ON THE 3D NAVIER-STOKES CONDITIONAL REGULARITY

ABSTRACT: This paper examines the global regularity problem for the three-dimensional incompressible NavierStokes equations. Starting from the exact energy and enstrophy balances, the nonlinear term is expressed in vorticity–strain form to isolate vortex stretching. A dimensionless quantity Q(t) = ‖∇u(t)‖₂² / ‖u(t)‖₂² is introduced, whose evolution satisfies a closed differential inequality. The analysis reduces regularity to the control of a single time-integrated geometric functional measuring stretching in regions where vorticity remains aligned with the maximal strain direction. Two complementary mechanisms: spectral anisotropy growth (Route A) and pressure-induced suppression of eigenframe rotation (Route B); are identified as sufficient to bound this functional. Under either route, the normalized enstrophy remains uniformly bounded, preventing finite-time blow-up and recovering classical regularity criteria such as Beale-Kato-Majda and ConstantinFefferman. The argument culminates in a single geometric lemma whose proof would imply global regularity