In this paper, we are concerned with the compressible viscous magnetohydrodynamic (MHD) system and investigate the large time behavior of strong solutions near constant equilibrium. In the eighties, Umeda, Kawashima and Shizuta initiated the dissipative mechanism for a rather general class of symmetric hyperbolic-parabolic systems. From the point of view of dissipativity, Kawashima in his doctoral dissertation established the optimal time-decay estimates of $L^{q}$-$L^{2}(q\in [1,2)$) type for solutions to the MHD system. Here, we shall improve Kawashima's efforts and give more precise description for the large time asymptotic behavior of solutions, not only in extra Lebesgue spaces but also in a full family of Besov norms with the negative regularity index. Precisely, we show that the $L^{p}$ norm (the slightly stronger $\dot{B}^{0}_{p,1}$ one in fact) of global solutions with the critical regularity, decays like $t^{-d(\frac{1}{p}-\frac{1}{4})}$ for $t\rightarrow\infty$. In particular, taking $p=2$ and $d=3$ goes back to the classical time decay $t^{-\frac{3}{4}}$. We derive new estimates which are used to deal with the strong coupling between the magnetic field and fluid dynamics.

The compressible Navier-Stokes-Poisson system takes the form of usual Navier-Stokes equations coupled with the self-consistent Poisson equation, which is used to simulate the transport of charged particles under the electric field of electrostatic potential force. In this paper, we focus on the large time behavior of global strong solutions in the $L^{p}$ Besov spaces of critical regularity. By exploring the dissipative effect arising from Poisson potential, we posed the new regularity assumption of low frequencies and then establish a \textit{sharp} time-weighted inequality, which leads to the optimal time-decay estimates of the solution. Indeed, we see that the decay of density is faster at the half rate than that of velocity, which is a different ingredient in comparison with the situation of usual Navier-Stokes equations. Our proof mainly depends on tricky and non classical Besov product estimates with respect to various Sobolev embeddings.

The present paper is dedicated to the large time asymptotic behavior of global strong solutions near constant equilibrium (away from vacuum) to the compressible Navier-Stokes-Poisson equations. Precisely, we present that under the same regularity assumptions as in \cite{SX2}, a \textit{different} time-decay framework of the $\dot{B}_{p,1}^{s}$ norm of the critical global solutions is established. The proof mainly depends on the pure energy argument \textit{without the spectral analysis}, which allows us to remove \textit{the usual smallness assumption of low frequencies of initial data}.