{
  "abstract": "We study a novel class of evolution variational inequalities (EVI) driven by bifunctions $\\mathsf{b}:\\mathrm{D} \\times \\mathrm{D} \\to \\mathbb{R}$ defined on a subset $\\mathrm{D}$ of a metric space $(\\mathrm{X},\\mathrm{d})$ and satisfying a natural dissipativity condition $$\\mathsf{b}(x,y)+\\mathsf{b}(y,x)\\le η\\mathrm{d}^2(x,y).$$ We provide general conditions for existence, stability, regularity, approximation and asymptotic behavior of solutions, by proposing a metric framework that generalizes the classical structure of evolutions driven by monotone operators in Hilbert spaces. A motivating application is the setting of minimax and multispecies coupled gradient flows, in which each of $N$ species evolves in the direction of steepest descent of its own energy functional and the joint system is not a gradient flow of any single energy. To construct solutions, we introduce a variational movement scheme (VMS), a time discretization scheme in which each update is a saddle point of a metrically penalized bifunction, generalizing the classical minimizing movement/ JKO scheme for single species gradient descent. In the coupled multispecies case, the VMS reduces to a Nash equilibrium problem. We prove existence of discrete solutions to the VMS via a general approach which combines dissipativity of $\\mathsf{b}$ with a new notion of convexity along suitable $\\textit{barycentric interpolations}$ (which is inspired by convexity along generalized geodesics, a crucial notion for gradient flows). Together, these conditions provide a zeroth-order notion of game-theoretic monotonicity applicable to general metric spaces without linear structure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20998",
  "author_total": 3,
  "authors": [
    {
      "name": "Lauren Conger"
    },
    {
      "name": "Franca Hoffmann"
    },
    {
      "name": "Giuseppe Savaré"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2scwqbdjzn390.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20998",
  "fulltext_url": null,
  "id": "2scwqbdjzn390",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20998",
  "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"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213886062",
  "pdf_url": "https://arxiv.org/pdf/2609.20998",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2scwqbdjzn390.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.20998",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20998"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Dissipative Evolutions in Metric Spaces",
  "title_text": "Dissipative Evolutions in Metric Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/2scwqbdjzn390",
  "venue": "arXiv",
  "versions": []
}