Global Regularity Bounds for the Hyper-Dissipative Navier-Stokes Equation in Critical Besov Spaces

This monograph establishes global well-posedness results for the hyper-dissipative incompressible Navier-Stokes equations in Rn where the diffusion operator is given by the fractional Laplacian (-) with (1, n2+1). We focus specifically on the critical regularity regime dictated by the scaling invariance of the equation. By employing a rigorous Littlewood-Paley decomposition and Bony's paraproduct calculus, we derive sharp a priori estimates in homogeneous Besov spaces sp,q(Rn). We prove that for initial data small in the critical space np - (2- 1)p,q, there exists a unique global-in-time mild solution. Furthermore, we analyze the analyticity of the solution and provide decay rates for the higher-order Sobolev norms, effectively extending the classical Fujita-Kato theory to the fractional dissipation setting. The results bridge the gap between the subcritical hyper-viscous regime and the supercritical standard Navier-Stokes framework.