Regularity of Weak Solutions to the Hyperdissipative Navier-Stokes System in Critical Besov Spaces

This monograph investigates the global regularity and uniqueness of weak solutions to the d-dimensional hyperdissipative Navier-Stokes equations, where the dissipation is governed by the fractional Laplacian operator (-) with 1. We focus specifically on the critical regime where the initial data belongs to the homogeneous Besov spaces sp,r(Rd) invariant under the natural scaling of the equation. By employing the Littlewood-Paley dyadic decomposition and Bony's paraproduct calculus, we establish a sharp regularity criterion extending the classical Prodi-Serrin-Ladyzhenskaya results. We prove that if a weak solution satisfies a specific integral condition in the critical Besov framework, it coincides with the unique strong solution and persists globally in time. Furthermore, we analyze the blow-up profile of solutions near the maximal time of existence, providing lower bounds on the blow-up rate in terms of the Besov norms. These results bridge the gap between the classical Navier-Stokes theory (=1) and the globally regular hyperviscous regime (1 + d/4).