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

TL;DR

This paper demonstrates the non-uniqueness of weak solutions for 2D hyper-dissipative Navier-Stokes equations in certain super-critical spaces, extending previous results and establishing sharpness at the endpoint.

Contribution

It introduces a new spatial-temporal construction and leverages intermittency to prove sharp non-uniqueness results for hyper-dissipative 2D Navier-Stokes equations.

Findings

Non-uniqueness at the endpoint $( abla,p)=(\infty, \frac{2}{2\alpha-1})$ is sharp.

Results extend to hyper-dissipative case with $oxed{\alpha \in(1,\frac{3}{2})}$.

Uses a novel building block and temporal intermittency to establish non-uniqueness.

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 $\alpha \in(1,\frac{3}{2})$.