Note on asymptotic profile of solutions to the linearized compressible Navier-Stokes flow

Abstract

We consider the asymptotic behavior as $t \to +\infty$ of the $L^{2}$-norm of the velocity of the linearized compressible Navier-Stokes equations in ${\bf R}^{n}$ ($n \geq 2$). As an application we shall study the optimality of the decay rate for the $L^{2}$-norm of the velocity by deriving a decay estimate from below as $t \to +\infty$. To get the estimates in the zone of high frequency we use a version of the energy method in the Fourier space combined with the Haraux-Komornik inequality and this seems much different from known techniques to study compressible Navier-Stokes system.