We prove the nonlinear inviscid damping for a class of monotone shear flows in T × [0,1] for initial perturbation in Gevrey-1/s class (1< 1/s < 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.
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Masmoudi et al. (2024) studied this question.