Linear stability analysis of the Couette flow for the two dimensional non-isentropic compressible Euler equations

TL;DR

This paper conducts a linear stability analysis of Couette flow within the non-isentropic compressible Euler equations, revealing instability in density, temperature, and compressible velocity, while demonstrating damping in the incompressible velocity component.

Contribution

It introduces a novel Lyapunov type instability analysis for key variables and identifies inviscid damping effects in the flow.

Findings

Instability in density, temperature, and compressible velocity components.

Inviscid damping observed in the incompressible velocity component.

Utilization of conservation laws from the system's structure.

Abstract

This note is devoted to the linear stability of the Couette flow for the non-isentropic compressible Euler equations in a domain $\mathbb{T}\times \mathbb{R}$. Exploiting the several conservation laws originated from the special structure of the linear system, we obtain a Lyapunov type instability for the density, the temperature, the compressible part of the velocity field, and also obtain an inviscid damping for the incompressible part of the velocity field.