Remark on a lower bound of perturbations in 2D inviscid shear flow in a periodic box

Bedrossian and Masmoudi (2015) showed nonlinear stability of 2D inviscid parallel shear flows in the infinite stripe $(\mathbb{R}\mathrm{/}2\mathbb{Z})\times \mathbb{R}$. Inspired by their result, combining a diffeomorphism group result by Misio{\l}ek (1993), in this paper we give a lower bound of perturbations in 2D inviscid shear flow in a periodic box.