{
  "id": "2210.10153",
  "version": "v1",
  "published": "2022-10-18T20:35:23.000Z",
  "updated": "2022-10-18T20:35:23.000Z",
  "title": "Long-time behaviour of interaction models on Riemannian manifolds with bounded curvature",
  "authors": [
    "Razvan C. Fetecau",
    "Hansol Park"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modelled by an interaction potential. We consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution's support. The analytical results are supported by numerical simulations for the equation set up on the rotation group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-18T20:35:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A30",
      "35B38",
      "35B40",
      "58J90"
    ],
    "keywords": [
      "long-time behaviour",
      "interaction models",
      "bounded curvature",
      "interaction potential",
      "nonlocal partial differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}