Optimal convergence rates for the compressible Navier-Stokes equations with potential forces

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 L ^p - L^q 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 L²-norm are obtained when the L¹-norm of the perturbation is bounded. © World Scientific Publishing Company.