Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non-constant density in a finite channel

We prove the nonlinear inviscid damping for a class of monotone shear flows with non-constant background density for the two-dimensional ideal inhomogeneous fluids in $\mathbb{T}\times [0,1]$ when the initial perturbation is in Gevrey-$\frac{1}{s}$ ($\frac{1}{2}<s<1$) class with compact support.