{
  "abstract": "We prove local regularity for axisymmetric suitable weak solutions of the 3D Navier-Stokes equations, which are smooth before the terminal time $t=0$, and satisfy Type~II pointwise bounds at a vanishing length scale $\\ell(t)$. We say $\\ell(t)$ is an Euler length if it is non-increasing, satisfies a doubling condition, and if $\\ell(t)\\to0$ and $(-t)/\\ell(t)^2\\to0$ as $t\\to0^-$. This includes power laws $\\ell(t)=(-t)^γ$ with $0<γ<1/2$, and logarithmic enlargements of the parabolic length. Our main result shows that local bounds of the type $|u(\\cdot,t)| \\leq C\\ell(t)/(-t)$ and $|\\nabla^2 u(\\cdot,t)|\\leq C/((-t)\\ell(t))$ for all $t\\in (-1,0)$ imply regularity. The proof adapts the circulation and potential-vorticity argument of our earlier paper~\\cite{CIV26} to ancient limits obtained from the rescaled vorticity system by zooming in. We show that the second derivative a priori assumption may be replaced by a Hölder bound on the azimuthal vorticity and a corresponding bound on its potential vorticity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20762",
  "author_total": 3,
  "authors": [
    {
      "name": "Peter Constantin"
    },
    {
      "name": "Mihaela Ignatova"
    },
    {
      "name": "Vlad Vicol"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8vjtgnd688v9k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20762",
  "fulltext_url": null,
  "id": "8vjtgnd688v9k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20762",
  "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": "W7213590921",
  "pdf_url": "https://arxiv.org/pdf/2609.20762",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8vjtgnd688v9k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.20762",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Regularity for axisymmetric Navier-Stokes with an Euler length",
  "title_text": "Regularity for axisymmetric Navier-Stokes with an Euler length",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/8vjtgnd688v9k",
  "venue": "arXiv",
  "versions": []
}