{
  "abstract": "We study the Hunter--Saxton equation on the real line and its global dissipative solution in the energy space $ L^\\infty(\\mathbb R)\\cap \\dot H^1(\\mathbb R). $ We prove strong ill-posedness through instantaneous failure of Sobolev regularity at and below the Lipschitz threshold. More precisely, for every $s\\in(1,\\nicefrac32]$, we construct $ u_0\\in L^\\infty(\\mathbb R)\\cap\\dot H^1(\\mathbb R)\\cap\\dot H^s(\\mathbb R) $ whose unique global dissipative solution satisfies $u\\notin C([0,T]; \\dot H^s (\\mathbb R))$, for every $T>0$. The constructions differ substantially in the subcritical and critical regimes. For $1<s<\\nicefrac32$, we superpose rescaled, localized bubbles with increasingly negative slopes; subcritical scaling preserves $\\dot H^s$-summability, while the explicit characteristic formula produces norm inflation. At $s=\\nicefrac32$, where scaling yields no smallness, we use logarithmically distributed compactly supported multiscale profiles whose breaking times converge to zero. A one-sided localization principle and an almost-orthogonality estimate then transfer the inflation of individual profiles to the full dissipative solution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02523",
  "author_total": 2,
  "authors": [
    {
      "name": "Billel Guelmame"
    },
    {
      "name": "Haroune Houamed"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/az70wchfxs4kp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02523",
  "fulltext_url": null,
  "id": "az70wchfxs4kp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02523",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7207713216",
  "pdf_url": "https://arxiv.org/pdf/2609.02523",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/az70wchfxs4kp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02523",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02523"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Strong Ill-Posedness in critical and subcritical regimes for the Hunter-Saxton equation",
  "title_text": "Strong Ill-Posedness in critical and subcritical regimes for the Hunter-Saxton equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/az70wchfxs4kp",
  "venue": "arXiv",
  "versions": []
}