{
  "id": "1510.04613",
  "version": "v1",
  "published": "2015-10-15T16:35:19.000Z",
  "updated": "2015-10-15T16:35:19.000Z",
  "title": "On the global existence and blowup of smooth solutions of 3-D compressible Euler equations with time-depending damping",
  "authors": [
    "Fei Hou",
    "Ingo Witt",
    "Huicheng Yin"
  ],
  "comment": "28 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D compressible Euler equation with time-depending damping $$ \\partial_t\\rho+\\operatorname{div}(\\rho u)=0, \\quad \\partial_t(\\rho u)+\\operatorname{div}\\left(\\rho u\\otimes u+p\\,I_{3}\\right)=-\\,\\frac{\\mu}{(1+t)^{\\lambda}}\\,\\rho u, \\quad \\rho(0,x)=\\bar \\rho+\\varepsilon\\rho_0(x),\\quad u(0,x)=\\varepsilon u_0(x), $$ where $x\\in\\mathbb R^3$, $\\mu>0$, $\\lambda\\geq 0$, and $\\bar\\rho>0$ are constants, $\\rho_0,\\, u_0\\in C_0^{\\infty}(\\mathbb R^3)$, $(\\rho_0, u_0)\\not\\equiv 0$, $\\rho(0,\\cdot)>0$, and $\\varepsilon>0$ is sufficiently small. For $0\\leq\\lambda\\leq1$, we show that there exists a global smooth solution $(\\rho, u)$ when $\\operatorname{curl} u_0\\equiv 0$, while for $\\lambda>1$, in general, the solution $(\\rho, u)$ will blow up in finite time. Therefore, $\\lambda=1$ appears to be the critical value for the global existence of small amplitude smooth solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-15T16:35:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L70",
      "35L65",
      "35L67",
      "76N15"
    ],
    "keywords": [
      "compressible euler equation",
      "global existence",
      "time-depending damping",
      "small amplitude smooth solutions",
      "global smooth solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151004613H"
    }
  }
}