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In this paper, both structural and dynamical stabilities of steady transonicshock solutions for one-dimensional Euler-Poission system are investigated.First, a steady transonic shock solution with supersonic backgroumd charge isshown to be structurally stable with respect to small perturbations of thebackground charge, provided that the electric field is positive at the shocklocation. Second, any steady transonic shock solution with the supersonicbackground charge is proved to be dynamically and exponentially stable withrespect to small perturbation of the initial data, provided the electric fieldis not too negative at the shock location. The proof of the first stabilityresult relies on a monotonicity argument for the shock position and thedownstream density, and a stability analysis for subsonic and supersonicsolutions. The dynamical stability of the steady transonic shock for theEuler-Poisson equations can be transformed to the global well-posedness of afree boundary problem for a quasilinear second order equation with nonlinearboundary conditions. The analysis for the associated linearized problem playsan essential role.

Stability of Transonic Shock Solutions for One-Dimensional Euler-Poisson Equations