  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 $t\to \infty$. To our knowledge, this is the
first global well-posedness result for the 2-D inhomogeneous incompressible
Euler equations.
