{
  "id": "math/0612168",
  "version": "v1",
  "published": "2006-12-06T23:55:06.000Z",
  "updated": "2006-12-06T23:55:06.000Z",
  "title": "Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole",
  "authors": [
    "P. Blue",
    "A. Soffer"
  ],
  "comment": "40 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We continue our study of the decoupled wave equation in the exterior of a spherically symmetric, Schwarzschild, black hole. Because null geodesics on the photon sphere orbit the black hole, extra effort must be made to show that the high angular momentum components of a solution decay sufficiently fast, particularly for low regularity initial data. Previous results are rapid decay for regular ($H^3$) initial data \\cite{BSterbenz} and slower decay for rough ($H^{1+\\epsilon}$) initial data \\cite{BlueSoffer3}. Here, we combine those methods to show boundedness of the conformal charge. From this, we conclude that there are bounds for global in time, space-time norms, in particular \\int_I |\\tilde\\phi|^4 d^4vol < C for $H^{1+\\epsilon}$ initial data with additional decay towards infinite and the bifurcation sphere. Here $\\tilde\\phi$ refers to a solution of the wave equation. $I$ denotes the exterior region of the Schwarzschild solution, which can be expressed in coordinates as $r>2M$, $t\\in\\Reals, \\omega\\in S^2$, and $d^4\\text{vol}$ is the natural 4-dimensional volume induced by the Schwarzschild pseudo-metric. We also demonstrate that the photon sphere has the same influence on the wave equation as a closed geodesic has on the wave equation on a Riemannian manifold. We demonstrate this similarity by extending our techniques to the wave equation on a class of Riemannian manifolds. Under further assumptions, the space-time estimates are sufficient to prove global bounds for small data, nonlinear wave equations on a class of Riemannian manifolds with closed geodesics. We must use global, space-time integral estimates since $L^\\infty$ estimates cannot hold at this level of regularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-12-06T23:55:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q75",
      "58J45"
    ],
    "keywords": [
      "wave equation",
      "small regularity loss",
      "schwarzschild black hole",
      "decay rates",
      "riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12168B"
    }
  }
}