Hardy Criticality in Axisymmetric Swirl
Abstract
For smooth axisymmetric Navier--Stokes flow with swirl, set F=uθ/r, Γ=ruθ, G=ωθ/r, and U=ur/r. Weighted swirl energies are classical; we determine the sharp two-parameter coercivity spectrum of the exact power-weight family ∫ |F|p rα dr dz, p>1, α>1. The full diffusion form has a positive gradient gap for α<2p+1, loses every uniform gap at α=2p+1, and is indefinite above it. Exactly at the same threshold the radial-strain term cancels, and the identity becomes the classical Lp circulation energy for Γ. Thus stretching cancellation and Hardy criticality coincide within the power family. We also characterize r3drdz 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.
Related
- A special Hardy estimate in the axisymmetric swirl problem
- Contact discontinuities for 3-D axisymmetric flows with nonzero swirl
- Axisymmetric Navier--Stokes with Swirl: Regularity and Blow-Up Constraints
- A double-logarithmic lifespan bound for axisymmetric Euler equations with a small initial swirl
- Critical Structure of Axisymmetric Navier--Stokes with Swirl
BibTeX
@misc{shahmurov2026hardy,
title = {{Hardy Criticality in Axisymmetric Swirl}},
author = {Rishad Shahmurov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.01875},
eprint = {2605.01875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.01875},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hardy Criticality in Axisymmetric Swirl
AU - Rishad Shahmurov
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2605.01875
UR - https://arxiv.org/abs/2605.01875
AB - For smooth axisymmetric Navier--Stokes flow with swirl, set F=u^θ/r, Γ=ru^θ, G=ω^θ/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 ∫ |F|^p r^α dr dz, p>1, α>1. The full diffusion form has a positive gradient gap for α<2p+1, loses every uniform gap at α=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 Γ. 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.
ER -