{
  "abstract": "The branched transport problem is a nonconvex and nonsmooth variational optimization problem on normal $1$-currents in $\\mathbb{R}^n$ with prescribed boundary. The optimality is with respect to some non-decreasing, lower semicontinuous, and subadditive function $τ:\\mathbb{R}_+\\to\\mathbb{R}_+$ with $τ(0)=0$ describing the cost $τ(m)$ to move an amount of mass $m$ per unit distance. The subadditivity leads to complicated, hierarchically ramified patterns in the support of (suboptimal) solutions. These network-like sets appear to have regularity properties similar to those of the singular surfaces arising in the so-called Brakke flow, a weak generalization of the mean curvature flow. We construct a geometric flow of transportation networks, which correspond to normal real $1$-rectifiable currents, by modifying Brakke's variational approximation scheme. We prove the existence of a limit that is Hölder continuous with respect to the flat norm and whose branched transport cost decreases along the geometric evolution. We further analyze a closely related geometric flow approximated by $1$-varifolds, whose weight measures model the branched transport cost, and establish its $1$-rectifiability together with a motion law analogous to Brakke's inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24451",
  "author_total": 2,
  "authors": [
    {
      "name": "Julius Lohmann"
    },
    {
      "name": "Yoshihiro Tonegawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ys7bq3q7agb7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24451",
  "fulltext_url": null,
  "id": "8ys7bq3q7agb7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24451",
  "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": "W7214048635",
  "pdf_url": "https://arxiv.org/pdf/2609.24451",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ys7bq3q7agb7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24451",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24451"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Geometric flows of branched transportation networks",
  "title_text": "Geometric flows of branched transportation networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/8ys7bq3q7agb7",
  "venue": "arXiv",
  "versions": []
}