{
  "abstract": "We study the linearized dynamics of the Kolmogorov flow $U^{(0)}=(a\\sin x_2,0)$ for the two-dimensional incompressible Navier--Stokes equations with horizontal dissipation $ν\\partial_1^2$. Unlike the fully dissipative case, this anisotropic system admits $U^{(0)}$ as an exact unforced steady state. On each horizontal Fourier mode $k$, the dissipation reduces to the scalar $-νk^2$ and commutes with the advection, so that the linearized semigroup factors exactly into $e^{-νk^2t}$ times the inviscid Euler group. For $|k|>1$, we prove two-sided bounds showing that the decay rate is exactly $νk^2$, uniformly in the shear amplitude $a$; hence no enhanced dissipation occurs. This reflects a fundamental mismatch: the shear transfers enstrophy to high vertical frequencies, which horizontal dissipation does not detect. At the critical modes $|k|=1$, the inviscid group grows like $\\sqrt{2at}$, producing a transient amplification of size $(a/eν)^{1/2}$ before decay at the rate $ν$. The horizontally independent modes form an infinite-dimensional undamped kernel. For $0<|k|<1$, the viscous spectrum is an exact translate of the inviscid one, and the mode is linearly unstable if and only if $aΛ(k)>νk^2$. In particular, when $L>2π$, the flow is linearly unstable for all sufficiently large $a$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34610",
  "author_total": 3,
  "authors": [
    {
      "name": "Wanrong Yang"
    },
    {
      "name": "Jiahong Wu"
    },
    {
      "name": "Xiaoping Zhai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46tja4ee4ybs9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34610",
  "fulltext_url": null,
  "id": "46tja4ee4ybs9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34610",
  "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",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7214705817",
  "pdf_url": "https://arxiv.org/pdf/2609.34610",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46tja4ee4ybs9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34610",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34610"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation",
  "title_text": "Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/46tja4ee4ybs9",
  "venue": "arXiv",
  "versions": []
}