{
  "id": "1803.07826",
  "version": "v1",
  "published": "2018-03-21T10:03:41.000Z",
  "updated": "2018-03-21T10:03:41.000Z",
  "title": "Singularity formation for Burgers equation with transverse viscosity",
  "authors": [
    "Charles Collot",
    "Tej-Eddine Ghoul",
    "Nader Masmoudi"
  ],
  "comment": "74 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider Burgers equation with transverse viscosity $$\\partial_tu+u\\partial_xu-\\partial_{yy}u=0, \\ \\ (x,y)\\in \\mathbb R^2, \\ \\ u:[0,T)\\times \\mathbb R^2\\rightarrow \\mathbb R.$$ We construct and describe precisely a family of solutions which become singular in finite time by having their gradient becoming unbounded. To leading order, the solution is given by a backward self-similar solution of Burgers equation along the $x$ variable, whose scaling parameters evolve according to parabolic equations along the $y$ variable, one of them being the quadratic semi-linear heat equation. We develop a new framework adapted to this mixed hyperbolic/parabolic blow-up problem, revisit the construction of flat blow-up profiles for the semi-linear heat equation, and the self-similarity in the shocks of Burgers equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-21T10:03:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35M10",
      "35L67",
      "35K58",
      "35Q35",
      "35A20",
      "35B35",
      "35B40",
      "35B44"
    ],
    "keywords": [
      "burgers equation",
      "transverse viscosity",
      "singularity formation",
      "quadratic semi-linear heat equation",
      "mixed hyperbolic/parabolic blow-up problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 74,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}