Critical Structure of Axisymmetric Navier--Stokes with Swirl
Abstract
We study the axisymmetric Navier--Stokes equations with swirl in the variables F=uθ/r, G=ωθ/r, and Γ=ruθ. Their distinct scaling degrees lead to several critical transition surfaces rather than a single scalar exponent. We prove mesoscopic and temporal critical-face identities, source/contact trace estimates, compact finite-depth preparation obstructions, an affine six-power flattening barrier, a fixed-old-current compression ceiling, and compactness for rescaled strain generators with bounded action and bounded variation. A sharp radial estimate gives exponential control of the time occupied by large swirl-energy records; on parabolic time intervals it yields logarithmic routing from total kinetic records to the meridional component, and an independent cone argument routes non-meridional Type-II records to a scale-critical signed compression action. The source-free circulation also satisfies an exact quadratic dissipation identity, while scalar-weight calculations show that no radial power simultaneously yields a compression-free continuity law and a scale-zero current action. The results exclude several compact and passively prepared recurrence mechanisms under explicit hypotheses; meridional Type-II records, signed critical compression, noncompact high-frequency behavior, and boundary-entry alternatives remain.
Related
- Axisymmetric Navier--Stokes with Swirl: Regularity and Blow-Up Constraints
- Self-similar stationary Navier--Stokes flows in a half-space with arbitrarily large swirl forcing
- Regularity of asymptotically axisymmetric Navier-Stokes flows with rotationally invariant curl of the force
- A logarithmic regularity criterion for the axisymmetric Navier-Stokes equations
- Regularity for axisymmetric Navier-Stokes with an Euler length
BibTeX
@misc{shahmurov2026critical,
title = {{Critical Structure of Axisymmetric Navier--Stokes with Swirl}},
author = {Rishad Shahmurov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.07869},
eprint = {2606.07869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.07869},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical Structure of Axisymmetric Navier--Stokes with Swirl
AU - Rishad Shahmurov
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2606.07869
UR - https://arxiv.org/abs/2606.07869
AB - We study the axisymmetric Navier--Stokes equations with swirl in the variables F=u^θ/r, G=ω^θ/r, and Γ=ru^θ. Their distinct scaling degrees lead to several critical transition surfaces rather than a single scalar exponent. We prove mesoscopic and temporal critical-face identities, source/contact trace estimates, compact finite-depth preparation obstructions, an affine six-power flattening barrier, a fixed-old-current compression ceiling, and compactness for rescaled strain generators with bounded action and bounded variation. A sharp radial estimate gives exponential control of the time occupied by large swirl-energy records; on parabolic time intervals it yields logarithmic routing from total kinetic records to the meridional component, and an independent cone argument routes non-meridional Type-II records to a scale-critical signed compression action. The source-free circulation also satisfies an exact quadratic dissipation identity, while scalar-weight calculations show that no radial power simultaneously yields a compression-free continuity law and a scale-zero current action. The results exclude several compact and passively prepared recurrence mechanisms under explicit hypotheses; meridional Type-II records, signed critical compression, noncompact high-frequency behavior, and boundary-entry alternatives remain.
ER -