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We study the 3D hyperdissipative Navier-Stokes equations on the torus, wherethe viscosity exponent $\alpha$ can be larger than the Lions exponent $5/4$. Itis well-known that, due to Lions [55], for any $L^2$ divergence-free initialdata, there exist unique smooth Leray-Hopf solutions when $\alpha \geq 5/4$. Weprove that even in this high dissipative regime, the uniqueness would fail inthe supercritical spaces $L^\gamma_tW^{s,p}_x$, in view of the generalizedLady\v{z}enskaja-Prodi-Serrin condition. The non-uniqueness is proved in thestrong 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 uniqueLeray-Hopf solutions near the initial time and, more delicately, admit thepartial regularity outside a fractal set of singular times with zero Hausdorff$\mathcal{H}^{\eta_*}$ measure, where $\eta_*>0$ is any given small positiveconstant. These results also provide the sharp non-uniqueness in thesupercritical Lebesgue and Besov spaces. Furthermore, the strong vanishingviscosity result is obtained for the hyperdissipative Navier-Stokes equations.

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