Asymptotic analysis of acoustic waves in a porous medium: Initial layers in time

This is the first of a series of three papers which study acoustic waves governed by the linearized compressible Navier-Stokes equations in a porous medium. In particular, we want to analyze the simultaneous inviscid and high frequency limits of fluid flows in a porous medium. In this paper, we focus on the case of strongly viscous flow, namely fluid flow without the presence of boundary layers in space. We study the behavior of the energy using two-scale expansions in space and reveal that initial layers in time trap the energy carried by the flow during the usual two-scale homogenization process. We examine the time-space boundary layer problem in our forthcoming works.