Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929402827243520 |
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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 solve Lions' open problem: {\it the uniqueness of weak solutions for the 2-D inhomogeneous Navier-Stokes equations (INS)}. We first prove the global existence of weak solutions to 2-D (INS) with bounded initial density and initial velocity in $L^2(\mathbb R^2)$. Moreover, if the initial density is bounded away from zero, then our weak solution equals to Lions' weak solution, which in particular implies the uniqueness of Lions' weak solution. We also extend a celebrated result by Fujita and Kato on the 3-D incompressible Navier-Stokes equations to 3-D (INS): {\it the global well-posedness of 3-D (INS) with bounded initial density and initial velocity being small in $\dot H^{1/2}(\mathbb R^3)$}.
The proof of the uniqueness is based on a surprising finding that the estimate $t^{1/2}\nabla u\in L^2(0,T; L^\infty(\mathbb R^d))$ instead of $\nabla u\in L^1(0, T; L^\infty(\mathbb R^d))$ is enough to ensure the uniqueness of the solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19907 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density Hao, Tiantian Shao, Feng Wei, Dongyi Zhang, Zhifei Analysis of PDEs In this paper, we solve Lions' open problem: {\it the uniqueness of weak solutions for the 2-D inhomogeneous Navier-Stokes equations (INS)}. We first prove the global existence of weak solutions to 2-D (INS) with bounded initial density and initial velocity in $L^2(\mathbb R^2)$. Moreover, if the initial density is bounded away from zero, then our weak solution equals to Lions' weak solution, which in particular implies the uniqueness of Lions' weak solution. We also extend a celebrated result by Fujita and Kato on the 3-D incompressible Navier-Stokes equations to 3-D (INS): {\it the global well-posedness of 3-D (INS) with bounded initial density and initial velocity being small in $\dot H^{1/2}(\mathbb R^3)$}. The proof of the uniqueness is based on a surprising finding that the estimate $t^{1/2}\nabla u\in L^2(0,T; L^\infty(\mathbb R^d))$ instead of $\nabla u\in L^1(0, T; L^\infty(\mathbb R^d))$ is enough to ensure the uniqueness of the solution. |
| title | Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.19907 |