{
  "abstract": "We investigate forward self-similar solutions to the incompressible Navier--Stokes equations with homogeneous initial data $u_0(x)=σ(x/|x|)/|x|$, $x\\in\\mathbb R^n$, focusing on the critical four-dimensional case. For the existence theory in dimensions $3\\leq n\\leq5$, we develop a Galerkin scheme for the Leray profile problem, featuring a decomposition of the self-similar data and nonlocal heat kernel effects across spatial scales. Exploiting the structural properties of the data and the coercivity of the Leray operator, we derive uniform \\textit{a priori} estimates, pass to the limit, and obtain weak forward self-similar solutions. In the 4D case, we prove new Stokes-type estimates for the linear Leray system. The inverse Leray operator admits an integral representation as a singular integral operator; its $L^p$-boundedness follows from Calderón--Zygmund theory. Combined with a 4D compactness estimate for the convective term, this closes the regularity iteration. For spatial decay, we reformulate the problem in weighted spaces. The non-integrability of the low-frequency kernel causes a logarithmic loss under $σ\\in W^{1,\\infty}(\\mathbb S^3)$. With $σ\\in C^{1,γ}(\\mathbb S^3)$, $0<γ<1$, frequency-localized estimates remove this loss and yield sharp decay.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33316",
  "author_total": 3,
  "authors": [
    {
      "name": "Baishun Lai"
    },
    {
      "name": "Changxing Miao"
    },
    {
      "name": "Xiaoxin Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vmsynmte69e7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33316",
  "fulltext_url": null,
  "id": "5vmsynmte69e7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33316",
  "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": "W7214769389",
  "pdf_url": "https://arxiv.org/pdf/2609.33316",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vmsynmte69e7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33316",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33316"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Forward self-similar solutions of the Navier-Stokes equations in R^n",
  "title_text": "Forward self-similar solutions of the Navier-Stokes equations in R^n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/5vmsynmte69e7",
  "venue": "arXiv",
  "versions": []
}