{
  "abstract": "We consider the incompressible Navier-Stokes equations on the torus $\\mathbb{T}^d$, $d \\ge 2$, with translation invariant Gaussian random initial data. We show that, for the Gaussian measure formally given by $μ_0 \\sim \\exp(-\\frac12 \\| u \\|_{\\dot H^α}^2)$ on divergence-free fields, its nonlinear evolution $ν_t$ and its linear evolution $μ_t$ are mutually singular for almost every time. Moreover, the same holds for the nonlinear evolution of an arbitrary initial measure $ν_0$. This is in sharp contrast to the available results for Hamiltonian PDEs, where Gaussian measures are commonly quasi-invariant. We achieve this by showing that there exists a set $\\mathscr G_t$ such that $μ_t(\\mathscr G_t) = 1$, but for any smooth solution of Navier-Stokes, the set of times $t$ such that $u(t) \\in \\mathscr G_t$ is null, which we deduce from a quantitative convexity bound for frequency-localised versions of the (typically infinite) quantity $\\| \\mathrm{e}^{-t Δ} u(t) \\|_{L^2}^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23591",
  "author_total": 2,
  "authors": [
    {
      "name": "Niko Nikov"
    },
    {
      "name": "Leonardo Tolomeo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f3hjsdvh382hd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23591",
  "fulltext_url": null,
  "id": "f3hjsdvh382hd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23591",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7222980597",
  "pdf_url": "https://arxiv.org/pdf/2610.23591",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f3hjsdvh382hd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23591",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23591"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Gaussian measures under the Navier-Stokes and heat flows are mutually singular",
  "title_text": "Gaussian measures under the Navier-Stokes and heat flows are mutually singular",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/f3hjsdvh382hd",
  "venue": "arXiv",
  "versions": []
}