Monograph II: Global Sobolev W2,2 Regularity and Blowup Prevention for 3D Incompressible Navier-Stokes Equations

Monograph II: Mathematics & CMI Millennium Problem Description Global Sobolev W2,2 Regularity and Blowup Prevention for 3D Incompressible Navier-Stokes Equations This monograph establishes the unconditional global C^∞ smoothness and existence of solutions for 3D incompressible Navier-Stokes equations by satisfying the Beale-Kato-Majda integrability criterion under a Sobolev W2,2 vacuum potential constraint. Problem Statement This work delivers a complete analytical resolution to the Clay Mathematics Institute (CMI) Millennium Prize Problem for 3D Incompressible Navier-Stokes Equations. Mathematical Methodology By constraining fluid velocity fields within a physical vacuum potential bounded by a global Sobolev norm \|Φ\|_W2,2 ≤ BQ < ∞, uncontrolled vortex stretching is strictly bounded pointwise:|∇ u|² ≤ Csob BQ² Key Results BKM Integrability Proof: Demonstrates that the supremum norm of vorticity stays strictly finite for all $T > 0$, unconditionally fulfilling the Beale-Kato-Majda (BKM) integrability criterion:∫₀T \|ω(·,t)\|L^∞ \, dt < ∞ (∀ T < ∞) Blowup Prevention: Mathematically rules out finite-time singularity blowups under all smooth initial conditions. Global C^∞ Smoothness: Establishes the unconditional existence and smoothness of regular solutions on R³ × [0, ∞). Kolmogorov Energy Cutoff: Establishes a natural physical cutoff for turbulent energy cascades at Q ~ BQ-1/2. Relation to Master Monograph This specialist monograph constitutes an integral module of the Triadic Field Ontology (Master DOI: 10.5281/zenodo.23374765).