{
  "abstract": "We prove that the solution to the Córdoba-Córdoba-Fontelos equation $θ_t-uθ_x=0$, $u=Hθ$, on the real line with analytic initial datum $θ_0(x)=(1+x^2)^{-1}$ develops a finite-time singularity while remaining uniformly bounded in $C^{2/5}$. We also prove that, as the solution approaches the singular time, the $C^{2/3}$ seminorms of both $θ$ and $u$ diverge. The main ingredient is an integral representation in terms of a probability measure that is preserved by the nonlinear evolution. From this representation we derive a pointwise quadratic estimate for the key nonlinear term in the equation for $u$, using a comparison of positive kernels. Together with evolution identities for the moments of the measure, this estimate yields both the uniform Hölder bound and the divergence of the higher seminorms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37232",
  "author_total": 3,
  "authors": [
    {
      "name": "Ángel Castro"
    },
    {
      "name": "Alberto Enciso"
    },
    {
      "name": "Antonio J. Fernández"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f9hc8hqqv51mb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37232",
  "fulltext_url": null,
  "id": "f9hc8hqqv51mb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37232",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7214952420",
  "pdf_url": "https://arxiv.org/pdf/2609.37232",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f9hc8hqqv51mb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37232",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37232"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hölder estimates up to the singular time for the Córdoba-Córdoba-Fontelos equation",
  "title_text": "Hölder estimates up to the singular time for the Córdoba-Córdoba-Fontelos equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/f9hc8hqqv51mb",
  "venue": "arXiv",
  "versions": []
}