{
  "id": "2304.01309",
  "version": "v1",
  "published": "2023-04-03T19:20:14.000Z",
  "updated": "2023-04-03T19:20:14.000Z",
  "title": "Oleĭnik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit",
  "authors": [
    "Giuseppe Maria Coclite",
    "Maria Colombo",
    "Gianluca Crippa",
    "Nicola De Nitti",
    "Alexander Keimer",
    "Elio Marconi",
    "Lukas Pflug",
    "Laura V. Spinolo"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:=\\mathbb{1}_{(-\\infty,0]}(\\cdot)\\exp(\\cdot) \\ast \\rho$ satisfy an Ole\\u{\\i}nik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a one-sided Lipschitz condition and that $V'(W) W \\partial_x W$ satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-03T19:20:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlocal conservation laws",
      "nonlocal-to-local limit",
      "oleĭnik-type estimates",
      "applications",
      "exponential kernel"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}