{
  "abstract": "We study two-dimensional steady incompressible Euler flows with finite total curvature. A distinguished family of such flows arises from finite-Morse-index solutions to semilinear elliptic equations in two dimensions. We establish three main properties of finite-total-curvature flows in the $L^\\infty\\cap H^1_{\\rm loc}$ class. First, in the plane and the half-plane, every such flow without stagnation points is a parallel shear flow. Second, in the plane, the half-plane, and the infinite strip, the pressure converges uniformly to a constant at each end at infinity. Third, in the plane, the half-plane, the infinite strip, and the periodic strip, we establish sharp lower bounds for the total curvature in terms of the pressure oscillation. The rigidity result in fact extends to a broader class of flows: in the plane and the half-plane, any steady Euler flow without stagnation points is a parallel shear flow provided that $\\nabla P\\in L^q$ for some $1\\le q\\le 2$. Combined with standard elliptic estimates for the pressure, this result yields a concise new proof of Hamel and Nadirashvili's rigidity theorems, with the regularity assumption sharpened optimally to $H^1_{\\rm loc}$. The pressure serves as a unifying quantity throughout the analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32647",
  "author_total": 4,
  "authors": [
    {
      "name": "Changfeng Gui"
    },
    {
      "name": "Qinfeng Li"
    },
    {
      "name": "Chunjing Xie"
    },
    {
      "name": "Huan Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ne990dttwf9z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32647",
  "fulltext_url": null,
  "id": "5ne990dttwf9z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32647",
  "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": "W7214739141",
  "pdf_url": "https://arxiv.org/pdf/2609.32647",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5ne990dttwf9z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32647",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32647"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Pressure estimates and classifications for steady $H^1_{\\rm loc}$-solutions to two-dimensional incompressible Euler equations",
  "title_text": "Pressure estimates and classifications for steady H^1_loc-solutions to two-dimensional incompressible Euler equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/5ne990dttwf9z",
  "venue": "arXiv",
  "versions": []
}