Abstract: In this paper, we investigate the linear stability/instability of the planar Couette flow to the two-dimensional compressible Euler-Euler system for $(\rho,{u})$ and $(n,{v})$ with the sound speeds $c_1$ and $c_2$ respectively, coupled each other through the drag force on $\mathbb{T} \times \mathbb{R}$. It is shown in general for the different sound speeds $c_1 \neq c_2$ that the perturbations of the densities $(\rho,n)$ and the velocities $({u},{v})$ demonstrate the stability in any fixed finite time interval $(0,T]$, besides, excluding the zero mode, the densities $(\rho, n)$ 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 $c_1=c_2$ (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 $(\rho, n)$ and the irrotational components of the velocities $({u},{v})$ is also shown for some large time in spite of the inviscid damping.

Key words: compressible Euler-Euler system, planar Couette flow, linear stability/instability