{
  "abstract": "We prove a rigidity theorem for stationary and uniformly rotating solutions of the two-dimensional incompressible Euler equation with vorticity $ ω_0\\in C^1(\\mathbb{R}^2)\\cap L^1(\\mathbb{R}^2) \\cap L^\\infty(\\mathbb{R}^2). $ If the angular velocity $Ω$ satisfies \\[ Ω\\leq\\frac12\\inf_{\\mathbb{R}^2}ω_0 \\qquad\\text{or}\\qquad Ω\\geq\\frac12\\sup_{\\mathbb{R}^2}ω_0, \\] then $ω_0$ is radially symmetric, in particular, every stationary vorticity of one sign in the class $C^1(\\mathbb{R}^2)\\cap L^1(\\mathbb{R}^2) \\cap L^\\infty(\\mathbb{R}^2)$ must be radially symmetric about some point. Furthermore, for $Ω\\neq0$, the center is necessarily the origin; The proof is based on the analysis for level-set geometry of a normalized stream function. A componentwise Bernoulli formula, together with the isoperimetric inequality and Pohozaev identities, yields a nonnegative defect measure encoding both geometric defects and topological branching. An analysis at infinity forces this measure to vanish, reducing the problem to a semilinear elliptic equation with a bounded nonnegative Borel nonlinearity. Radial symmetry then follows from a theorem of P.-L.\\ Lions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30664",
  "author_total": 4,
  "authors": [
    {
      "name": "Boquan Fan"
    },
    {
      "name": "Yuchen Wang"
    },
    {
      "name": "Chunjing Xie"
    },
    {
      "name": "Weicheng Zhan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2241tkk2c317k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30664",
  "fulltext_url": null,
  "id": "2241tkk2c317k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30664",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.30664",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2241tkk2c317k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30664",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30664"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Rigidity of stationary and uniformly rotating planar Euler flows in the $C^1$ Yudovich class",
  "title_text": "Rigidity of stationary and uniformly rotating planar Euler flows in the C^1 Yudovich class",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/2241tkk2c317k",
  "venue": "arXiv",
  "versions": []
}