{
  "abstract": "We establish nonlinear stability of three-dimensional Couette flow for the incompressible Navier--Stokes equations on $\\mathbb T\\times\\mathbb R^2$. Divergence-free perturbations satisfying $\\|u_{\\rm in}\\|_{H^2}+\\|u^\\perp_{0,\\rm in}\\|_{\\dot B^{-2}_{2,\\infty}}\\leq c_*ν$ generate unique global $H^2$ solutions that remain close to Couette flow, where $u^\\perp_{0,\\rm in}$ denotes the streamwise average of the transverse velocity. This result retains the $O(ν)$ stability threshold despite the absence of a spanwise spectral gap. The nonzero streamwise modes exhibit enhanced dissipation, their wall-normal component satisfies a viscosity-uniform time-integrated inviscid-damping estimate, and the transverse zero mode decays in $L^2$ at rate $(1+νt)^{-1}$. The endpoint space $\\dot B^{-2}_{2,\\infty}$ controls the transverse low frequencies, accommodates divergence-form nonlinear forcing, and is critical for linear lift-up from the wall-normal zero mode. The proof combines propagation of this endpoint norm, Fourier splitting, and a two-level roll-to-streak representation with an adaptation of the nonzero-mode framework of Wei--Zhang (CPAM, 2021).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08288",
  "author_total": 3,
  "authors": [
    {
      "name": "Yubo Chen"
    },
    {
      "name": "Guoxu Yang"
    },
    {
      "name": "Jianbo Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/11n05n1qafk29.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08288",
  "fulltext_url": null,
  "id": "11n05n1qafk29",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08288",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7220951671",
  "pdf_url": "https://arxiv.org/pdf/2610.08288",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/11n05n1qafk29.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08288",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08288"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Nonlinear stability of 3D Couette flow on $\\mathbb T\\times\\mathbb R^2$ under critical transverse low-frequency control",
  "title_text": "Nonlinear stability of 3D Couette flow on 𝕋×ℝ^2 under critical transverse low-frequency control",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/11n05n1qafk29",
  "venue": "arXiv",
  "versions": []
}