{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.07869",
  "author_total": 1,
  "authors": [
    {
      "name": "Rishad Shahmurov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dc9pbbmx2w2ae.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.07869",
  "fulltext_url": null,
  "id": "dc9pbbmx2w2ae",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.07869",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7164074616",
  "pdf_url": "https://arxiv.org/pdf/2606.07869",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dc9pbbmx2w2ae.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2606.07869",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.07869"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Critical Structure of Axisymmetric Navier--Stokes with Swirl",
  "title_text": "Critical Structure of Axisymmetric Navier--Stokes with Swirl",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/dc9pbbmx2w2ae",
  "venue": "arXiv",
  "versions": []
}