{
  "id": "1809.06803",
  "version": "v1",
  "published": "2018-09-18T15:37:52.000Z",
  "updated": "2018-09-18T15:37:52.000Z",
  "title": "Approximate solutions of vector fields and an application to Denjoy-Carleman regularity of solutions of a nonlinear PDE",
  "authors": [
    "Nicholas Braun Rodrigues",
    "Antonio V. da Silva Jr"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we study microlocal regularity of a $\\mathcal{C}^2$ solution $u$ of the equation \\begin{equation*} u_t = f(x,t,u,u_x), \\end{equation*} where $f(x,t,\\zeta_0, \\zeta)$ is ultradifferentiable in the variables $(x,t)\\in \\mathbb{R}^{N} \\times \\mathbb{R}$ and holomorphic in the variables $(\\zeta_0,\\zeta) \\in \\mathbb{C} \\times \\mathbb{C}^{N}$. We proved that if $\\mathcal{C}^{\\mathcal{M}}$ is a regular Denjoy-Carleman class (including the quasianalytic case) then: \\begin{equation*} \\mathrm{WF}_\\mathcal{M} (u)\\subset \\mathrm{Char}(L^u), \\end{equation*} where $\\mathrm{WF}_\\mathcal{M}(u)$ is the Denjoy-Carleman wave-front set of $u$ and $\\mathrm{Char}(L^u)$ is the characteristic set of the linearized operator $L^u$: \\begin{equation*} L^u = \\dfrac{\\partial}{\\partial t} - \\sum_{j=1}^{N}\\frac{\\partial f}{\\partial\\zeta_j}(x,t,u,u_x)\\dfrac{\\partial}{\\partial x_j}. \\end{equation*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-18T15:37:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear pde",
      "denjoy-carleman regularity",
      "vector fields",
      "approximate solutions",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}