Nonlinear inviscid damping for 2-D inhomogeneous incompressible Euler equations

We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressible Euler equations are globally well-posed and the velocity converges strongly to a shear flow close to the Couette flow, and the vorticity will be driven to small scales by a linear evolution and weakly converges as $t\to \mathrm{\infty }$. To our knowledge, this is the first global well-posedness result for the 2-D inhomogeneous incompressible Euler equations.