{
  "id": "1212.1694",
  "version": "v4",
  "published": "2012-12-07T19:53:53.000Z",
  "updated": "2013-12-22T16:30:55.000Z",
  "title": "Regularity of the Boltzmann Equation in Convex Domains",
  "authors": [
    "Yan Guo",
    "Chanwoo Kim",
    "Daniela Tonon",
    "Ariane Trescases"
  ],
  "comment": "116 pages, 1. correct some typos, 2. change some notations, 3. improve the non-existence part",
  "categories": [
    "math.AP"
  ],
  "abstract": "A basic question about regularity of Boltzmann solutions in the presence of physical boundary conditions has been open due to characteristic nature of the boundary as well as the non-local mixing of the collision operator. Consider the Boltzmann equation in a strictly convex domain with the specular, bounce-back and diffuse boundary condition. With the aid of a distance function toward the grazing set, we construct weighted classical $C^{1}$ solutions away from the grazing set for all boundary conditions. For the diffuse boundary condition, we construct $W^{1,p}$ solutions for $1< p<2$ and weighted $W^{1,p}$ solutions for $2\\leq p\\leq \\infty $ as well. On the other hand, we show second derivatives do not exist up to the boundary in general by constructing counterexamples for all boundary conditions.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-12-22T16:30:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boltzmann equation",
      "diffuse boundary condition",
      "regularity",
      "grazing set",
      "physical boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 116,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.1694G"
    }
  }
}