Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: above the Lions exponent

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent $\alpha$ can be larger than the Lions exponent $5/4$. It is well-known that, due to Lions [55], for any $L^2$ divergence-free initial data, there exist unique smooth Leray-Hopf solutions when $\alpha \geq 5/4$. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces $L^\gamma_tW^{s,p}_x$, in view of the generalized Lady\v{z}enskaja-Prodi-Serrin condition. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints $(3/p+1-2\alpha, \infty, p)$ and $(2\alpha/\gamma+1-2\alpha, \gamma, \infty)$. Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff $\mathcal{H}^{\eta_*}$ measure, where $\eta_*>0$ is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, the strong vanishing viscosity result is obtained for the hyperdissipative Navier-Stokes equations.