{
  "abstract": "We prove local well-posedness in the Hadamard sense for the three-dimensional compressible Euler equations governing a liquid with a free surface and no surface tension. Under the Taylor sign condition and the boundary compatibility conditions, we establish existence, uniqueness, and strong continuous dependence in a hybrid Sobolev class for every $s>3$. The velocity and interface have $H^s$ regularity, while the enthalpy and velocity divergence belong to $H^{s+1/2}$ and $H^{s-1/2}$, respectively. This separation reflects the coupling of a nondegenerate acoustic equation with the free-surface and vorticity dynamics. The proof uses regularization operators that smooth the moving domain and extend the nonlinear compatibility hierarchy, together with $L^2$ and partial $H^2$ distance estimates for solutions on different domains. Regular solutions are constructed by a regularized forward Euler scheme. High-order energy estimates and frequency envelopes then yield strong convergence of smooth approximations and continuity of the data-to-solution map. Integer Sobolev indices are treated using a Lions--Magenes endpoint compatibility condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40163",
  "author_total": 2,
  "authors": [
    {
      "name": "Tao Luo"
    },
    {
      "name": "Ruixi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/de61c5cycs6jj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40163",
  "fulltext_url": null,
  "id": "de61c5cycs6jj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40163",
  "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": "W7215059297",
  "pdf_url": "https://arxiv.org/pdf/2609.40163",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/de61c5cycs6jj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40163",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40163"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hadamard Local Well-Posedness for Compressible Liquids with a Free Surface",
  "title_text": "Hadamard Local Well-Posedness for Compressible Liquids with a Free Surface",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/de61c5cycs6jj",
  "venue": "arXiv",
  "versions": []
}