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In this paper, we investigate linear stability properties of the 2Disentropic compressible Euler equations linearized around a shear flow given bya monotone profile, close to the Couette flow, with constant density, in thedomain $\mathbb{T}\times \mathbb{R}$. We begin by directly investigating theCouette shear flow, where we characterize the linear growth of the compressiblepart of the fluid while proving time decay for the incompressible part(inviscid damping with slower rates). Then we extend the analysis to monotoneshear flows near Couette, where we are able to give an upper bound, superlinearin time, for the compressible part of the fluid. The incompressible part enjoysan inviscid damping property, analogous to the Couette case. In the pureCouette case, we exploit the presence of an additional conservation law (whichconnects the vorticity and the density on the moving frame) in order to reducethe number of degrees of freedom of the system. The result then follows byusing weighted energy estimates. In the general case, unfortunately, thisconservation law no longer holds. Therefore we define a suitable weightedenergy functional for the whole system, which can be used to estimate theirrotational component of the velocity but does not provide sharp bounds on thesolenoidal component. However, even in the absence of the aforementionedadditional conservation law, we are still able to show the existence of afunctional relation which allows us to recover somehow the vorticity from thedensity, on the moving frame. By combining the weighted energy estimates withthe functional relation we also recover the inviscid damping for the solenoidalcomponent of the velocity.

Linear stability analysis for 2D shear flows near Couette in the isentropic Compressible Euler equations