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

TL;DR

This paper investigates the structural and dynamical stability of steady transonic shock solutions in the one-dimensional Euler-Poisson system, demonstrating conditions under which these shocks remain stable under small perturbations.

Contribution

It provides new stability results for transonic shocks in Euler-Poisson equations, including conditions for structural and exponential dynamical stability.

Findings

Structural stability with positive electric field at shock

Exponential dynamical stability under small initial perturbations

Stability analysis via linearized problem and free boundary formulation

Abstract

In this paper, both structural and dynamical stabilities of steady transonic shock solutions for one-dimensional Euler-Poission system are investigated. First, a steady transonic shock solution with supersonic backgroumd 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 the supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbation 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 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.