{
  "abstract": "We prove that strong solutions to the incompressible, homogeneous Navier-Stokes system on $\\mathbb{R}^3$ are global in time for initial data in the Sobolev space $H^1_σ(\\mathbb{R}^3)$. The central ingredient of the proof is a new energy estimate that rules out finite-time blow-up in $L^r(\\mathbb{R}^3)$ for $r = 2 + 2 / \\sqrt{3}$. This estimate is based crucially on properties of the material derivative in combination with the divergence-free constraint, together with an intricate estimate that harnesses the delicate interplay between the viscous coercivity of the new energy functional and the control of a perturbing pressure term. Notably, the estimate does not carry over to systems that merely share the standard energy structure, such as T. Tao's averaged Navier-Stokes system. As a consequence of the new energy estimate, every Leray-Hopf weak solution $u$ to the homogeneous Navier-Stokes system is globally unique whenever the initial value $u(0, \\cdot)$ belongs to $L^2_σ(\\mathbb{R}^3) \\cap L^3(\\mathbb{R}^3)$. Even for $u(0, \\cdot) \\in L^2_σ(\\mathbb{R}^3)$, the only possible non-uniqueness in the class of Leray-Hopf weak solutions is initial branching, and every such solution is $C^\\infty$-smooth on $\\left( 0, \\infty \\right) \\times \\mathbb{R}^3$. Moreover, if $u(0, \\cdot)$ is smooth with derivatives of all orders decaying rapidly at infinity, we show that $u$ is in fact $C^\\infty$-smooth on all of $\\left[ 0, \\infty \\right) \\times \\mathbb{R}^3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18808",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Ruf"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/50ypdfnj0yrfg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18808",
  "fulltext_url": null,
  "id": "50ypdfnj0yrfg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18808",
  "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-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7213512435",
  "pdf_url": "https://arxiv.org/pdf/2609.18808",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/50ypdfnj0yrfg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:19Z",
      "native_id": "2609.18808",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18808"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "A very short proof of a new energy identity for strong solutions to the Navier-Stokes system in 3D",
  "title_text": "A very short proof of a new energy identity for strong solutions to the Navier-Stokes system in 3D",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/50ypdfnj0yrfg",
  "venue": "arXiv",
  "versions": []
}