{
  "abstract": "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 \\(\\int |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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.01875",
  "author_total": 1,
  "authors": [
    {
      "name": "Rishad Shahmurov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/47kfj7b4wg99c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.01875",
  "fulltext_url": null,
  "id": "47kfj7b4wg99c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.01875",
  "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": "W7160293050",
  "pdf_url": "https://arxiv.org/pdf/2605.01875",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/47kfj7b4wg99c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2605.01875",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.01875"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hardy Criticality in Axisymmetric Swirl",
  "title_text": "Hardy Criticality in Axisymmetric Swirl",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/47kfj7b4wg99c",
  "venue": "arXiv",
  "versions": []
}