{
  "id": "2008.06704",
  "version": "v1",
  "published": "2020-08-15T11:40:59.000Z",
  "updated": "2020-08-15T11:40:59.000Z",
  "title": "$L^1$-convergence to generalized Barenblatt solution for compressible Euler equations with time-dependent damping",
  "authors": [
    "Geng Shifeng",
    "Huang Feimin",
    "Wu Xiaochun"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The large time behavior of entropy solution to the compressible Euler equations for polytropic gas (the pressure $p(\\rho)=\\kappa\\rho^{\\gamma}, \\gamma>1$) with time dependent damping like $-\\frac{1}{(1+t)^\\lambda}\\rho u$ ($0<\\lambda<1$) is investigated. By introducing an elaborate iterative method and using the intensive entropy analysis, it is proved that the $L^\\infty$ entropy solution of compressible Euler equations with finite initial mass converges strongly in the natural $L^1$ topology to a fundamental solution of porous media equation (PME) with time-dependent diffusion, called by generalized Barenblatt solution. It is interesting that the $L^1$ decay rate is getting faster and faster as $\\lambda$ increases in $(0, \\frac{\\gamma}{\\gamma+2}]$, while is getting slower and slower in $[ \\frac{\\gamma}{\\gamma+2}, 1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-15T11:40:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compressible euler equations",
      "generalized barenblatt solution",
      "time-dependent damping",
      "entropy solution",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}