Time periodic problem for the compressible Navier-Stokes equation on R2 with antisymmetry

The compressible Navier Stokes equation is considered on the two dimensional whole space when the external force is periodic in the time variable. The existence of a time periodic solution is proved for sufficiently small time periodic external force with antisymmetry condition. The proof is based on using the time-T-map associated with the linearized problem around the motionless state with constant density. In some weighted L-infinity and Sobolev spaces the spectral properties of the time-T-map are investigated by a potential theoretic method and an energy method. The existence of a stationary solution to the stationary problem is also shown for sufficiently small time-independent external force with antisymmetry condition on R-2.