Sharp non-uniqueness for the 2D hyper-dissipative Navier–Stokes equations

In this paper, we study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier–Stokes equations (NSE) in the super-critical spaces $L_t^{\gamma }{W}_x^{s,p}$ when the viscosity exponent $\alpha \in [1,\frac{3}{2})$, 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 $(s,\gamma ,p)=(s,\infty ,\frac{2}{2\alpha -1+s})$ and $(s,\frac{2\alpha }{2\alpha -1+s},\infty )$. By using the intermittency of the temporal concentrated function in an almost optimal way, we extend the recent elegant works on the non-uniqueness of 2D NSE in Cheskidov and Luo [Invent. Math. 229 (2022), no. 3, 987–1054] and [Ann. PDE, 9 (2023), no. 2, Paper No. 13] to the hyper-dissipative case $\alpha \in (1,\frac{3}{2})$. In particularly, the viscosity exponent $\alpha =\frac{3}{2}$ is the upper limit for the one endpoint case $(s,\infty ,\frac{2}{2\alpha -1+s})$ when $s=0$.