{
  "abstract": "We establish instantaneous $H^2$ norm explosion for the two-dimensional Boussinesq equations around the linearly stable background profile $-x_2$. While previous ill-posedness results for this system relied on $L^\\infty$-type spaces, this work provides the first strong ill-posedness result in the energy-critical space, where stable background stratification acts to suppress deformation growth. We adapt an idea from our previous work on the IPM equation \\cite{BCMZ2025} and select an initial density perturbation vanishing near the origin to neutralize the stable stratification. Crucially, unlike the IPM setting, the evolution is here driven by the non-linear coupling between density and vorticity. Since the initial vorticity vanishes identically, the instantaneous loss of regulari\\-ty is driven by density-vorticity feedback - a mechanism fundamentally distinct from both $H^1$ Euler ill-posedness of Bourgain \\& Li and the IPM evolution. Through an iterative scheme yielding a sequence of approximations with strong deformation at the origin, we construct a strong solution $(ρ(t), ω(t))$ originating from data with arbitrarily small $H^2 \\times (H^1\\cap \\dot H^{-1})$ norm, which immediately exhibits $H^2 \\times H^1$ norm explosion for all $t > 0$ while maintaining uniqueness within the energy class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38584",
  "author_total": 2,
  "authors": [
    {
      "name": "Roberta Bianchini"
    },
    {
      "name": "Luis Martínez-Zoroa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0v2a9x253m8qe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38584",
  "fulltext_url": null,
  "id": "0v2a9x253m8qe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38584",
  "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": "W7215037970",
  "pdf_url": "https://arxiv.org/pdf/2609.38584",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0v2a9x253m8qe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38584",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38584"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Non Existence of the Stable 2D Boussinesq Equations in $H^2$ via Density-Vorticity Feedback",
  "title_text": "Non Existence of the Stable 2D Boussinesq Equations in H^2 via Density-Vorticity Feedback",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/0v2a9x253m8qe",
  "venue": "arXiv",
  "versions": []
}