Subsonic Flow for the Multidimensional Euler–Poisson System

Abstract

We establish the existence and stability of subsonic potential flow for the steady Euler–Poisson system in a multidimensional nozzle of a finite length when prescribing the electric potential difference on a non-insulated boundary from a fixed point at the exit, and prescribing the pressure at the exit of the nozzle. The Euler–Poisson system for subsonic potential flow can be reduced to a nonlinear elliptic system of second order. In this paper, we develop a technique to achieve a priori $${C^{1,\alpha}}$$ estimates of solutions to a quasi-linear second order elliptic system with mixed boundary conditions in a multidimensional domain enclosed by a Lipschitz continuous boundary. In particular, we discovered a special structure of the Euler–Poisson system which enables us to obtain $${C^{1,\alpha}}$$ estimates of the velocity potential and the electric potential functions, and this leads us to establish structural stability of subsonic flows for the Euler–Poisson system under perturbations of various data.