{
  "id": "2108.11741",
  "version": "v1",
  "published": "2021-08-26T12:38:36.000Z",
  "updated": "2021-08-26T12:38:36.000Z",
  "title": "The singularities for a periodic transport equation",
  "authors": [
    "Yong Zhang",
    "Fei Xu",
    "Fengquan Li"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider a 1D periodic transport equation with nonlocal flux and fractional dissipation $$ u_{t}-(Hu)_{x}u_{x}+\\kappa\\Lambda^{\\alpha}u=0,\\quad (t,x)\\in R^{+}\\times S, $$ where $\\kappa\\geq0$, $0<\\alpha\\leq1$ and $S=[-\\pi,\\pi]$. We first establish the local-in-time well-posedness for this transport equation in $H^{3}(S)$. In the case of $\\kappa=0$, we deduce that the solution, starting from the smooth and odd initial data, will develop into singularity in finite time. If adding a weak dissipation term $\\kappa\\Lambda^{\\alpha}u$, we also prove that the finite time blowup would occur.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-26T12:38:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singularity",
      "1d periodic transport equation",
      "odd initial data",
      "weak dissipation term",
      "finite time blowup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}