A Classical Two-Part First-Threshold Proof of Global Smoothness for Navier--Stokes: Axisymmetric Swirl Closure and Full-System Reduction

Abstract

We prove global smooth continuation for smooth finite-energy solutions of the three-dimensional incompressible Navier--Stokes equations by a two-part first-threshold argument. Part I proves the axisymmetric-with-swirl theorem in the exact five-dimensional lifted formulation. The central variables are the lifted vorticity ratio \(G=ω_θ/r\), the regularized swirl derivative \(F=u^θ/r\), and the squared source density \(H=F^2\). In these variables the derivative source in the \(G\)-equation and the compressive feedback generated by the recovered strain \(U=u^r/r\) form a single pair-transfer mechanism. The proof combines localized energy identities, Hardy--Littlewood--Sobolev and Sobolev interpolation estimates, pair-threshold absorption, finite-overlap descendant exclusion, localized temporal source-to-score estimates, compactness of endpoint profiles, projected Pohozaev--Morawetz strictness, and an auxiliary recovery estimate for \(F\). Part II gives a full three-dimensional finite-threshold front-end. Starting from a hypothetical singular terminal packet, it removes leakage, shell, pressure, tail, fragmentation, passive-strain, angular phase-lock, and transfer-active temporal channels by finite-overlap descendants or strict terminal loss. A zero final defect forces the active frame measure into either a constant-frame locally two-dimensional class or a physical azimuthal orbit around one fixed axis. The first alternative is excluded by the classical two-dimensional Navier--Stokes theory, and the second is precisely the axisymmetric-with-swirl class proved in Part I.