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

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