{
  "abstract": "We prove global nonlinear stability near the mass Kolmogorov--Zakharov spectrum $f_M(ω)=ω^{-7/6}$ for the isotropic cubic wave kinetic equation, including small time-dependent external forces. For sufficiently small initial perturbations and forcing, the solution remains uniformly close to $f_M$ for all time in a weighted norm. This norm allows $f(t,ω)\\sim a_0(t)ω^{-7/6}$ as $ω\\to0$ and $f(t,ω)\\sim a_\\infty(t)ω^{-7/6}$ as $ω\\to\\infty$. When the external force $ϕ$ is independent of time, the solution converges at an algebraic rate to a stationary solution in weaker weighted norms. We also give a necessary and sufficient condition on the initial perturbation for convergence in the original weighted norm. These results resolve, for small perturbations of the Kolmogorov--Zakharov spectrum, the question of convergence to a stationary solution under time-independent forcing raised by Collot, Dietert, and Germain (arXiv:2208.00947, Section 1.3). To establish this stability, the collision integral is expanded around $a_0(t)f_M$ at low frequencies and $a_\\infty(t)f_M$ at high frequencies, separating it into a linear term with time-dependent coefficients and a nonlinear remainder. This decomposition yields improved control of the nonlinear remainder. The main difficulty is that $a_0(t)$ may vary in time, so the linear operator itself changes with time and its evolution cannot be represented by the semigroup of a single fixed operator. To address this difficulty, we use a parametrix construction to explicitly construct the evolution operator $U_{a_0,a_\\infty}$ and establish the required uniform bounds. Moreover, our arguments provide a general framework for proving global stability that may also apply to other evolution equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34359",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiajun Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ka1phsmecq0a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34359",
  "fulltext_url": null,
  "id": "9ka1phsmecq0a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34359",
  "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",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7223890357",
  "pdf_url": "https://arxiv.org/pdf/2610.34359",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ka1phsmecq0a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34359",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34359"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Global Stability of the Kolmogorov--Zakharov Spectrum for the Wave Kinetic Equation",
  "title_text": "Global Stability of the Kolmogorov--Zakharov Spectrum for the Wave Kinetic Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/9ka1phsmecq0a",
  "venue": "arXiv",
  "versions": []
}