Critical Compression in Axisymmetric Swirl

Abstract

We study two scale-critical inward-compression mechanisms in smooth axisymmetric Navier--Stokes flow with swirl. For the weighted swirl family Y_d=r^d u^theta, -1<=d<1, an exact L^p balance yields a signed radial-compression action. If p>=3/(1-d), finite L^p mass and a bounded signed action imply continuation. On the critical diagonal Y_q=r^(1-3/q)u^theta, q>=3/2, the integrating factor is universal, exp(3 int Ubar_q); H^1 data automatically supply the critical mass for 2<=q<=3, while bounded circulation extends this to every finite q>3. Independently, a sliding Petrovskii displacement criterion reaches the endpoint coefficient 2 and admits a quantified iterated-log correction, while a pulse train shows terminal-only control is insufficient. For profiles with the corresponding weighted moments, the instantaneous Hodge response to partial_z(F^2) is non-amplifying for power-weighted compressive F^2-work throughout 1<=alpha<=5. Thus a finite-time singularity must escape a continuum of critical signed-compression detectors and recurrently cross the Petrovskii wall.