{
  "id": "2210.15886",
  "version": "v1",
  "published": "2022-10-28T04:28:50.000Z",
  "updated": "2022-10-28T04:28:50.000Z",
  "title": "Wellposedness of nonlinear flows on manifolds of bounded geometry",
  "authors": [
    "Eric Bahuaud",
    "Christine Guenther",
    "James Isenberg",
    "Rafe Mazzeo"
  ],
  "comment": "36 pages",
  "categories": [
    "math.AP",
    "math.DG",
    "math.FA"
  ],
  "abstract": "We present simple conditions which ensure that a strongly elliptic operator $L$ generates an analytic semigroup on H\\\"older spaces on an arbitrary complete manifold of bounded geometry. This is done by establishing the equivalent property that $L$ is \"sectorial\", a condition that specifies the decay of the resolvent $(\\lambda I - L)^{-1}$ as $\\lambda$ diverges from the H\\\"older spectrum of $L$. As one step, we prove existence of this resolvent if $\\lambda$ is sufficiently large, and on this general class of manifolds, use a geometric microlocal version of the semiclassical pseudodifferential calculus. The properties of $L$ and $e^{-tL}$ we obtain can then be used to prove wellposedness of a wide class of nonlinear flows. We illustrate this by proving wellposedness on H\\\"older spaces of the flow associated to the ambient obstruction tensor on complete manifolds of bounded geometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-28T04:28:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J35",
      "35J05"
    ],
    "keywords": [
      "bounded geometry",
      "nonlinear flows",
      "wellposedness",
      "geometric microlocal version",
      "arbitrary complete manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}