{
  "abstract": "We prove magnetic-field norm inflation at the origin for the viscous, non-resistive incompressible magnetohydrodynamic equations on $\\R^d$, $d\\ge2$, in the endpoint space \\[ \\dot B^{-1}_{\\infty,1}(\\R^d)\\times \\dot B^{0}_{\\infty,1}(\\R^d). \\] For every sufficiently small $\\varepsilon>0$, we construct smooth divergence-free initial data $(u_0,b_0)$, with $b_0$ compactly supported and \\[ \\norm[\\dot B^{-1}_{\\infty,1}]{u_0} +\\norm[\\dot B^{0}_{\\infty,1}]{b_0}<\\varepsilon, \\] such that the corresponding classical solution satisfies \\[ \\norm[\\dot B^{0}_{\\infty,1}]{b(t_\\varepsilon)}>\\varepsilon^{-1} \\] for some $0<t_\\varepsilon<\\varepsilon$. Consequently, no solution map agreeing with classical solutions can be continuous at the origin in the magnetic endpoint topology. The low--high paraproduct mechanism available when the dyadic summability exponent is greater than one gives no gain at the $\\ell^1$ endpoint. We instead combine a resonant four-wave velocity interaction with magnetic stretching to produce uniformly sized contributions on a growing family of dyadic shells. Frequency-localized test functionals extract the endpoint lower bound, while uniform Lagrangian estimates in weighted Fourier $L^1$ spaces and a spatial localization argument control the remainders and remove the auxiliary constant magnetic field.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17192",
  "author_total": 3,
  "authors": [
    {
      "name": "Yingzhe Ban"
    },
    {
      "name": "Boling Guo"
    },
    {
      "name": "Maotuo Guo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/enn79n076f1hp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17192",
  "fulltext_url": null,
  "id": "enn79n076f1hp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17192",
  "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": "W7213407113",
  "pdf_url": "https://arxiv.org/pdf/2609.17192",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/enn79n076f1hp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17192",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17192"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Magnetic norm inflation at the $\\ell^1$ Besov endpoint for viscous non-resistive incompressible MHD",
  "title_text": "Magnetic norm inflation at the ℓ^1 Besov endpoint for viscous non-resistive incompressible MHD",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/enn79n076f1hp",
  "venue": "arXiv",
  "versions": []
}