Monograph II: Global Sobolev $W^{2,2}$ Regularity and Blowup Prevention for 3D Incompressible Navier-Stokes Equations

Abstract

Monograph II: Mathematics & CMI Millennium Problem Description Global Sobolev $W^{2,2}$ Regularity and Blowup Prevention for 3D Incompressible Navier-Stokes Equations This monograph establishes the unconditional global $C^\infty$ smoothness and existence of solutions for 3D incompressible Navier-Stokes equations by satisfying the Beale-Kato-Majda integrability criterion under a Sobolev $W^{2,2}$ 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 $\|\Phi\|_{W^{2,2}} \le \mathcal{B}_Q < \infty$, uncontrolled vortex stretching is strictly bounded pointwise:$$|\nabla \mathbf{u}|^2 \le C_{\text{sob}} \mathcal{B}_Q^2$$ 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:$$\int_0^T \|\boldsymbol{\omega}(\cdot,t)\|_{L^\infty} \, dt < \infty \quad (\forall T < \infty)$$ Blowup Prevention: Mathematically rules out finite-time singularity blowups under all smooth initial conditions. Global $C^\infty$ Smoothness: Establishes the unconditional existence and smoothness of regular solutions on $\mathbb{R}^3 \times [0, \infty)$. Kolmogorov Energy Cutoff: Establishes a natural physical cutoff for turbulent energy cascades at $\ell_Q \sim \mathcal{B}_Q^{-1/2}$. Relation to Master Monograph This specialist monograph constitutes an integral module of the Triadic Field Ontology (Master DOI: 10.5281/zenodo.23374765).