{
  "abstract": "We prove strong ill-posedness for the two-dimensional incompressible Euler equation in the endpoint velocity space $W^{3,1}_σ(\\mathbb R^2)$. Given any compactly supported smooth divergence-free background and any $δ>0$, we construct a single compactly supported divergence-free datum within $δ$ of the background in $W^{3,1}$. The perturbation can be localized inside any prescribed nonempty open neighborhood of the background support while remaining disjoint from that support. The associated unique global finite-energy solution $u$ satisfies ${\\rm curl} u\\in C([0,\\infty);W^{2,1}(\\mathbb R^2))$, but \\[ \\operatorname*{ess\\,sup}_{0<t<T}\\|u(t)\\|_{W^{3,1}}=\\infty \\qquad\\text{for every }T>0. \\] The proof combines localized hyperbolic insertion, the Bonami--Poornima nonmultiplier mechanism, and many-packet dilution. Moment cancellation and positive material separation control the nonlocal Biot--Savart interactions, allowing a nested construction inside one fixed compact region. Without the compact-support requirement on the limiting datum, a complementary far-field argument yields a dense $G_δ$ set of data whose solutions leave $W^{3,1}$ at every positive rational time and on a residual set of times, with unbounded essential supremum on every nonempty open time interval.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04872",
  "author_total": 3,
  "authors": [
    {
      "name": "Yingzhe Ban"
    },
    {
      "name": "Maotuo Guo"
    },
    {
      "name": "Qi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8wmqa7ncvynmm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04872",
  "fulltext_url": null,
  "id": "8wmqa7ncvynmm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04872",
  "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": "W7220699284",
  "pdf_url": "https://arxiv.org/pdf/2610.04872",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8wmqa7ncvynmm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04872",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04872"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Compactly Supported Strong Ill-Posedness of the Two-Dimensional Euler Flow in $W^{3,1}(\\mathbb R^2)$",
  "title_text": "Compactly Supported Strong Ill-Posedness of the Two-Dimensional Euler Flow in W^3,1(ℝ^2)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/8wmqa7ncvynmm",
  "venue": "arXiv",
  "versions": []
}