{
  "id": "2401.07922",
  "version": "v1",
  "published": "2024-01-15T19:16:54.000Z",
  "updated": "2024-01-15T19:16:54.000Z",
  "title": "Measure-based approach to mesoscopic modeling of optimal transportation networks",
  "authors": [
    "Jan Haskovec",
    "Peter Markowich",
    "Simone Portaro"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We propose a mesoscopic modeling framework for optimal transportation networks with biological applications. The network is described in terms of a joint probability measure on the phase space of tensor-valued conductivity and position in physical space. The energy expenditure of the network is given by a functional consisting of a pumping (kinetic) and metabolic power-law term, constrained by a Poisson equation accounting for local mass conservation. We establish convexity and lower semicontinuity of the functional on approriate sets. We then derive its gradient flow with respect to the 2-Wasserstein topology on the space of probability measures, which leads to a transport equation, coupled to the Poisson equation. To lessen the mathematical complexity of the problem, we derive a reduced Wasserstein gradient flow, taken with respect to the tensor-valued conductivity variable only. We then construct equilibrium measures of the resulting PDE system. Finally, we derive the gradient flow of the constrained energy functional with respect to the Fisher-Rao (or Hellinger-Kakutani) metric, which gives a reaction-type PDE. We calculate its equilibrium states, represented by measures concentrated on a hypersurface in the phase space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-15T19:16:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal transportation networks",
      "mesoscopic modeling",
      "measure-based approach",
      "gradient flow",
      "phase space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}