Asymptotic behavior of solutions to the compressible Navier-Stokes equation in a cylindrical domain

Abstract

Asymptotic behavior of solutions to the compressible Navier-Stokes equation around a given constant state is investigated on a cylindrical domain in $\mathbf{R}^{3}$, under the no slip boundary condition for the velocity field. The $L^{2}$ decay estimate is established for the perturbation from the constant state. It is also shown that the time-asymptotic leading part of the perturbation is given by a function satisfying a 1 dimensional heat equation. The proof is based on an energy method and asymptotic analysis for the associated linearized semigroup.