On Lions' density patch problem at a critical level of regularity
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915942114525184 |
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| author | Škondrić, Stefan Violini, Alessandro |
| author_facet | Škondrić, Stefan Violini, Alessandro |
| contents | In this article, we study Lions' density patch problem in two space dimensions at critical regularity. We prove global existence, uniqueness, and stability for a fluid occupying a bounded Lipschitz region surrounded by vacuum and evolving according to the incompressible Navier--Stokes equations, with initial velocity in $\dot{B}^0_{2,1}(\mathbb{R}^2)$. Moreover, we show that the Lipschitz regularity of the patch is preserved, and that its long-time dynamics is a rigid motion leading to the emergence of an asymptotic domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16017 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Lions' density patch problem at a critical level of regularity Škondrić, Stefan Violini, Alessandro Analysis of PDEs In this article, we study Lions' density patch problem in two space dimensions at critical regularity. We prove global existence, uniqueness, and stability for a fluid occupying a bounded Lipschitz region surrounded by vacuum and evolving according to the incompressible Navier--Stokes equations, with initial velocity in $\dot{B}^0_{2,1}(\mathbb{R}^2)$. Moreover, we show that the Lipschitz regularity of the patch is preserved, and that its long-time dynamics is a rigid motion leading to the emergence of an asymptotic domain. |
| title | On Lions' density patch problem at a critical level of regularity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.16017 |