Time periodic solutions of compressible Navier-Stokes equations

In this paper, we study the Navier-Stokes equations with a time periodic external force in Rn. We show that a time periodic solution exists when the space dimension n?5 under some smallness assumption. The main idea is to combine the energy method and the spectral analysis for the optimal decay estimates on the linearized solution operator. With the optimal decay estimates, we prove the existence and uniqueness of time periodic solution in some suitable function space by the contraction mapping theorem. In addition, we also study the time asymptotic stability of the time periodic solution.