On the Global Regularity of the 3D Incompressible Navier–Stokes Equations: A Four-Pillar Conditional Framework and Regularity Elevation

On the Global Regularity of the 3D Incompressible Navier–Stokes Equations: A Four-Pillar Conditional Framework and Regularity Elevation

Michael Allan A. Bahtaji, Ph.D.

Department of Physics, College of Science

Technological University of the Philippines

Ayala Boulevard, Ermita, Manila, Philippines, 1000

Email: michaelallan_bahtaji@tup.edu.ph | michaelallanbahtaji336@gmail.com

Abstract

We establish a rigorous, self-contained mathematical framework addressing the global existence and smoothness of solutions to the three-dimensional incompressible Navier–Stokes equations. The formulation is structured around four technical pillars: (1)  Sobolev energy estimates () and local energy inequalities for suitable weak solutions; (2) Littlewood–Paley dyadic frequency analysis and the Beale–Kato–Majda (BKM) logarithmic criterion governing the non-linear vortex stretching operator; (3) critical scaling invariance () and anomalous energy dissipation bounds in the zero-viscosity limit (); and (4) the Caffarelli–Kohn–Nirenberg (CKN) -regularity theory establishing that the space-time singular set  has 1D Hausdorff measure zero (). We demonstrate how elevating a Leray–Hopf weak solution to a global  smooth solution reduces strictly to proving high-frequency energy decay across Littlewood–Paley blocks. Finally, we explicitly isolate the three open Millennium-level barriers that must be surpassed for an unconditional proof.