{
  "abstract": "For square-integrable divergence-free vector field $\\boldsymbol{v}$ on $\\mathbb{R}^d$ we prove that the following properties are equivalent: 1) the operator $A_0 ρ= \\boldsymbol{v} \\cdot \\nabla ρ$ (where $ρ\\in C^\\infty_c(\\mathbb{R}^d)$) is essentially skew-adjoint on $L^2(\\mathbb{R}^d)$; 2) square-integrable (with respect to spatial variables) generalized solutions of the continuity equation are renormalized; 3) generalized square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique both forward an backward in time. We also construct a compactly supported bounded divergence-free vector field $\\boldsymbol{v}\\colon \\mathbb{R}^3 \\to \\mathbb{R}^3$ for which square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique forward, but not backward in time.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03831",
  "author_total": 4,
  "authors": [
    {
      "name": "Nikolay A. Gusev"
    },
    {
      "name": "Mikhail V. Korobkov"
    },
    {
      "name": "Evgeny Yu. Panov"
    },
    {
      "name": "Konstantin Yu. Zamana"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xs8hyvjz6dh2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03831",
  "fulltext_url": null,
  "id": "0xs8hyvjz6dh2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03831",
  "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",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7208780619",
  "pdf_url": "https://arxiv.org/pdf/2609.03831",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xs8hyvjz6dh2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03831",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03831"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On flows generated by square-integrable vector fields",
  "title_text": "On flows generated by square-integrable vector fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/0xs8hyvjz6dh2",
  "venue": "arXiv",
  "versions": []
}