{
  "id": "2412.09863",
  "version": "v1",
  "published": "2024-12-13T05:07:13.000Z",
  "updated": "2024-12-13T05:07:13.000Z",
  "title": "Sharp $L^1$-convergence rates to the Barenblatt solutions for the compressible Euler equations with time-varying damping",
  "authors": [
    "Jun-Ren Luo",
    "Ti-Jun Xiao"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the asymptotic behavior of compressible isentropic flow when the initial mass is finite and the friction varies with time, which is modeled by the compressible Euler equation with time-dependent damping. In this paper, we obtain the best $L^1$-convergence rates to date, for any $\\gamma\\in(1,+\\infty)$ and $\\nu\\in[0,1)$. Here, $\\gamma$ is the adiabatic gas exponent, and $\\nu$ is the physical parameter in the damping term. The key to the analysis lies in a new perspective on the relationship between the density function and the Barenblatt solution of the porous medium equation, and finding the relevant lower bound for the case of $\\gamma<2$ is a tricky problem. Specialized to $\\nu=0$, these convergence rates also show an essential improvement over the original rates. Moreover, for all $\\gamma\\in(1,+\\infty)$, the results in this work are the first to present a unified form of $L^1$-convergence rates. Indeed, even for $\\nu=0$, as noted in 2011, ``the current rate is difficult to improve with the current method\". Our results are therefore an encouraging advancement.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-13T05:07:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convergence rates",
      "compressible euler equation",
      "barenblatt solution",
      "time-varying damping",
      "adiabatic gas exponent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}