{
  "abstract": "An investigation of local aspects of the space of axisymmetric traveling-wave solutions to the three-dimensional Euler equations near Hill's zero-flux spherical vortex and Norbury's small-flux vortex rings is the endeavor undertaken herein. The aforementioned family consists of idealized, swirl-free, and nonsmooth objects, their azimuthal relative vorticity being prescribed by a scaled characteristic function of the compact vortex core. We confront the desingularization question, both within the swirl-free solution subspace and upon admitting swirl, and prove that each member of the Hill--Norbury family of sufficiently small flux is the limit of traveling-wave solutions to the Euler equations that are smooth to infinite order and have compact toroidal vorticity support. In the latter swirling regime, however, a curious flexibility is revealed: total vorticity may diverge along sequences converging in a natural desingularization topology. Our construction, founded upon a five-dimensional reformulation of the Stokes stream function, is an amalgamation of delicate thin-domain analysis, singular-parameter Newton-type nonlinear inversion, and topological fixed-point arguments. To the best of the authors' knowledge, this constitutes the first verification that the small-flux Hill--Norbury family is nowhere isolated from the collection of smooth vortex rings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22513",
  "author_total": 2,
  "authors": [
    {
      "name": "Răzvan-Octavian Radu"
    },
    {
      "name": "Noah Stevenson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dt8f06tedt27d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22513",
  "fulltext_url": null,
  "id": "dt8f06tedt27d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22513",
  "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": "W7214058297",
  "pdf_url": "https://arxiv.org/pdf/2609.22513",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dt8f06tedt27d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22513",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22513"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Smooth and swirling steady vortex rings near the Hill--Norbury family",
  "title_text": "Smooth and swirling steady vortex rings near the Hill--Norbury family",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/dt8f06tedt27d",
  "venue": "arXiv",
  "versions": []
}