Global Existence and Optimal Decay Rate of the Compressible Magnetohydrodynamic Equation with Potential Force

In this paper, we consider the three dimensional compressible viscous magnetohydrodynamic equations (MHD) with the external potential force. We first derive the corresponding non-constant stationary solutions. Next, we show global wellposedness of the initial value problem for the three dimensional compressible viscous magnetohydrodynamic equations, provided that the initial data is close to the stationary solution. Finally, based on the elaborate energy estimates for the nonlinear system and L^p - L^q decay estimates of the linearized equation, we show the optimal convergence rates of the solution in L^q-norm with 2?q?6 and its first derivative in L²-norm when the initial perturbation is bounded in L^p-norm with 1? p<6 5.