{
  "abstract": "We develop a unified Lagrangian framework for the uniqueness of weak solutions to a broad class of two-dimensional inviscid active scalar equations in generalized mean oscillation spaces. Under common structural assumptions on the Biot--Savart kernels, we identify mean oscillation spaces that ensure an Osgood condition for the velocity, and establish uniqueness in these classes. For the two-dimensional Euler equations, we establish uniqueness for vorticities whose mean oscillation grows mildly at small scales, extending beyond $\\mathrm{BMO}$. For the surface quasi-geostrophic (SQG) and incompressible porous media (IPM) equations, we obtain uniqueness in spaces logarithmically rougher than $C^{0,1}$. We also obtain what are (to our knowledge) the strongest known uniqueness results for the $α$-SQG equation, $0<α<1$, in spaces logarithmically rougher than critical Hölder spaces $C^α$. The framework also applies to various logarithmically perturbed models, yielding uniqueness in the corresponding sharp regularity classes. The proof combines a sharp regularity estimate for the velocity with a stability estimate for integral operators under measure-preserving rearrangements. This provides a single uniqueness argument for the entire family under common structural assumptions on the kernels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07834",
  "author_total": 4,
  "authors": [
    {
      "name": "Hyunjin In"
    },
    {
      "name": "Dong-ha Kim"
    },
    {
      "name": "Junha Kim"
    },
    {
      "name": "Seungjae Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7yk1wvg6ge1t6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07834",
  "fulltext_url": null,
  "id": "7yk1wvg6ge1t6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07834",
  "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/2611.07834",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7yk1wvg6ge1t6.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07834",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07834"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Lagrangian Approach to Uniqueness for Active Scalars in Generalized Mean Oscillation Spaces",
  "title_text": "A Lagrangian Approach to Uniqueness for Active Scalars in Generalized Mean Oscillation Spaces",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/7yk1wvg6ge1t6",
  "venue": "arXiv",
  "versions": []
}