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

TL;DR

This paper establishes global smoothness for solutions of the 3D Navier--Stokes equations using a novel two-part threshold approach, combining axisymmetric swirl analysis with full-system reduction.

Contribution

It introduces a new two-part proof method that combines axisymmetric swirl analysis with a full 3D reduction to prove global regularity.

Findings

Proves axisymmetric-with-swirl theorem in a five-dimensional lifted formulation.

Develops a full three-dimensional finite-threshold front-end.

Eliminates various leakage and transfer channels to establish smoothness.

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 \(G=\omega_\theta/r\), the regularized swirl derivative \(F=u^\theta/r\), 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.