Existence and stability of time periodic solution to the compressible Navier-Stokes equation for time periodic external force with symmetry

Time periodic problem for the compressible Navier-Stokes equation for barotropic flow on the whole space is studied. The existence of a time periodic solution is proved for sufficiently small time periodic external force with some symmetry 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 Sobolev space. The stability of the time periodic solution is also proved and the decay estimate of the perturbation is established.