{
  "abstract": "We establish global existence, uniqueness and quantitative stability for one-dimensional pressureless flow governed by \\[ \\partial_tρ+\\partial_x\\bigl(ρf(u)\\bigr)=0, \\qquad \\partial_t(ρu)+\\partial_x\\bigl(ρu f(u)\\bigr)=0, \\] where $ρ$ is the mass density, possibly measure-valued, $u$ is the physical velocity, and $f\\in C^1(\\mathbb R)$, with $f'>0$, determines the transport velocity $f(u)$. The initial mass has finite positive mass and finite second moment, allowing atoms and unbounded support, and the initial physical velocity is bounded. The difficulty at a cluster collision is that mass and momentum conservation determine a mass-weighted average of physical velocity, whereas transport is governed by the nonlinear law $f$ applied to that average. We construct global entropy solutions by a Bregman variational time discretization whose cost is chosen to reproduce the transport velocity determined by this mass-weighted average, coupled with conditional-expectation updates of physical velocity. For every entropy solution satisfying the Oleinik condition on the transport velocity and initial weak continuity of the kinetic energy measure, we establish a Lagrangian representation and prove uniqueness within this class by comparing flows with possibly different cluster intervals. For two such solutions with the same velocity law and common mass, we obtain a $W_\\infty$ estimate for the normalized mass measures in terms of the initial position and physical-velocity discrepancies in mass coordinates, whenever the initial $W_\\infty$ distance is finite. The Lagrangian evolution has the semigroup property.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16341",
  "author_total": 3,
  "authors": [
    {
      "name": "Gaowei Cao"
    },
    {
      "name": "Feimin Huang"
    },
    {
      "name": "Guirong Tang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29t3a6a73n4ja.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16341",
  "fulltext_url": null,
  "id": "29t3a6a73n4ja",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16341",
  "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|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7221798377",
  "pdf_url": "https://arxiv.org/pdf/2610.16341",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29t3a6a73n4ja.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16341",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16341"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "One-dimensional pressureless flow with a nonlinear velocity law: Bregman variational construction and well-posedness",
  "title_text": "One-dimensional pressureless flow with a nonlinear velocity law: Bregman variational construction and well-posedness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/29t3a6a73n4ja",
  "venue": "arXiv",
  "versions": []
}