Non-uniqueness of weak solutions to 2D generalized Navier-Stokes equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915050645618688 |
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| author | Li, Xinliang Tan, Zhong |
| author_facet | Li, Xinliang Tan, Zhong |
| contents | We study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^γL_{x}^{p}$ when $α\in[1,\frac{3}{2})$, and obtain the conclusion that the non-uniqueness of the weak solutions at the endpoint $(γ,p)=(\infty, \frac{2}{2α-1})$ is sharp in view of the generalized Ladyženskaja-Prodi-Serrin condition by using a different spatial-temporal building block from [Cheskidov-Luo, Ann. PDE, 9:13 (2023)] and taking advantage of the intermittency of the temporal concentrated function $g_{(k)}$ in an almost optimal way. Our results recover the above 2D non-uniqueness conclusion and extend to the hyper-dissipative case $α\in(1,\frac{3}{2})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20754 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of weak solutions to 2D generalized Navier-Stokes equations Li, Xinliang Tan, Zhong Analysis of PDEs 35A02, 35Q30, 76D05 We study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^γL_{x}^{p}$ when $α\in[1,\frac{3}{2})$, and obtain the conclusion that the non-uniqueness of the weak solutions at the endpoint $(γ,p)=(\infty, \frac{2}{2α-1})$ is sharp in view of the generalized Ladyženskaja-Prodi-Serrin condition by using a different spatial-temporal building block from [Cheskidov-Luo, Ann. PDE, 9:13 (2023)] and taking advantage of the intermittency of the temporal concentrated function $g_{(k)}$ in an almost optimal way. Our results recover the above 2D non-uniqueness conclusion and extend to the hyper-dissipative case $α\in(1,\frac{3}{2})$. |
| title | Non-uniqueness of weak solutions to 2D generalized Navier-Stokes equations |
| topic | Analysis of PDEs 35A02, 35Q30, 76D05 |
| url | https://arxiv.org/abs/2405.20754 |