{
  "abstract": "We construct a nonzero divergence-free initial velocity on $\\mathbb{R}^3$ and a continuum of pairwise distinct suitable Leray--Hopf solutions of the unforced three-dimensional Navier--Stokes equations on a common time interval. The initial datum is axisymmetric, smooth away from the origin, and belongs to every $L^{\\mathfrak q}$ with $1\\leq \\mathfrak q<3$, but not to $L^3$. All the solutions attain this datum strongly in $L^2$, are smooth at positive times, and satisfy both the global and local energy equalities through time zero. The same family belongs simultaneously to every $L^{\\mathfrak p}_tL^{\\mathfrak q}_x$ with $1\\leq \\mathfrak p\\leq\\infty$, $2\\leq \\mathfrak q\\leq\\infty$, and $2/\\mathfrak p+3/\\mathfrak q>1$. The same solutions also belong to the critical Lorentz space $L^\\infty_tL^{3,\\infty}_x$. This endpoint is sharp for uniqueness in the Leray--Hopf class within the Lorentz scale. The proof is entirely analytic and uses a cascade through geometrically separated Gaussian swirls. We establish growth for the time-dependent viscous linearization and determine the sign of its complete quadratic stress, including the interaction of the reference velocity with the second variation. The integrated stress generates a controlled perturbation at the next larger spatial scale. At each finite cutoff, a prescribed terminal coordinate selects an initial perturbation whose physical $L^2$ norm tends to zero as the cutoff is removed. Uniform energy, moment, and transfer estimates yield the common strong initial trace while preserving a nonzero observable at a fixed positive time.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06117",
  "author_total": 4,
  "authors": [
    {
      "name": "Thomas Y. Hou"
    },
    {
      "name": "Xiang Qin"
    },
    {
      "name": "Peicong Song"
    },
    {
      "name": "Wenhan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9qnwmc86zzb6a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06117",
  "fulltext_url": null,
  "id": "9qnwmc86zzb6a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06117",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06117",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9qnwmc86zzb6a.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06117",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06117"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Viscous Cascade Construction of Nonunique Leray--Hopf Solutions to the Unforced 3D Navier--Stokes Equations",
  "title_text": "A Viscous Cascade Construction of Nonunique Leray--Hopf Solutions to the Unforced 3D Navier--Stokes Equations",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/9qnwmc86zzb6a",
  "venue": "arXiv",
  "versions": []
}