Abstract: The interest in studying stability of shear flows dates back to the origin of fluid dynamics. In the talk, we will briefly introduce the problem in the incompressible setting. Then we consider the 2D isothermal compressible Euler equations in T×R, where a steady solution is given by the Couette flow, i.e. a shear flow with a linear velocity profile, with constant density. For this problem, in the physics literature, the linearization around the Couette flow was considered by Chagelishvili et al. By some heuristic argument they claim that there is a linear growth for a sort of linearized energy. Our result regards the characterization on the frequency space of the solution of the linearization around the Couette flow. Thanks to standard Fourier transform techniques, by proper weighting, we are able to reduce the system to a 2D non-autonomous dynamical system, where we study the dynamics. As a consequence we are able to prove in a rigorous way the claim about the linear growth, being more precise on the dependece of compressible and incompressible effects. In the end, if time permits, we discuss also about the stability of monotone shear flows “close” to Couette.