On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914832724262912 |
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| author | Hao, Tiantian Shao, Feng Wei, Dongyi Zhang, Zhifei |
| author_facet | Hao, Tiantian Shao, Feng Wei, Dongyi Zhang, Zhifei |
| contents | In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumption that the initial density is only bounded and the initial velocity is in $H^1(\mathbb{R}^2)$. With suitable assumptions on the initial density, which includes the case of density patch and vacuum bubbles, we prove that Lions' s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions: {\it for the density patch data $ρ_0=1_{D}$ with a smooth bounded domain $D\subset\mathbb{R}^2$, the regularity of $D$ is preserved by the time evolution of Lions's weak solution.} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07984 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations Hao, Tiantian Shao, Feng Wei, Dongyi Zhang, Zhifei Analysis of PDEs In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumption that the initial density is only bounded and the initial velocity is in $H^1(\mathbb{R}^2)$. With suitable assumptions on the initial density, which includes the case of density patch and vacuum bubbles, we prove that Lions' s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions: {\it for the density patch data $ρ_0=1_{D}$ with a smooth bounded domain $D\subset\mathbb{R}^2$, the regularity of $D$ is preserved by the time evolution of Lions's weak solution.} |
| title | On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.07984 |