On decay properties of the linearized compressible Navier–Stokes equations around time-periodic flows in an infinite layer

Abstract

We investigate decay properties of solutions to the linearized compressible Navier–Stokes equation around time-periodic parallel flow. We show that if the Reynolds and Mach numbers are sufficiently small, solutions of the linearized problem decay in $L^2$ norm as an $n-1$ dimensional heat kernel. Furthermore, we prove that the asymptotic leading part of solutions is given by solutions of an $n-1$ dimensional linear heat equation with a convective term multiplied by time-periodic function.