Large time behavior of solutions to the compressible Navier-Stokes equations with potential force

The compressible Navier-Stokes equation with a potential external force is considered in R3 in the present paper. Under the smallness assumption on both the external force and the initial perturbation of the stationary solution in some Sobolev spaces, the existence theory of global solutions to the stationary profile is established. Furthermore, when the H˙−s norm (s∈(0,12)) of initial perturbation is finite, we obtain the optimal time decay rates of the solutions in L2-norm. As a corollary, the Lp-Lq(3/2