{
  "abstract": "We study finite-time blow-up, critical norm inflation, and global well-posedness for the Geng--Xue system, an integrable two-component extension of the Novikov equation. For initial velocities $(u_0,v_0)\\in(H^s)^2$, $s>3/2$, with second momentum $n_0=v_0-v_{0,xx}\\in L^1_{\\rm uloc}$, we derive explicit blow-up conditions and upper bounds on the lifespan. The blow-up proof uses a cancellation in the equation for $nu_x$ to close the convolution estimates without an a priori bound on $\\|u_x\\|_{L^\\infty}$. Combined with a continuation criterion involving only $u$, these estimates yield critical Besov norm inflation around the initial data in $(H^{3/2})^2$. For initial momenta in $(H^σ)^2$, $σ>-1/2$, each with a fixed sign, we prove global well-posedness. The blow-up and norm-inflation results also extend to the three-component Degasperis--Procesi system.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34267",
  "author_total": 2,
  "authors": [
    {
      "name": "Song Liu"
    },
    {
      "name": "Zhaoyang Yin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9xsxrn4vba5r5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34267",
  "fulltext_url": null,
  "id": "9xsxrn4vba5r5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34267",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.34267",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9xsxrn4vba5r5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34267",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34267"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Blow-up, critical norm inflation, and global well-posedness for the Geng--Xue system",
  "title_text": "Blow-up, critical norm inflation, and global well-posedness for the Geng--Xue system",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/9xsxrn4vba5r5",
  "venue": "arXiv",
  "versions": []
}