Asymptotic Behavior of the Semigroup Associated with the Linearized Compressible Navier–Stokes Equation in an Infinite Layer

Abstract

Asymptotic behavior of solutions to the linearized compressible Navier–Stokes equation around a given constant state is considered in an infinite layer ${\mathbf{R}}^{n-1}\times (0,a)$, $n\ge 2$, under the no slip boundary condition for the momentum. The $L^p$ decay estimates of the associated semigroup are established for all $1\le p\le \infty$. It is also shown that the time-asymptotic leading part of the semigroup is given by an $n-1$ dimensional heat semigroup.