On the Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes Equation on the Whole Space

Abstract

The existence of a time periodic solution of the compressible Navier–Stokes equation on the whole space is proved for a sufficiently small time periodic external force 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 L∞ and Sobolev spaces. The time periodic solution is shown to be asymptotically stable under sufficiently small initial perturbations and the L∞ norm of the perturbation decays as time goes to infinity.