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

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