{
  "id": "1909.13269",
  "version": "v1",
  "published": "2019-09-29T12:32:00.000Z",
  "updated": "2019-09-29T12:32:00.000Z",
  "title": "Lower Bound and Space-time Decay Rates of Higher Order Derivatives of Solution for the Compressible Navier-Stokes and Hall-MHD Equations",
  "authors": [
    "Jincheng Gao",
    "Zeyu Lyu",
    "Zheng-an Yao"
  ],
  "comment": "37 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we address the lower bound and space-time decay rates for the compressible Navier-Stokes and Hall-MHD equations under $H^3-$framework in $\\mathbb{R}^3$. First of all, the lower bound of decay rate for the density, velocity and magnetic field converging to the equilibrium status in $L^2$ is $(1+t)^{-\\frac{3}{4}}$; the lower bound of decay rate for the first order spatial derivative of density and velocity converging to zero in $L^2$ is $(1+t)^{-\\frac{5}{4}}$, and the $k(\\in [1, 3])-$th order spatial derivative of magnetic field converging to zero in $L^2$ is $(1+t)^{-\\frac{3+2k}{4}}$. Secondly, the lower bound of decay rate for time derivatives of density and velocity converging to zero in $L^2$ is $(1+t)^{-\\frac{5}{4}}$; however, the lower bound of decay rate for time derivatives of magnetic field converging to zero in $L^2$ is $(1+t)^{-\\frac{7}{4}}$. Finally, we address the decay rate of solution in weighted Sobolev space $H^3_{\\gamma}$. More precisely, the upper bound of decay rate of the $k(\\in [0, 2])$-th order spatial derivatives of density and velocity converging to the $k(\\in [0, 2])$-th order derivatives of constant equilibrium in weighted space $L^2_{\\gamma}$ is $t^{-\\frac{3}{4}+{\\gamma}-\\frac{k}{2}}$; however, the upper bounds of decay rate of the $k(\\in [0, 3])$-th order spatial derivatives of magnetic field converging to zero in weighted space $L^2_{\\gamma}$ is $t^{-\\frac{3}{4}+\\frac{{\\gamma}}{2}-\\frac{k}{2}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-29T12:32:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower bound",
      "space-time decay rates",
      "higher order derivatives",
      "th order spatial derivative",
      "hall-mhd equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}