On the Regularity of Navier-Stokes Equations in Critical Space
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908860801875968 |
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| author | Xiong, Shiyang Zhang, Liqun |
| author_facet | Xiong, Shiyang Zhang, Liqun |
| contents | This paper focuses on the regularity of the Navier-Stokes equations in critical space. Let $ u(x,t) $ and $ p(x,t) $ denote suitable weak solution of the Navier-Stokes equations in $Q_T=\mathbb{R}^3\times(-T, 0)$. We prove that if $u(x,t)$ is in the scaling invariant spaces $L_t^{\infty}L_{x_3}^{p_1}L_{x_h}^{p_2}(Q_T)$ , where $ \frac{1}{p_1}+\frac{2}{p_2}=1 $ , $p_1\geq 2$ and $ x_h = (x_1, x_2) $ , then $ u $ is a smooth solution in $ Q_T $ and doesn't blow up at $ t = 0 $. In particular, if $ u(x,t) \in L_t^{\infty}L_{x_3}^{\infty}L_{x_h}^{2}(Q_T)$, then $u(x,t)$ is a smooth solution in $ Q_T $ and regular up to $ t = 0 $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03881 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Regularity of Navier-Stokes Equations in Critical Space Xiong, Shiyang Zhang, Liqun Analysis of PDEs This paper focuses on the regularity of the Navier-Stokes equations in critical space. Let $ u(x,t) $ and $ p(x,t) $ denote suitable weak solution of the Navier-Stokes equations in $Q_T=\mathbb{R}^3\times(-T, 0)$. We prove that if $u(x,t)$ is in the scaling invariant spaces $L_t^{\infty}L_{x_3}^{p_1}L_{x_h}^{p_2}(Q_T)$ , where $ \frac{1}{p_1}+\frac{2}{p_2}=1 $ , $p_1\geq 2$ and $ x_h = (x_1, x_2) $ , then $ u $ is a smooth solution in $ Q_T $ and doesn't blow up at $ t = 0 $. In particular, if $ u(x,t) \in L_t^{\infty}L_{x_3}^{\infty}L_{x_h}^{2}(Q_T)$, then $u(x,t)$ is a smooth solution in $ Q_T $ and regular up to $ t = 0 $. |
| title | On the Regularity of Navier-Stokes Equations in Critical Space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.03881 |