{
  "id": "math/0202031",
  "version": "v3",
  "published": "2002-02-04T20:20:06.000Z",
  "updated": "2002-10-10T14:09:50.000Z",
  "title": "Global existence for nonlinear wave equations with multiple speeds",
  "authors": [
    "Christopher D. Sogge"
  ],
  "comment": "14 pages, to appear in Proceedings of the 2001 Mount Holyoke Conference on Harmonic Analysis. Corrected a couple of typos",
  "categories": [
    "math.AP"
  ],
  "abstract": "We shall be concerned with the Cauchy problem for quasilinear systems in three space dimensions of the form \\label{i.1} \\partial^2_tu^I-c^2_I\\Delta u^I = C^{IJK}_{abc}\\partial_c u^J\\partial_a\\partial_b u^K + B^{IJK}_{ab}\\partial_a u^J\\partial_b u^K, \\quad I=1,..., D. Here we are using the convention of summing repeated indices, and $\\partial u$ denotes the space-time gradient, $\\partial u=(\\partial_0 u, \\partial_1 u, \\partial_2 u, \\partial_3u)$, with $\\partial_0=\\partial_t$, and $\\partial_j=\\partial_{x_j}$, $j=1,2,3$. We shall be in the nonrelativistic case where we assume that the wave speeds $c_k$ are all positive but not necessarily equal. Using a new pointwise estimate of the M. Keel, H. Smith and the author we shall prove global existence of small amplitude solutions for such equations satisfying a null condition. This generalizes the earlier result of Christodoulou and Klainerman where all the wave speeds are the same. Our approach is related to that of Klainerman; however, since we are in the non-relativistic case we cannot use the Lorentz boost vector fields or the Morawetz vector fields. Instead we exploit both the 1/t decay of linear solutions as well as the much easier to prove 1/|x| decay.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2002-10-10T14:09:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L70",
      "42B99"
    ],
    "keywords": [
      "nonlinear wave equations",
      "global existence",
      "multiple speeds",
      "wave speeds",
      "lorentz boost vector fields"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......2031S"
    }
  }
}