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

Abstract

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 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 ℝ².