2022/05/20 by Yachun Li, Li, Yachun, Peng Qu +5 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Cosmology and Gravitation Theories #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2205.10260
openalex publication_date 2022/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 [55], for any L2 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 LγtWs,px, in view of the generalized Ladyženskaja-Prodi-Serrin condition. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints (3/p+1-2α, ∞, p) and (2α/γ+1-2α, 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 Hη_* 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, the strong vanishing viscosity result is obtained for the hyperdissipative Navier-Stokes equations.