{
  "abstract": "We consider the two-dimensional generalized surface quasi-geostrophic equations in $\\mathbb{R}^2$, given by \\[\\partial_t θ+u\\cdot \\nabla θ=0,\\quad u=-\\nabla^{\\perp}Λ^{β-2}θ,\\,β\\in [1,2).\\] When $β=1$, the equation defines the SQG equation and for $β>1$, it defines a family of more singular active scalar equations. We prove that if the interval of existence of the smooth solution to the generalized SQG equations for some $β_0\\in[1,2)$ is $[0,T]$, then with the same initial data, the interval of existence of the generalized SQG equations for $β$ close to $β_0$ also contains $[0,T]$. To prove these results, we develop estimates for a conservation law with flux modified around the generalized SQG equations. Furthermore, we also prove some new commutator estimates with bounds uniformly bounded as $β\\to 1$ that may be of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39580",
  "author_total": 1,
  "authors": [
    {
      "name": "Anuj Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dp94agzqse1y3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39580",
  "fulltext_url": null,
  "id": "dp94agzqse1y3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39580",
  "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": "W7215042975",
  "pdf_url": "https://arxiv.org/pdf/2609.39580",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dp94agzqse1y3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39580",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39580"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On wellposedness and limiting behavior of generalized SQG equations",
  "title_text": "On wellposedness and limiting behavior of generalized SQG equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/dp94agzqse1y3",
  "venue": "arXiv",
  "versions": []
}