Paper
Nonlinear inviscid damping near monotonic shear flows
Authors
Alexandru D. Ionescu, Hao Jia
Abstract
We prove nonlinear 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 nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved.