{
  "id": "1710.07762",
  "version": "v1",
  "published": "2017-10-21T06:41:02.000Z",
  "updated": "2017-10-21T06:41:02.000Z",
  "title": "Ill-posedness for the Hamilton-Jacobi equation in Besov spaces $B^0_{\\infty,q}$",
  "authors": [
    "Jinlu Li",
    "Weipeng Zhu",
    "Zhaoyang Yin"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the Cauchy problem for the following Hamilton-Jacobi equation \\bbal\\bca \\pa_tu-\\De u=|\\na u|^2,\\quad t>0, \\ x\\in \\R^d,\\\\ u(0,x)=u_0, \\quad \\quad x\\in \\R^d. \\eca\\end{align*} We show that the solution map in Besov spaces $B^0_{\\infty,q}(\\R^d),1\\leq q\\leq \\infty$ is discontinuous at origin. That is, we can construct a sequence initial data $\\{u^N_0\\}$ satisfying $||u^N_0||_{B^0_{\\infty,q}(\\R^d)}\\rightarrow 0, \\ N\\rightarrow \\infty$ such that the corresponding solution $\\{u^N\\}$ with $u^N(0)=u^N_0$ satisfies \\bbal ||u^N||_{L^\\infty_T(B^0_{\\infty,q}(\\R^d))}\\geq c_0, \\qquad \\forall \\ T>0, \\quad N\\gg 1, \\end{align*} with a constant $c_0>0$ independent of $N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-21T06:41:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35F21"
    ],
    "keywords": [
      "hamilton-jacobi equation",
      "besov spaces",
      "ill-posedness",
      "sequence initial data",
      "solution map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}