{
  "abstract": "For the 3D Navier-Stokes equations with sufficiently smooth initial data having zero swirl $u_{0θ}\\equiv 0$, Ladyzhenskaya \\cite{L1968} and Ukhovskii and Yudovich \\cite{UY1968} established global regularity. It has since remained open as to whether well-posedness still holds for smooth axisymmetric initial data, with the initial swirl being smaller than an \\textit{absolute} constant in a critical space. In this remark, we prove \\textit{norm inflation} occurs for the Navier-Stokes scalar swirl component, with initial smallness in the critical space $\\dot{H}^{\\frac{1}{2}}$ or even subcritical Sobolev spaces. We also obtain norm inflation for the 3D Euler swirl \\textit{vector} in the Euler subcritical space $\\dot{H}^3$. Our results are achieved by introducing an anisotropic scale in Luo's supercritical norm inflation construction \\cite{Luo24}, as well as performing an anisotropic bootstrap argument to get a sufficiently small swirl remainder term.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33973",
  "author_total": 2,
  "authors": [
    {
      "name": "Tobias Barker"
    },
    {
      "name": "Ruilin Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/daayn5wnpvfh0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33973",
  "fulltext_url": null,
  "id": "daayn5wnpvfh0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33973",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7223947641",
  "pdf_url": "https://arxiv.org/pdf/2610.33973",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/daayn5wnpvfh0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33973",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33973"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Swirl ill-posedness for incompressible fluids in (sub)critical spaces",
  "title_text": "Swirl ill-posedness for incompressible fluids in (sub)critical spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/daayn5wnpvfh0",
  "venue": "arXiv",
  "versions": []
}