{
  "abstract": "We develop a direct paradifferential approach to nonlinear graph integral operators arising in contour dynamics. The main methodological result is an abstract structure-preserving paralinearization theorem, valid in arbitrary dimension, for a class of homogeneous kernels whose nonlinear dependence on the graph is expressed through normalized finite differences. The theorem covers both critical principal-value kernels and subcritical locally integrable kernels, gives explicit formulas for the principal symbols, and separates the finite-order symbolic contribution generated by the homogeneous singularity from order-zero far-field terms and arbitrarily smoothing remainders. A structural feature of the reduction is the cancellation of the intermediate symbolic order; at the critical endpoint the construction also preserves the exact skew-adjoint structure of the original singular integral. As a main application, we consider the three-dimensional two-phase free-boundary Euler equations with constant interfacial background vorticity. We first derive an autonomous contour-dynamics equation in canonical surface variables and then apply the abstract theorem directly to the singular integral operators generated by the Birkhoff--Rott formulation. This yields the complete paradifferential structure of the system without taking the Dirichlet--Neumann reduction as a starting point. After introducing an Alinhac good unknown and diagonalizing the resulting system, we obtain local well-posedness for small Sobolev perturbations in the stable Kelvin--Rayleigh--Taylor regime, uniformly on compact subsets of the stable parameter set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03091",
  "author_total": 3,
  "authors": [
    {
      "name": "Xingyu Li"
    },
    {
      "name": "Emeric Roulley"
    },
    {
      "name": "Stefano Scrobogna"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9kz9qr1354mys.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03091",
  "fulltext_url": null,
  "id": "9kz9qr1354mys",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03091",
  "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": "W7220363386",
  "pdf_url": "https://arxiv.org/pdf/2610.03091",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9kz9qr1354mys.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03091",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03091"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Singular Contour Dynamics and Paradifferential Reduction",
  "title_text": "Singular Contour Dynamics and Paradifferential Reduction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/9kz9qr1354mys",
  "venue": "arXiv",
  "versions": []
}