Global Regularity for Axisymmetric Navier--Stokes Flows with Swirl

Abstract

We prove global smoothness for smooth finite-energy axisymmetric solutions of the three-dimensional incompressible Navier--Stokes equations with arbitrary swirl. The proof is organized around the circulation \(Γ=ru^θ\), the lifted azimuthal vorticity ratio \(G=ω^θ/r\), and the axis-compatible circulation-gradient pair \[ Ξ=(A,W)=\left(\frac{Γ_r}{r},\frac{Γ_z}{r}\right). \] The principal near-axis difficulty is the source term \(\partial_z(F^2)\), where \(F=u^θ/r=Γ/r^2\), in the lifted \(G\)-equation. The first key observation is the exact identity \[ \partial_z(F^2)=\frac{2ΓW}{r^3}, \qquad dμ_5=r^3\,dr\,dz, \] which converts the source pairing into \(2\int GΓW\,drdz\). This term is controlled by an axis Hardy formula for \(Γ\), one-dimensional Sobolev estimates in the axial variable for radial energy densities, and the positive \(W/r\)-Hardy term in the \(Ξ\)-dissipation. The second key point is that the typed zero-output endpoint is no longer treated as an abstract bridge-profile problem. After all source, collar, macro, motion, projection, cascade, and backward-ancestor channels vanish, a small-threshold energy-seeding lemma gives \[ G\in L_t^\infty L^2(dμ_5)\cap L_t^2\dot H^1(dμ_5). \]