{
  "abstract": "For a $d$-dimensional closed hypersurface in $\\mathbb{R}^{d+1}$, we reformulate the (stationary) tangent Stokes, transient tangent Stokes, and transient tangent Navier-Stokes equations by replacing the divergence equation with a pressure Poisson equation, thereby circumventing their saddle-point structure. The pressure Poisson equation hinges on a commutator identity between the divergence and the surface diffusion operator, whose lower-order term depends on the Ricci curvature, and requires no boundary conditions for the pressure. We show that the reformulated operator is the composition of a linear, block-triangular data map with the primal operator. Consequently, each reformulation is well-posed if and only if the primal problem is, and solving it with arbitrary data amounts to solving the primal problem with a modified divergence datum. For hypersurfaces of class $C^2$ and data of the pressure Poisson equation in $H^{-1}$, the tangent Stokes reformulations can be posed in $H^1$-based spaces, which paves the way for conforming discretizations that do not require a discrete inf-sup condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36641",
  "author_total": 2,
  "authors": [
    {
      "name": "Ricardo H. Nochetto"
    },
    {
      "name": "Mansur Shakipov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fgca8yjan8qbj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36641",
  "fulltext_url": null,
  "id": "fgca8yjan8qbj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36641",
  "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",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7224123893",
  "pdf_url": "https://arxiv.org/pdf/2610.36641",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fgca8yjan8qbj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36641",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36641"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Fluid Problems Posed on Closed Hypersurfaces: Circumventing the Saddle-Point Structure",
  "title_text": "Fluid Problems Posed on Closed Hypersurfaces: Circumventing the Saddle-Point Structure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/fgca8yjan8qbj",
  "venue": "arXiv",
  "versions": []
}