Time periodic problem for the compressible Navier–Stokes equation on $\mathbb{R}^2$ 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 $L^\infty$ 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 $\mathbb{R}^2$.