Linear stability of the Couette flow for the non-isentropic compressible fluid

TL;DR

This paper analyzes the linear stability of Couette flow in a non-isentropic compressible fluid, revealing instability in some variables and damping in others, with conditions for enhanced dissipation.

Contribution

It provides the first analysis of linear stability for non-isentropic compressible Navier-Stokes equations with vanishing shear viscosity, including instability and damping results.

Findings

Lyapunov instability of density, temperature, and compressible velocity components

Inviscid damping of the incompressible velocity component

Enhanced dissipation under specific initial data conditions

Abstract

We are concerned with the linear stability of the Couette flow for the non-isentropic compressible Navier-Stokes equations with vanished shear viscosity in a domain $\mathbb{T}\times \mathbb{R}$. For a general initial data settled in Sobolev spaces, we obtain a Lyapunov type instability of 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. Moreover, if the initial density, the initial temperature and the incompressible part of the initial velocity field satisfy some quality relation, we can prove the enhanced dissipation phenomenon for the velocity field.