Critical Structure of Axisymmetric Navier--Stokes with Swirl

Abstract

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