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 $t^{ -\frac{ 3}{ 4}+\frac{ { \gamma}}{ 2}-\frac{ k}{ 2}}$.
Categories: math.AP