Spectral Properties of the Linearized Semigroup of the Compressible Navier–Stokes Equation on a Periodic Layer

Abstract

The linearized problem for the compressible Navier-Stokes equation around a given constant state is considered in a periodic layer of ${\mathbb{R}}^n$ with $n\ge 2$, and spectral properties of the linearized semigroup is investigated. It is shown that the linearized operator generates a $C_0$-semigroup in $L^2$ over the periodic layer and the time-asymptotic leading part of the semigroup is given by a $C_0$-semigroup generated by an $n-1$ dimensional elliptic operator with constant coefficients that are determined by solutions of a Stokes system over the basic period domain.