Sharp non-uniqueness for the Navier-Stokes equations in R^3
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916520416772096 |
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| author | Miao, Changxing Nie, Yao Ye, Weikui |
| author_facet | Miao, Changxing Nie, Yao Ye, Weikui |
| contents | In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the incompressible Navier-Stokes equations in $\R^3$. To be more precise, we exhibit the non-uniqueness result in a strong sense, that is, any weak solution is non-unique in L^p([0,T];L^\infty(\R^3)) with 1\le p<2. Moreover, this non-uniqueness result is sharp with regard to the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint (2, \infty), which extends the sharp nonuniqueness for the Navier-Stokes equations on torus $\TTT^3$ in the recent groundbreaking work (Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987-1054) to the setting of the whole space. The key ingredient is developing a new iterative scheme that balances the compact support of the Reynolds stress error with the non-compact support of the solution via introducing incompressible perturbation fluid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_09637 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp non-uniqueness for the Navier-Stokes equations in R^3 Miao, Changxing Nie, Yao Ye, Weikui Analysis of PDEs In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the incompressible Navier-Stokes equations in $\R^3$. To be more precise, we exhibit the non-uniqueness result in a strong sense, that is, any weak solution is non-unique in L^p([0,T];L^\infty(\R^3)) with 1\le p<2. Moreover, this non-uniqueness result is sharp with regard to the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint (2, \infty), which extends the sharp nonuniqueness for the Navier-Stokes equations on torus $\TTT^3$ in the recent groundbreaking work (Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987-1054) to the setting of the whole space. The key ingredient is developing a new iterative scheme that balances the compact support of the Reynolds stress error with the non-compact support of the solution via introducing incompressible perturbation fluid. |
| title | Sharp non-uniqueness for the Navier-Stokes equations in R^3 |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.09637 |