Global Regularity and Blow-Up Criteria for the Hyper-Dissipative Navier-Stokes System in Critical Besov Spaces

This monograph investigates the well-posedness and regularity properties of the hyper-dissipative Navier-Stokes system in Rn, characterized by the fractional diffusion operator (-) with > 1. We establish global-in-time existence and uniqueness of mild solutions for initial data belonging to the scaling-invariant critical Besov spaces sp,q(Rn), where s = n/p - 2+ 1, under a smallness assumption. Furthermore, we derive rigorous blow-up criteria of Beale-Kato-Majda type, extending classical results to the fractional dissipation setting. By leveraging Littlewood-Paley theory and Bony's paraproduct decomposition, we provide optimal estimates for the nonlinear transport term, thereby characterizing the precise threshold between global regularity and potential finite-time singularity formation in the supercritical regime.