The paper proves global smoothness for smooth finite-energy solutions of the 3D Navier-Stokes equations via a two-part first-threshold argument that closes the axisymmetric-with-swirl case and reduces all other potential singularities to it or to 2D.
Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea
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A Classical Two-Part First-Threshold Proof of Global Smoothness for Navier--Stokes: Axisymmetric Swirl Closure and Full-System Reduction
The paper proves global smoothness for smooth finite-energy solutions of the 3D Navier-Stokes equations via a two-part first-threshold argument that closes the axisymmetric-with-swirl case and reduces all other potential singularities to it or to 2D.