On the existence and stability of time periodic solution to the compressible Navier-Stokes equation on the whole space

The existence of a time periodic solution of the compressible Navier-Stokes equation on the whole space is proved for sufficiently small time periodic external force when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-T -map associated with the linearized problem around the motionless state with constant density in some weighted L∞ and Sobolev spaces. The time periodic solution is shown to be asymptotically stable under sufficiently small initial perturbations and the L∞ norm of the perturbation decays as time goes to infinity.