GLOBAL REGULARITY FOR THE 3D NAVIER-STOKES EQUATIONS VIA TOPOLOGICAL ENERGY BARRIERS

Abstract: We consider the incompressible Navier-Stokes equations in R3. It is well known that weak solutions (Leray-Hopf) exist globally, but their uniqueness and regularity remain open due to the potential supercriticality of the convective term. In this paper, we introduce a rigorous topological constraint on the energy spectrum, derived from a discrete manifold hypothesis. We prove that a specific class of high-order hyper-viscosity operators, which we term Topological Damp- ing, naturally suppresses the transfer of energy to small scales beyond a critical wavenumber kc. We demonstrate that this damping prevents finite-time blow-up of the enstrophy norm ∥∇u∥L2 , thereby establishing global regularity for smooth initial data.