{
  "abstract": "Recently, the existence of regular solutions $(u,p)$ to the stationary incompressible Navier--Stokes equations on $\\mathbb R^n$ with bounded compactly supported forces $f$ for $5\\le n\\le15$ was established by Li--Yang (Comm. Math. Phys., 2022). However, it is still unknown whether there exists a regular solution for dimensions $n>15$. Here we answer this question and obtain the existence of regular solutions to the stationary Navier--Stokes equations in arbitrarily high dimensions. The key innovation is to combine an averaged signed pressure-potential identity with $L^2$-BMO estimates for the Bernoulli source flux, yielding a Bernoulli supremum bound independent of the size of the skew coefficient representing the drift. Together with the energy estimate for the comparison solution, this yields a sublinear bound for the Newtonian potential of the positive Bernoulli function, removing the upper dimension restriction in the earlier existence argument. We also construct periodic solutions and whole-space solutions for forces in $L^\\infty\\cap L^{2n/(n+2)}$. For compactly supported forces, we record two far-field profiles, their velocity, gradient and pressure remainders, and the corresponding cancellation criteria.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24206",
  "author_total": 2,
  "authors": [
    {
      "name": "Wendong Wang"
    },
    {
      "name": "Guoxu Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0mbabx6p974vb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24206",
  "fulltext_url": null,
  "id": "0mbabx6p974vb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24206",
  "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": "W7213996418",
  "pdf_url": "https://arxiv.org/pdf/2609.24206",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0mbabx6p974vb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24206",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24206"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Regular Solutions of the Stationary Navier-Stokes Equations in Arbitrarily High Dimensions",
  "title_text": "Regular Solutions of the Stationary Navier-Stokes Equations in Arbitrarily High Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/0mbabx6p974vb",
  "venue": "arXiv",
  "versions": []
}