Hypothetical Singularity of 3D Navier-Stokes in Clay Institute set up Reduces to Axisymmetric with Swirl class

Abstract

We prove a singular-endpoint reduction for the three-dimensional incompressible Navier--Stokes large-data regularity problem in the smooth finite-energy class. The result is a reduction theorem: any hypothetical first singularity of a general three-dimensional solution generates, after scale normalization, threshold selection, and endpoint extraction, a terminal singular endpoint in the axisymmetric-with-swirl class. Thus a proof of the companion axisymmetric-with-swirl endpoint theorem closes the corresponding full three-dimensional finite-energy regularity statement. The proof is organized around the scalar vorticity-amplitude identity. Let \(ω=\nabla\times u\), \(A=|ω|\), \(ξ=ω/|ω|\), and \(S=\sym\nabla u\). The regularized identity \[ (\partial_t+u\cdot\nabla-νΔ)A =A\bigl(ξ\cdot Sξ-ν|\nablaξ|^2\bigr) \] splits the endpoint into a zero-production branch and a positive-production branch. The zero-production branch is closed by a Nash--Liouville argument after the amplitude-tail alternatives are removed. In the positive-production branch, the vortex-stretching density is first rebased onto comparable active donors. The central point of the present formulation is that this rebase is signed and stable: positive-signed donor mass is selected by a Carleson procedure, rebased into an exact orthogonal solenoidal hull, and then either produces a nonzero finite-band hull energy or activates a routed output.