In this paper, we investigate the linear stability/instability of the planar Couette flow to the two-dimensional compressible Euler-Euler system for (ρ, u) and (n, v) with the sound speeds c1 and c2 respectively, coupled each other through the drag force on \(\mathbb{T}\times\mathbb{R}\) . It is shown in general for the different sound speeds c1 ≠ c2 that the perturbations of the densities (ρ, n) and the velocities (u, v) demonstrate the stability in any fixed finite time interval (0, T], besides, excluding the zero mode, the densities (ρ, n) and the irrotational components of the velocities (u, v) obey the algebraic time-growth rates, while the rotational components of the velocities (u, v) exhibit the algebraic time-decay rates due to the inviscid damping. For the case c1 = c2 (same sound speeds), it is proved that the relative velocity u – v decays faster than those of the velocities u, v caused by the inviscid damping, but the linear instability of the densities (ρ, n) and the irrotational components of the velocities (u, v) is also shown for some large time in spite of the inviscid damping.
