{
  "abstract": "In this paper, we revisit a multi-dimensional nonlocal transport equation of the form \\begin{eqnarray*} \\partial_tρ+g\\mathcal{R}_aρ\\cdot \\nablaρ= 0,~ρ(\\cdot,0)=ρ_{0}, \\end{eqnarray*} where the transform $\\mathcal{R}_a$ is defined by \\begin{eqnarray*} \\mathcal{R}_aρ(x,t)=\\frac{Γ(\\frac{n+1}{2})}{π^{\\frac{n+1}{2}}}P.V.\\int\\limits_{\\mathbb{R}^n}\\Big(\\frac{x-y}{|x-y|^{n+1}}-\\frac{x-y}{(|x-y|^2+a^2)^{\\frac{n+1}{2}}}\\Big)ρ(y,t)dy. \\end{eqnarray*} This model was first introduced in [W. Zhang, J. Differential Equations 467 (2026) 114275] as a natural generalization of the well-known Kiselev-Sarsam equation proposed in [A. Kiselve and N.A. Sarsam, Nonlinearity 38 (2025) 055018]. By using an improved pointwise inequality for the transform $\\mathcal{R}_a$, we obtain a better and more natural version of a nonlinear integral inequality proved in Zhang [31] for the transform $\\mathcal{R}_a$. In addition, we also establish a new weighted nonlinear inequality for the transform $\\mathcal{R}_a$ implying the finite-time blow-up for a class of smooth radial initial data.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04632",
  "author_total": 1,
  "authors": [
    {
      "name": "Wanwan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bay17zscwgf34.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04632",
  "fulltext_url": null,
  "id": "bay17zscwgf34",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04632",
  "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|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7220702743",
  "pdf_url": "https://arxiv.org/pdf/2610.04632",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bay17zscwgf34.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04632",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04632"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Remarks on a multi-dimensional transport equation with nonlocal velocity",
  "title_text": "Remarks on a multi-dimensional transport equation with nonlocal velocity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/bay17zscwgf34",
  "venue": "arXiv",
  "versions": []
}