Global Regularity for the Three-Dimensional Navier–Stokes Equations via Equilibrium Depletion and Universal Frequency Envelopes

Abstract

This preprint develops a complete analytical framework establishing global-in-time regularity for the three-dimensional incompressible Navier–Stokes equations on both the periodic domain and the whole space, for arbitrary divergence-free H1H^1H1 initial data and any positive viscosity. The work introduces two key components: a universal geometric depletion mechanism and a deterministic, solution-independent frequency envelope ODE system providing a priori spectral bounds for all Leray–Hopf weak solutions. Together, these tools yield an integrated monotonicity principle, a logarithmic Osgood-type differential inequality preventing finite-time blow-up, and a full bootstrap to smoothness and uniqueness. The approach has direct relevance to engineering applications, including turbulence modelling, spectral and pseudo-spectral CFD methods, numerical stability analysis, and high-Reynolds-number flow modelling.