Convergence rate for the compressible Navier-Stokes equations with external force

Communicated by (xxxxxxxxxx) For the viscous and heat-conductive fluids governed by the compressible Navier-Stokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the Lp −Lq estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space, when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L 2-norm are obtained when the L1-norm of the perturbation is bounded