A localized criterion for the regularity of solutions to Navier-Stokes equations
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866912154131628032 |
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| author | Li, Congming Liu, Chenkai Zhuo, Ran |
| author_facet | Li, Congming Liu, Chenkai Zhuo, Ran |
| contents | The Serrin-Prodi-Ladyzhenskaya type $L^{p,q}$ criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. In this article, we establish some localized $L^{p,q}$ criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local $L^{p,q}$ type norms. These local $L^{p,q}$ type norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local $L^{p,q}$ type norms is necessary and sufficient to affirmatively answer the millennium problem. Our work provides an interesting and plausible approach to study the millennium problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_00405 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A localized criterion for the regularity of solutions to Navier-Stokes equations Li, Congming Liu, Chenkai Zhuo, Ran Analysis of PDEs The Serrin-Prodi-Ladyzhenskaya type $L^{p,q}$ criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. In this article, we establish some localized $L^{p,q}$ criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local $L^{p,q}$ type norms. These local $L^{p,q}$ type norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local $L^{p,q}$ type norms is necessary and sufficient to affirmatively answer the millennium problem. Our work provides an interesting and plausible approach to study the millennium problem. |
| title | A localized criterion for the regularity of solutions to Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2212.00405 |