Sharp non-uniqueness for the 2D hyper-dissipative Navier-Stokes equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929670724780032 |
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| author | Du, Lili Li, Xinliang |
| author_facet | Du, Lili Li, Xinliang |
| contents | In this article, we study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^γW_{x}^{s,p}$ when $α\in[1,\frac{3}{2})$, and obtain the conclusion that the non-uniqueness of the weak solutions at the two endpoints is sharp in view of the generalized Ladyženskaya-Prodi-Serrin condition with the triplet $(s,γ,p)=(s,\infty, \frac{2}{2α-1+s})$ and $(s, \frac{2α}{2α-1+s}, \infty)$. As a good observation, we use the intermittency of the temporal concentrated function in an almost optimal way. The research results extend the recent elegant works on 2D Navier-Stokes equations in [Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987--1054; Cheskidov and Luo, Ann. PDE, 9:13 (2023)] to the hyper-dissipative case $α\in(1,\frac{3}{2})$, and are also applicable in Lebesgue and Besov spaces. It is proved that even in the case of high viscosity, the behavior of the solution remains unpredictable and stochastic due to the lack of integrability and regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_06880 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp non-uniqueness for the 2D hyper-dissipative Navier-Stokes equations Du, Lili Li, Xinliang Analysis of PDEs 35A02, 35D30, 76D05 In this article, we study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^γW_{x}^{s,p}$ when $α\in[1,\frac{3}{2})$, and obtain the conclusion that the non-uniqueness of the weak solutions at the two endpoints is sharp in view of the generalized Ladyženskaya-Prodi-Serrin condition with the triplet $(s,γ,p)=(s,\infty, \frac{2}{2α-1+s})$ and $(s, \frac{2α}{2α-1+s}, \infty)$. As a good observation, we use the intermittency of the temporal concentrated function in an almost optimal way. The research results extend the recent elegant works on 2D Navier-Stokes equations in [Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987--1054; Cheskidov and Luo, Ann. PDE, 9:13 (2023)] to the hyper-dissipative case $α\in(1,\frac{3}{2})$, and are also applicable in Lebesgue and Besov spaces. It is proved that even in the case of high viscosity, the behavior of the solution remains unpredictable and stochastic due to the lack of integrability and regularity. |
| title | Sharp non-uniqueness for the 2D hyper-dissipative Navier-Stokes equations |
| topic | Analysis of PDEs 35A02, 35D30, 76D05 |
| url | https://arxiv.org/abs/2407.06880 |