Recurrence in 2D inviscid channel flow

I will prove a recurrence theorem which says that any H s ( s > 2 ) solution to the 2D inviscid channel flow returns repeatedly to an arbitrarily small H 0 neighborhood. The periodic boundary condition is imposed along the stream-wise direction. The result is an extension of an early result of Li [Y. Li, A recurrence theorem on the solutions to the 2D Euler equation, Asian J. Math. 13 (1) (2009) 1–6] on the 2D Euler equation under periodic boundary conditions along both directions.