{
  "id": "1212.3754",
  "version": "v2",
  "published": "2012-12-16T06:19:40.000Z",
  "updated": "2012-12-18T13:40:37.000Z",
  "title": "Decay of the solution to the bipolar Euler-Poisson system with damping in $\\mathbb{R}^3$",
  "authors": [
    "Zhigang Wu",
    "Weike Wang"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We construct the global solution to the Cauchy's problem of the bipolar Euler-Poisson equations with damping in $\\mathbb{R}^3$ when $H^3$ norm of the initial data is small. If further, the $\\dot{H}^{-s}$ norm ($0\\leq s<3/2)$ or $\\dot{B}_{2,\\infty}^{-s}$ norm ($0<s\\leq3/2$) of the initial data is bounded, we give the optimal decay rates of the solution. As a byproduct, the decay results of the $L^p-L^2$ ($1\\leq p\\leq2$) type hold without the smallness of the $L^p$ norm of the initial data. In particular, we deduce that $\\|\\nabla^k(\\rho_1-\\rho_2)\\|_{L^2} \\sim(1+t)^{-5/4-\\frac{k}{2}}$ and $\\|\\nabla^k(\\rho_i-\\bar{\\rho},u_i,\\nabla\\phi)\\|_{L^2} \\sim(1+t)^{-3/4-\\frac{k}{2}}$. We improve the decay results in Li and Yang \\cite{Li3}(\\emph{J.Differential Equations} 252(2012), 768-791), where they showed the decay rates as $\\|\\nabla^k(\\rho_i-\\bar{\\rho})\\|_{L^2} \\sim(1+t)^{-3/4-\\frac{k}{2}}$ and $\\|\\nabla^k(u_i,\\nabla\\phi)\\|_{L^2} \\sim(1+t)^{-1/4-\\frac{k}{2}}$, when the $H^3\\cap L^1$ norm of the initial data is small. Our analysis is motivated by the technique developed recently in Guo and Wang \\cite{Guo}(\\emph{Comm. Partial Differential Equations} 37(2012), 2165-2208) with some modifications.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-12-18T13:40:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "35B40"
    ],
    "keywords": [
      "bipolar euler-poisson system",
      "initial data",
      "decay results",
      "partial differential equations",
      "optimal decay rates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.3754W"
    }
  }
}