{
  "id": "2102.06400",
  "version": "v1",
  "published": "2021-02-12T08:58:25.000Z",
  "updated": "2021-02-12T08:58:25.000Z",
  "title": "Well-posedness theory for nonlinear scalar conservation laws on networks",
  "authors": [
    "Ulrik Skre Fjordholm",
    "Markus Musch",
    "Nils Henrik Risebro"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We consider nonlinear scalar conservation laws posed on a network. We establish $L^1$ stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply standard finite volume methods and show stability and convergence to the unique entropy solution, thus establishing existence of a solution in the process. Both our existence and stability/uniqueness theory is centred around families of stationary states for the equation. In one important case -- for monotone fluxes with an upwind difference scheme -- we show that the set of (discrete) stationary solutions is indeed sufficiently large to suit our general theory. We demonstrate the method's properties through several numerical experiments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-12T08:58:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D30",
      "65M12",
      "65M08"
    ],
    "keywords": [
      "nonlinear scalar conservation laws",
      "well-posedness theory",
      "apply standard finite volume methods",
      "unique entropy solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}