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We prove nonlinear asymptotic stability of a large class of monotonic shearflows among solutions of the 2D Euler equations in the channel$\mathbb{T}\times[0,1]$. More precisely, we consider shear flows $(b(y),0)$given by a function $b$ which is Gevrey smooth, strictly increasing, and linearoutside a compact subset of the interval $(0,1)$ (to avoid boundarycontributions which are incompatible with inviscid damping). We also assumethat the associated linearized operator satisfies a suitable spectralcondition, which is needed to prove linear inviscid damping.  Under these assumptions, we show that if $u$ is a solution which is a smalland Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$,then the velocity field $u$ converges strongly to a nearby shear flow as thetime goes to infinity. This is the first nonlinear asymptotic stability resultfor Euler equations around general steady solutions for which the linearizedflow cannot be explicitly solved.

Nonlinear inviscid damping near monotonic shear flows