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 α ≥ 5 / 4. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces L t γ 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 α , ∞ , p) and (2 α / γ + 1 − 2 α , γ , ∞). 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 η ⁎ measure, where η ⁎ > 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. [ABSTRACT FROM AUTHOR]