{
  "abstract": "We establish maximal local $L^p$-well-posedness, for $p>3$, for the stochastic tamed Navier--Stokes equations on $\\mathbb R^3$ driven simultaneously by multiplicative cylindrical Wiener noise and a compensated Poisson random measure. For divergence-free initial data $ u_0\\in L^p\\bigl(Ω,\\mathcal F_0; L^p(\\mathbb R^3;\\mathbb R^3)\\bigr), $ we prove existence and pathwise uniqueness of a local strong solution with $L^p$-valued càdlàg trajectories and the local $L^p$-energy regularity. For solutions driven by the same noises, we establish localized Lipschitz dependence on the initial datum in the path supremum and space--time norms. The discontinuous forcing makes the whole-space Gaussian construction non formal as stopping levels may be overshot by jumps, while convergence of the compensated-Poisson term requires simultaneous control of its quadratic and $p$th integrability modes. We resolve these difficulties by a jump-compatible localization based on strict pre-exit bounds and predictable left limits. We further establish bounded-time restart and stochastic pasting on the prescribed stochastic basis. The resulting family of attainable lifetimes is upward directed and yields a unique maximal local strong solution, which inherits the localized dependence estimate. Continuation criteria, blow-up alternatives, and global well-posedness under stronger finite-energy hypotheses are treated in the companion Part~II.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.03734",
  "author_total": 2,
  "authors": [
    {
      "name": "Bikram Podder"
    },
    {
      "name": "Surendra Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6ars80hkb2rem.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.03734",
  "fulltext_url": null,
  "id": "6ars80hkb2rem",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.03734",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7160446995",
  "pdf_url": "https://arxiv.org/pdf/2605.03734",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6ars80hkb2rem.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2605.03734",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.03734"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Stochastic Tamed Navier--Stokes Equations with Wiener and Jump Noise on $\\mathbb R^3$. I. Maximal Local $L^p$-Well-Posedness",
  "title_text": "Stochastic Tamed Navier--Stokes Equations with Wiener and Jump Noise on ℝ^3. I. Maximal Local L^p-Well-Posedness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/6ars80hkb2rem",
  "venue": "arXiv",
  "versions": []
}