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