Asymptotic profile of solutions to the linearized compressible Navier-Stokes flow

TL;DR

This paper investigates the long-term decay behavior of solutions to the linearized compressible Navier-Stokes equations in multiple dimensions, establishing optimal decay rates for the velocity's $L^2$-norm.

Contribution

It provides a detailed analysis of the asymptotic decay rates and demonstrates their optimality for solutions in the linearized compressible Navier-Stokes framework.

Findings

Derived decay estimates for the $L^2$-norm of velocity

Proved the optimality of decay rates

Extended results to higher dimensions (n ≥ 2)

Abstract

We consider the asymptotic behavior as time goes to infinity 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 the decay estimate from below as time goes to infinity.