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

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.