On the spectral properties for the linearized problem around space-time periodic states of the compressible Navier-Stokes equations

TL;DR

This paper investigates the spectral properties of the linearized compressible Navier-Stokes equations around space-time periodic states, revealing asymptotic behaviors of solutions under small Reynolds and Mach numbers.

Contribution

It provides the first analysis of Floquet exponents near the imaginary axis for this problem, offering insights into the long-term behavior of solutions.

Findings

Asymptotic expansions of Floquet exponents near the imaginary axis.

Leading part of the linearized solution operator as time approaches infinity.

Spectral analysis under small Reynolds and Mach numbers.

Abstract

This paper studies the linearized problem for the compressible Navier-Stokes equation around space-time periodic state in an infinite layer of $\mathbb{R}^n$ ($n=2,3$), and the spectral properties of the linearized evolution operator is investigated. It is shown that if the Reynolds and Mach numbers are sufficiently small, then the asymptotic expansions of the Floquet exponents near the imaginary axis for the Bloch transformed linearized problem are obtained for small Bloch parameters, which would give the asymptotic leading part of the linearized solution operator as $t\rightarrow\infty$.