{
  "abstract": "We consider distributional weak solutions $u\\in L^\\infty_tL^2_x$ of the two-dimensional Navier--Stokes equations with horizontal viscosity on $\\mathbb T^2$ or $\\mathbb R^2$, without assuming $\\partial_1u\\in L^2_tL^2_x$. We show that the Lions integrability condition $u\\in L^4_tL^4_x$ together with Onsager-critical regularity in the nondissipative direction, $u\\in L^3_tB^{1/3,v}_{3,\\infty}$ with the Besov regularity imposed only in the vertical direction, implies $\\partial_1u\\in L^2_tL^2_x$, with a quantitative bound on the horizontal dissipation. Thus the dissipative regularity is a consequence of the equation and need not be part of the definition of the solution. If moreover $u\\in L^3_tB^{1/3,v}_{3,c_0}$, then $u$ belongs to the isotropic critical Onsager space $L^3_tB^{1/3}_{3,c_0}$, satisfies the energy equality, and is continuous in $L^2$. Here $c_0$ indicates that the corresponding dyadic Besov sequence tends to zero at high frequencies. The proof combines absorption of the horizontal energy flux into the dissipation with a one-dimensional commutator estimate for the vertical energy flux.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11488",
  "author_total": 1,
  "authors": [
    {
      "name": "Siyu Liang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9fchnpn3pbwk1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11488",
  "fulltext_url": null,
  "id": "9fchnpn3pbwk1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11488",
  "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": "W7212287504",
  "pdf_url": "https://arxiv.org/pdf/2609.11488",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9fchnpn3pbwk1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11488",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11488"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "An Anisotropic Onsager Criterion for the Two-Dimensional Navier--Stokes Equations with Horizontal Viscosity",
  "title_text": "An Anisotropic Onsager Criterion for the Two-Dimensional Navier--Stokes Equations with Horizontal Viscosity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/9fchnpn3pbwk1",
  "venue": "arXiv",
  "versions": []
}