{
  "id": "1408.3780",
  "version": "v1",
  "published": "2014-08-17T00:38:39.000Z",
  "updated": "2014-08-17T00:38:39.000Z",
  "title": "A geometric approach for sharp Local well-posedness of quasilinear wave equations",
  "authors": [
    "Qian Wang"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "The commuting vector fields approach, devised for strichartz estimates in [13], was developed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\\frac{2-\\sqrt{3}}{2}$ for general quasi-linear wave equation in ${\\mathbb R}^{1+3}$ by Klainerman and Rodnianski. Via this approach they obtained the local well-posedness in $H^s$ with $s>2$ for $(1+3)$ vacuum Einstein equations, by taking advantage of the vanishing Ricci curvature. The sharp, $H^{2+\\epsilon}$, local well-posedness result for general quasilinear wave equation was achieved by Smith and Tataru by constructing a parametrix using wave packets. Using the vector fields approach, one has to face the major hurdle caused by the Ricci tensor of the metric for the quasi-linear wave equations. This posed a question that if the geometric approach can provide the sharp result for the non-geometric equations. In this paper, based on geometric normalization and new observations on the mass aspect function, we prove the sharp local well-posedness of general quasilinear wave equation in ${\\Bbb R}^{1+3}$ by a vector field approach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-17T00:38:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sharp local well-posedness",
      "geometric approach",
      "general quasilinear wave equation",
      "vector fields approach",
      "general quasi-linear wave equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3780W"
    }
  }
}