Classification of the steady states for 2D Euler equation is a classical topic in fluid mechanics. In this talk, we consider the rigidity of the analytic steady states in  bounded simply-connected domains. By studying an over-determined elliptic problem in Serrin type, we show the stream functions for the steady state are either radial functions or solutions to semi-linear elliptic equations.  This is the joint work with Tarek Elgindi, Ayman Said and Chunjing Xie.