The topologicial behaviors of generic steady states of 2D Euler equations are closely related to the long-time behavior of solutions. 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. In the end, I will mention an application of our result. This is the joint work with Tarek Elgindi, Ayman Said and Chunjing Xie.