{
  "id": "math/0601164",
  "version": "v6",
  "published": "2006-01-09T05:28:30.000Z",
  "updated": "2008-06-20T21:08:35.000Z",
  "title": "Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions",
  "authors": [
    "Terence Tao"
  ],
  "comment": "18 pages, no figures. Some corrections",
  "journal": "Dynamics of PDE 3 (2006), 93-110",
  "categories": [
    "math.AP"
  ],
  "abstract": "Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah, Bahouri-G\\'erard and Nakanishi have established global wellposedness, regularity, and scattering in the energy class for the energy-critical nonlinear wave equation $\\Box u = u^5$ in $\\R^{1+3}$, together with a spacetime bound $$ \\| u \\|_{L^4_t L^{12}_x(\\R^{1+3})} \\leq M(E(u))$$ for some finite quantity M(E(u)) depending only on the energy E(u) of u. We reprove this result, and show that this quantity obeys a bound of at most exponential type in the energy, and specifically $M(E) \\leq C (1+E)^{C E^{105/2}}$ for some absolute constant C > 0. The argument combines the quantitative local potential energy decay estimates of these previous papers with arguments used by Bourgain and the author for the analogous nonlinear Schr\\\"odinger equation.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2008-06-20T21:08:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L15"
    ],
    "keywords": [
      "energy-critical nonlinear wave equation",
      "spacetime bound",
      "spatial dimensions",
      "local potential energy decay estimates"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1164T"
    }
  }
}