{
  "id": "2008.01850",
  "version": "v1",
  "published": "2020-08-04T21:49:56.000Z",
  "updated": "2020-08-04T21:49:56.000Z",
  "title": "Well-posedness and global in time behavior for mild solutions to the Navier-Stokes equation on the hyperbolic space with initial data in $L^p$",
  "authors": [
    "Braden Balentine"
  ],
  "comment": "40 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study mild solutions to the Navier-Stokes equation on the $n$-dimensional hyperbolic space $\\mathbb{H}^n$, $n \\geq 2$. We use dispersive and smoothing estimates proved by Pierfelice on a class of complete Riemannian manifolds to extend the Fujita-Kato theory of mild solutions from $\\mathbb{R}^n$ to $\\mathbb{H}^n$. This includes well-posedness results for $L^p$ initial data in the range $1 < p < \\infty$, global in time results for small initial data, and $L^p$ norm decay results for both $u$ and $\\nabla u$. As part of this, we extend to the hyperbolic space $\\mathbb{H}^n$ known facts in Euclidean space concerning the strong continuity and contractivity of the semigroup generated by the Laplacian. Also, we establish necessary boundedness and commutation properties for a certain projection operator in the setting of $\\mathbb{H}^n$ using spectral theory. This work, together with Pierfelice's, contributes to providing a full theory for mild solutions on $\\mathbb{H}^n$. While the statements of the results are the same as in the Euclidean case, the methods of the proofs are at times different.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-04T21:49:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "76D05",
      "76D03"
    ],
    "keywords": [
      "navier-stokes equation",
      "time behavior",
      "well-posedness",
      "dimensional hyperbolic space",
      "complete riemannian manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}