{
  "abstract": "We prove finite-time gradient blow-up for the unforced incompressible porous media equation on the two-dimensional torus, removing the forcing used in the IPM blow-up constructions of Alpöge, Buckmaster, and Coiculescu \\cite{lit:ABC} and Córdoba and Martínez-Zoroa \\cite{lit:CMIPM}. The key is an identity that expresses the logarithmic growth of a layer's gradient as the negative pressure curvature transverse to its transported phase. A nearly vertical parent layer rotates and amplifies a new phase, with a gain exponential in the reciprocal parent tilt. An asymmetric periodic profile gives a uniform upper bound for the pressure Hessian, preventing rapid decay of the gradient magnitude. Each layer is seeded at time zero; its phase means are selected by a terminal displacement problem, and a finite order expansion is corrected to an exact unforced solution. The scale induction then shows blow-up of two derivative components at the origin and convergence to a terminal density in every $C^η$, $η<1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06544",
  "author_total": 1,
  "authors": [
    {
      "name": "Fan Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ct8k4wz3acm4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06544",
  "fulltext_url": null,
  "id": "0ct8k4wz3acm4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06544",
  "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": "W7220738691",
  "pdf_url": "https://arxiv.org/pdf/2610.06544",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ct8k4wz3acm4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06544",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06544"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Finite-time blowup of the IPM equation from smooth initial data without forcing",
  "title_text": "Finite-time blowup of the IPM equation from smooth initial data without forcing",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/0ct8k4wz3acm4",
  "venue": "arXiv",
  "versions": []
}