{
  "abstract": "This paper studies time-dependent semi-similar Prandtl flows with outer flow $U(x)=x^m$, where $m\\geq1$. The normal coordinate is scaled by the local boundary-layer thickness, and time by the associated diffusion scale. Every time-monotone solution in the Oleinik class extends globally in time. The Falkner--Skan boundary layer globally attracts every such unsteady semi-similar profile. The convergence rate is exponential for $m=1$ and algebraic for $m>1$. We also prove nonlinear stability of each admissible time-dependent semi-similar trajectory under small compatible perturbations that break the semi-similar ansatz. Together, these results provide a dynamical justification of the time-dependent semi-similar class within the full Prandtl theory. For semi-similar flows, the Crocco transformation reduces the problem to a one-dimensional degenerate parabolic equation. Combined with weighted barriers, comparison principles propagate time monotonicity and lead to global continuation. Zero-order convergence and a one-sided gradient estimate then produce uniform strict concavity after a transient period; this concavity supplies the damping needed for derivative decay. The same damping mechanism extends to the full ultraparabolic perturbation equation, allowing us to close the nonlinear stability estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25807",
  "author_total": 5,
  "authors": [
    {
      "name": "Chen Gao"
    },
    {
      "name": "Tao Tao"
    },
    {
      "name": "Zhouping Xin"
    },
    {
      "name": "Shing-Tung Yau"
    },
    {
      "name": "Chuankai Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5d30mhs9g9mxn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25807",
  "fulltext_url": null,
  "id": "5d30mhs9g9mxn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25807",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7222996719",
  "pdf_url": "https://arxiv.org/pdf/2610.25807",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5d30mhs9g9mxn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25807",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25807"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Semi-similar Prandtl Flows near a Stagnation Point: Dynamical Selection and Nonlinear Stability",
  "title_text": "Semi-similar Prandtl Flows near a Stagnation Point: Dynamical Selection and Nonlinear Stability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/5d30mhs9g9mxn",
  "venue": "arXiv",
  "versions": []
}