{
  "abstract": "We consider the Cauchy problem for the 2D inhomogeneous incompressible Euler equations. We show that the Cauchy problem is locally well-posed in a class of $m$-fold symmetric functions at critical regularity and allowing the velocity field to grow linearly at spatial infinity. This was a problem posed by P.L. Lions in the homogeneous, not necessarily symmetric case. A key ingredient is a variable-coefficient elliptic theory developed using quasiconformal mappings and Schauder estimates in certain scale-invariant Hölder spaces. Local well-posedness allows us to derive the one-dimensional system on $\\mathbb{S}^1$ satisfied by solutions with $0$-homogeneous vorticity and density. For this system, we can obtain global well-posedness for a class of initial data away from equilibrium. We are unaware of any previous nonperturbative class of time-dependent solutions to the two-dimensional inhomogeneous Euler equations for which global well-posedness has been established. Finally, for some signed global solutions, we show a dichotomy between density homogenization and averaged relaxation in infinite time.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30096",
  "author_total": 1,
  "authors": [
    {
      "name": "Ibrahim Suleiman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5pjxcs1vqa3mp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30096",
  "fulltext_url": null,
  "id": "5pjxcs1vqa3mp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30096",
  "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": "W7223251497",
  "pdf_url": "https://arxiv.org/pdf/2610.30096",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5pjxcs1vqa3mp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.30096",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30096"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A well-posedness theory for the 2D inhomogeneous incompressible Euler equations and a class of global-in-time solutions",
  "title_text": "A well-posedness theory for the 2D inhomogeneous incompressible Euler equations and a class of global-in-time solutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/5pjxcs1vqa3mp",
  "venue": "arXiv",
  "versions": []
}