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

Abstract 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 when the viscosity exponent , 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 and . 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 . In particularly, the viscosity exponent is the upper limit for the one endpoint case when .