Global Regularity and Finite-Time Blow-up Scenarios for the Hyper-Dissipative Navier-Stokes Equations in Critical Besov Spaces

This monograph investigates the well-posedness and singularity formation of the hyper-dissipative Navier-Stokes equations in three-dimensional Euclidean space. We analyze the system with fractional dissipation (-) for [1, ). By leveraging the Littlewood-Paley dyadic decomposition and the machinery of Bony's paraproducts, we establish global regularity results in critical homogeneous Besov spaces sp,q(R3) for the regime 54. Furthermore, we rigorously examine finite-time blow-up scenarios in the hypo-dissipative and slightly supercritical regimes, deriving precise Beale-Kato-Majda type criteria extended to the Besov framework. We discuss the implications of spectral truncation and "averaged" models, providing a comprehensive overview of the current boundary between global well-posedness and potential singularity formation in fluid dynamics.