20162016-05-06 , Faculty of Mathematics, Kyushu University
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^∞ and Sobolev spaces the spectral properties of the time-T-map is 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.