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

Abstract

We are concerned with the linear stability of the Couette flow for the nonisentropic compressible Navier-Stokes equations with vanished shear viscosity in a domain T × 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.

An eigen-representation of the Navier–Stokes equations

“…In the non-isentropic regime, Duck et al [8] investigated the linear stability of the plane Couette flow for the full compressible Navier-Stokes equations. Recently, Zhai [43] studied the linear stability of the Couette flow for the non-isentropic compressible Navier-Stokes equations with vanished shear viscosity. However, to the best of our knowledge, there is few result on the nonlinear stability of the plane Couette flow for the full compressible Navier-Stokes equations.…”