{
  "abstract": "We construct nonzero compactly supported stationary distributional solutions $u\\in L^2(\\mathbb{R}^3;\\mathbb{R}^3)$ of the unforced incompressible Navier--Stokes equations, together with compactly supported pressures in $L^1$, with both norms arbitrarily small. The construction addresses the three-dimensional finite-energy endpoint at which the usual single-scale intermittent Mikado mechanism loses its algebraic gain against the Laplacian. The main new ingredient is a logarithmic Mikado profile, built from a truncated two-dimensional harmonic dipole and spread over logarithmically many transverse scales. The same perturbation yields a localized $h$-principle: nonzero stationary solutions are strongly dense in the localized $L^p$ classes for every $1\\leq p<2$ and in $H^{-1}$, and weakly dense at $p=2$, while strong $L^2$ density fails because of an exact isotropic quadratic-moment constraint. We also prescribe arbitrary positive $L^2$ norms, periodize the construction to $\\mathbb{T}^3$, and obtain a stationary-versus-Leray nonuniqueness mechanism for the evolutionary equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.17383",
  "author_total": 1,
  "authors": [
    {
      "name": "Xuanxuan Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dfavtx0arfvff.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.17383",
  "fulltext_url": null,
  "id": "dfavtx0arfvff",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.17383",
  "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": "W7203745964",
  "pdf_url": "https://arxiv.org/pdf/2608.17383",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dfavtx0arfvff.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.17383",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.17383"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations",
  "title_text": "Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/dfavtx0arfvff",
  "venue": "arXiv",
  "versions": []
}