In this talk, we discuss about the problem concerning the motion of the barotropic compressible Navier-Stokes equations (CNS) in the smooth exterior domain with some free surface. By applying the method of Lagrangian coordinates, the nonlinear problem reduces to the study of the Lam'e system with the free boundary condition. Then the classical theory of the $L_p$-$L_q$ decay estimate is established for such linearized system, by taking advantage of the local energy approach. At last, we apply the $L_p$-$L_q$ decay theory to construct the global solution of (CNS).