{
  "abstract": "We examine the proof of Dongyi Wei's logarithmic circulation criterion. A trace estimate specific to $\\nabla(u_r/r)$ replaces a generic squared-swirl form bound by one with one logarithm rather than two. Repeating the coupled $J$--$Ω$ energy and comparison argument then gives a proposed threshold of order $(\\log D)^{-5/4}$. Under the hypotheses stated below, the proof is intended to establish regularity when $|r u_θ|\\le C|\\log r|^{-α}$ uniformly near the axis with $α>5/4$, and at $α=5/4$ with an absolute sufficiently small coefficient. Every analytic step, including the cutoff core and exterior terms, is included for independent checking.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31435",
  "author_total": 1,
  "authors": [
    {
      "name": "Daoguo Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cc4ammg1pd4tw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31435",
  "fulltext_url": null,
  "id": "cc4ammg1pd4tw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31435",
  "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": "W7223408302",
  "pdf_url": "https://arxiv.org/pdf/2610.31435",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cc4ammg1pd4tw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31435",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31435"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A special Hardy estimate in the axisymmetric swirl problem",
  "title_text": "A special Hardy estimate in the axisymmetric swirl problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/cc4ammg1pd4tw",
  "venue": "arXiv",
  "versions": []
}