Some remarks on large-time behaviors for the linearized compressible Navier-Stokes equations

TL;DR

This paper investigates the large-time behavior of solutions to the linearized compressible Navier-Stokes equations in or different initial data thresholds, establishing growth and decay estimates and asymptotic profiles.

Contribution

It introduces a new threshold condition for initial data to distinguish between growth and decay behaviors in large-time dynamics.

Findings

Optimal growth estimates in low dimensions when |B_0|>0

Optimal decay estimates when |B_0|=0

Asymptotic profiles derived for solutions with weighted L^1 data

Abstract

In this paper, we consider the linearized compressible Navier-Stokes equations in the whole space $\mathbb{R}^n$. Concerning initial datum with suitable regularities, we introduce a new threshold $|\mathbb{B}_0|=0$ to distinguish different large-time behaviors. Particularly in the lower-dimensions, optimal growth estimates ($n=1$ polynomial growth, $n=2$ logarithmic growth) hold when $|\mathbb{B}_0|>0$, whereas optimal decay estimates hold when $|\mathbb{B}_0|=0$. Furthermore, we derive asymptotic profiles of solutions with weighted $L^1$ datum as large-time.