Non-linear inviscid damping near monotonic shear flows

We prove non-linear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel. More precisely, we consider shear flows given by a function which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if is a solution which is a small and Gevrey smooth perturbation of such a shear flow at time then the velocity field converges strongly to a nearby shear flow as the time goes to infinity. This is the first non-linear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved.