{
  "abstract": "We prove a critical Morrey rigidity theorem for the five-dimensional stationary Navier--Stokes equations. More precisely, every smooth solution on $\\mathbb R^5\\setminus\\{0\\}$ satisfying \\[ \\sup_{R>0}R^{-2}\\int_{B_R}|u|^3\\,dx<\\infty \\] is identically zero, up to an additive constant in the pressure. This replaces the pointwise Type-I control in the known higher-dimensional rigidity theory by a velocity-only, scale-invariant averaged condition that allows spatial concentration. The proof develops a weak head-pressure mechanism that does not rely on pointwise pressure estimates or classical normal traces. We reconstruct a canonical pressure from the velocity, derive a renormalized inequality for the positive head pressure, and introduce two monotone radial fluxes. Annular energy estimates, suitable-weak compactness, and blow-up and blow-down limits are then used to identify the endpoint fluxes and force rigidity. As an application, we obtain a removable-singularity criterion in dimension five: if a suitable weak solution is smooth away from one point and either its scale-invariant Dirichlet energy or its cubic velocity Morrey quantity remains bounded near that point, then the singularity is removable. Thus, within the isolated-singularity class, the smallness assumption in the classical stationary regularity criterion is replaced by boundedness. We also prove the corresponding velocity-only cubic Morrey rigidity theorem in dimension four by a different finite-energy argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29664",
  "author_total": 5,
  "authors": [
    {
      "name": "Yubo Chen"
    },
    {
      "name": "Wendong Wang"
    },
    {
      "name": "Xiao Wang"
    },
    {
      "name": "Guoxu Yang"
    },
    {
      "name": "Jianbo Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9kw3e3070t4pz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29664",
  "fulltext_url": null,
  "id": "9kw3e3070t4pz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29664",
  "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": "W7204909102",
  "pdf_url": "https://arxiv.org/pdf/2608.29664",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9kw3e3070t4pz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29664",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29664"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows",
  "title_text": "Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/9kw3e3070t4pz",
  "venue": "arXiv",
  "versions": []
}