Critical Structure of Axisymmetric Navier--Stokes with Swirl

Rishad Shahmurov

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arXiv · 2606.07869

Critical Structure of Axisymmetric Navier--Stokes with Swirl

Rishad Shahmurov

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Abstract

We study the axisymmetric Navier--Stokes equations with swirl in the variables $F=u^\theta/r$, $G=\omega^\theta/r$, and $\Gamma=ru^\theta$. 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.

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Rishad Shahmurov. 2026-06-05. Critical Structure of Axisymmetric Navier--Stokes with Swirl. https://arxiv.org/abs/2606.07869

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