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.
Shahmurov, Large-data global regularity for three-dimensional Navier–Stokes I: a direct first-threshold continuation proof for the axisymmetric swirl class, preprint
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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.