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.