Archive for Rational Mechanics and Analysis

, Volume 202, Issue 3, pp 787–827 | Cite as

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

  • Tao Luo
  • Jeffrey Rauch
  • Chunjing Xie
  • Zhouping Xin
Article

Abstract

In this paper, both structural and dynamical stabilities of steady transonic shock solutions for one-dimensional Euler–Poisson systems are investigated. First, a steady transonic shock solution with a supersonic background charge is shown to be structurally stable with respect to small perturbations of the background charge, provided that the electric field is positive at the shock location. Second, any steady transonic shock solution with a supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbations of the initial data, provided the electric field is not too negative at the shock location. The proof of the first stability result relies on a monotonicity argument for the shock position and the downstream density, and on a stability analysis for subsonic and supersonic solutions. The dynamical stability of the steady transonic shock for the Euler–Poisson equations can be transformed to the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions. The analysis for the associated linearized problem plays an essential role.

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Tao Luo
    • 1
  • Jeffrey Rauch
    • 2
  • Chunjing Xie
    • 2
  • Zhouping Xin
    • 3
  1. 1.Department of Mathematics and StatisticsGeorgetown UniversityWashingtonUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.The Institute of Mathematical Sciences and Department of MathematicsThe Chinese University of Hong KongShatinHong Kong

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