Lower Bound and Space-time Decay Rates of Higher Order Derivatives of Solution for the Compressible Navier-Stokes and Hall-MHD Equations

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 $t^{-\frac{3}{4}+\frac{{\gamma}}{2}-\frac{k}{2}}$.