{
  "id": "1510.01539",
  "version": "v1",
  "published": "2015-10-06T11:52:56.000Z",
  "updated": "2015-10-06T11:52:56.000Z",
  "title": "Global existence and smoothness for solutions of viscous Burgers equation. (2) The unbounded case: a characteristic flow study",
  "authors": [
    "Jeremie Unterberger"
  ],
  "comment": "37 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We show that the homogeneous viscous Burgers equation $(\\partial_t-\\eta\\Delta) u(t,x)+(u\\cdot\\nabla)u(t,x)=0,\\ (t,x)\\in{\\mathbb{R}}_+\\times{\\mathbb{R}}^d$ $(d\\ge 1, \\eta>0)$ has a globally defined smooth solution if the initial condition $u_0$ is a smooth function growing like $o(|x|)$ at infinity. The proof relies mostly on estimates of the random characteristic flow defined by a Feynman-Kac representation of the solution. Viscosity independent a priori bounds for the solution are derived from these. The regularity of the solution is then proved for fixed $\\eta>0$ using Schauder estimates. The result extends with few modifications to initial conditions growing abnormally large in regions with small relative volume, separated by well-behaved bulk regions, provided these are stable under the characteristic flow with high probability. We provide a large family of examples for which this loose criterion may be verified by hand.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-06T11:52:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35B45",
      "35B50",
      "35K15",
      "35Q30",
      "35Q35",
      "35L65",
      "76N10"
    ],
    "keywords": [
      "viscous burgers equation",
      "characteristic flow study",
      "global existence",
      "unbounded case",
      "smoothness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151001539U"
    }
  }
}