv.org/abs/2604.26906 |