{
  "abstract": "We study a nonlocal continuity equation on $\\mathbb{R}^d$ in which a probability density is driven by the competition between attraction toward a prescribed background measure $ω$ and self-repulsion among particles, governed respectively by the power-law kernels $ψ_a(x) = |x|^{1+a}$ and $ψ_r(x) = |x|^{1+r}$ with exponents $a, r \\in [0,1)$. We establish global Lagrangian well-posedness via a squared-radius regularization, obtaining {finite-time $L^\\infty$ and moment bounds}, $W^{n,\\infty}$ regularity, and uniqueness in the Lagrangian class. When the initial data is compactly supported and attraction dominates ($a > r$, or $a = r$ with $ω(\\mathbb{R}^d) > 1$), we prove that the support remains uniformly bounded at all time; a counterexample shows this fails for $a = r > 1$. For the attractive-dominant nonquadratic range $0 \\leq r \\leq a < 1$, we characterize zero-flux stationary states via a free-boundary problem involving a fractional Laplacian operator, reducing the stationarity condition to a fractional exterior Dirichlet problem. This characterization allows us to exhibit explicit examples of stationary measures in dimensions $d \\in \\{1,2,3\\}$. Numerical particle simulations {are consistent} with the theoretical stationary profiles. Finally, for $0 \\le r \\le a < 1$, we prove that every global solution whose energy is bounded from below and whose moments are bounded uniformly in time converges, along a sequence of times, to a zero-flux stationary state. Full narrow convergence follows whenever the omega-limit set contains a single stationary state.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04424",
  "author_total": 3,
  "authors": [
    {
      "name": "Massimo Fornasier"
    },
    {
      "name": "Hui Huang"
    },
    {
      "name": "Lukang Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1jmcb06ntrk6j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04424",
  "fulltext_url": null,
  "id": "1jmcb06ntrk6j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04424",
  "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": "W7167625813",
  "pdf_url": "https://arxiv.org/pdf/2607.04424",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1jmcb06ntrk6j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2607.04424",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04424"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The nonlocal attraction-repulsion transport equation with power kernels",
  "title_text": "The nonlocal attraction-repulsion transport equation with power kernels",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/1jmcb06ntrk6j",
  "venue": "arXiv",
  "versions": []
}