{
  "id": "1610.05346",
  "version": "v1",
  "published": "2016-10-17T20:34:10.000Z",
  "updated": "2016-10-17T20:34:10.000Z",
  "title": "A $L^{2}$ to $L^{\\infty}$ approach for the Landau Equation",
  "authors": [
    "Jinoh Kim",
    "Yan Guo",
    "Hyung Ju Hwang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $L^{2}\\rightarrow L^{\\infty }$ framework to construct global unique solutions near Maxwellian with small $L^{\\infty }\\ $norm. The first step is to establish global $L^{2}$ estimates with strong velocity weight and time decay, under the assumption of $L^{\\infty }$ bound, which is further controlled by such $L^{2}$ estimates via De Giorgi's method \\cite{golse2016harnack} and \\cite{mouhot2015holder}. The second step is to employ estimates in $S_{p}$ spaces to control velocity derivatives to ensure uniqueness, which is based on Holder estimates via De Giorgi's method \\cite{golse2016harnack}, \\cite{golse2015holder}, and \\cite{mouhot2015holder}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-17T20:34:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "landau equation",
      "giorgis method",
      "construct global unique solutions",
      "strong velocity weight",
      "control velocity derivatives"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}