Linear Stability and Inviscid Damping of Monotone Shear Flows for 2D Compressible Euler Equations

We study 2D compressible Euler equations linearized around monotone shear flows $(U(y),0)$ on $\mathbb{T} \times \mathbb{R}$. The shear rate $U'$ is strictly positive, not necessarily close to any constant, and varies sufficiently slowly. For every fixed Mach number $M > 0$, we prove that the densit