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

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1], for any $L^2$ divergence-free initial data, there exist unique smooth Leray-Hopf solutions when $\alpha \ge 5/4$. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces $L_t^{\gamma }{W}_x^{s,p}$, in view of the Ladyženskaja-Prodi-Serrin criteria. 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 $H^{{\eta }_{⁎}}$ 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, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.