Nonlinear inviscid damping for a class of monotone shear flows in a finite channel

Abstract

We prove the nonlinear inviscid damping for a class of monotone shear flows in $\mathbb{T} \times [0,1]$ for initial perturbation in Gevrey-$\frac{1}{s}$ class $(1\lt \frac{1}{s} \lt 2)$ with compact support. The main new idea of the proof is to construct and use the wave operator of a slightly modified Rayleigh operator in a well-chosen coordinate system.