Hardy Criticality in Axisymmetric Swirl

Rishad Shahmurov

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arXiv · 2605.01875

Hardy Criticality in Axisymmetric Swirl

Rishad Shahmurov

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Abstract

For smooth axisymmetric Navier--Stokes flow with swirl, set $F=u^\theta/r$, $\Gamma=ru^\theta$, $G=\omega^\theta/r$, and $U=u^r/r$. Weighted swirl energies are classical; we determine the sharp two-parameter coercivity spectrum of the exact power-weight family $\int |F|^p r^\alpha\,dr\,dz$, $p>1$, $\alpha>1$. The full diffusion form has a positive gradient gap for $\alpha<2p+1$, loses every uniform gap at $\alpha=2p+1$, and is indefinite above it. Exactly at the same threshold the radial-strain term cancels, and the identity becomes the classical $L^p$ circulation energy for $\Gamma$. Thus stretching cancellation and Hardy criticality coincide within the power family. We also characterize $r^3drdz$ by conservative $F$-transport, self-adjoint five-dimensional diffusion, and a flat $G$-source pairing, and obtain the optimal strain form-bound threshold within the Hardy-coercive powers.

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Rishad Shahmurov. 2026-05-03. Hardy Criticality in Axisymmetric Swirl. https://arxiv.org/abs/2605.01875

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