{
  "abstract": "We prove linear ill-posedness in tangential Sobolev spaces for the three-dimensional resistive MHD boundary layer equations. We assume that a suitable linear combination of the initial tangential velocity components has a non-degenerate critical point and that both initial tangential magnetic components have a double zero at the same point. Compared with the corresponding two-dimensional MHD result, our construction requires only this double-zero condition, rather than a higher-order degeneracy of the magnetic shear. The proof combines the three-dimensional Prandtl critical layer construction with a vector correction to the two-dimensional magnetic quotient. This correction cancels the leading terms in both induction equations while preserving the magnetic divergence constraint. The resulting approximate solutions grow like $\\exp(σt/\\sqrt{\\varepsilon})$, and their residuals have a prefactor of order $O(\\varepsilon^{-3/8})$, up to logarithmic factors. A Duhamel argument then yields linear ill-posedness for every tangential derivative loss $μ<\\frac18$. Thus the three-dimensional Prandtl instability persists when the stabilizing tangential magnetic field degenerates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14532",
  "author_total": 2,
  "authors": [
    {
      "name": "Mingxue Zhang"
    },
    {
      "name": "Zhonger Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4r30q9y3n9pmf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14532",
  "fulltext_url": null,
  "id": "4r30q9y3n9pmf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14532",
  "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": "W7213294893",
  "pdf_url": "https://arxiv.org/pdf/2609.14532",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4r30q9y3n9pmf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14532",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14532"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Linear ill-posedness of three-dimensional MHD boundary layer equations",
  "title_text": "Linear ill-posedness of three-dimensional MHD boundary layer equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/4r30q9y3n9pmf",
  "venue": "arXiv",
  "versions": []
}