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arXiv · 2405.20754

Non-uniqueness of weak solutions to 2D generalized Navier-Stokes equations

Xinliang Li·Zhong Tan
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Abstract

We study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^{\gamma}L_{x}^{p}$ when $\alpha\in[1,\frac{3}{2})$, and obtain the conclusion that the non-uniqueness of the weak solutions at the endpoint $(\gamma,p)=(\infty, \frac{2}{2\alpha-1})$ is sharp in view of the generalized Lady\v{z}enskaja-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 $\alpha \in(1,\frac{3}{2})$.

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Xinliang Li, Zhong Tan. 2024-05-31. Non-uniqueness of weak solutions to 2D generalized Navier-Stokes equations. https://arxiv.org/abs/2405.20754

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