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In this talk,we will discuss the structural stability for subsonic,supersonic,and transonic solutions for the Euler-Poisson equations.The Euler-Poisson equations for subsonic flows contain an elliptic system.We find a nice structure for this elliptic system so that the energy estimate helps us to get the existence of solutions.The stability of supersonic and transonic solutions for the Euler-Poisson system is obtained via energy method with delicate choice for the multipliers.