{
  "abstract": "The global well-posedness for Lions' weak solutions to the inhomogeneous incompressible Navier-Stokes equations in two-dimensional bounded domains is concerned. When the initial density $0\\leq ρ_0\\in L^\\infty(\\Om)$ (allowing for vacuum) and initial velocity $u_0\\in L^2_σ(Ω)$, the global existence, regularity, and exponential decay-in-time rates of weak solutions are established via a priori estimates, which rely crucially on the duality of BMO and Hardy space for the pressure term, and the classical Sobolev extension theorem. In particular, for the non-vacuous case where $0< ρ_0\\in L^\\infty(\\Om)$ is bounded away from zero, we further prove the uniqueness of weak solutions in the sense of approximation by strong solutions, based on the key estimate \\(t^{1/2}\\nabla u\\in L^2(0,T;L^\\infty(Ω))\\), which essentially follows from a delicate localized point-wise bound for \\(\\nabla u\\) near the boundary. The latter bound is derived by the Green functions estimate for the nonhomogeneous Dirichlet Stokes problem, and the uniform \\(C^1\\) boundary estimate on the remaining homogeneous Stokes problem, through a Stokes decomposition on suitably constructed local \\(C^2\\) domains.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41637",
  "author_total": 1,
  "authors": [
    {
      "name": "Hongyu Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29v3erqtsdkjc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41637",
  "fulltext_url": null,
  "id": "29v3erqtsdkjc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41637",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41637",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29v3erqtsdkjc.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41637",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41637"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Global existence and uniqueness of Lions' weak solutions for the 2D inhomogeneous Navier-Stokes equations in bounded domains",
  "title_text": "Global existence and uniqueness of Lions' weak solutions for the 2D inhomogeneous Navier-Stokes equations in bounded domains",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/29v3erqtsdkjc",
  "venue": "arXiv",
  "versions": []
}