Subsonic flow for multidimensional Euler-Poisson system

TL;DR

This paper proves the existence and stability of subsonic potential flows in multidimensional Euler-Poisson systems within a finite nozzle, using advanced elliptic PDE techniques to handle boundary conditions and system structure.

Contribution

It introduces a novel method for obtaining a priori estimates for solutions of a nonlinear elliptic system derived from the Euler-Poisson equations, establishing structural stability of subsonic flows.

Findings

Established unique existence of subsonic potential flow

Proved stability of solutions under data perturbations

Developed a technique for a priori $C^{1,eta}$ estimates

Abstract

We establish unique existence and stability of subsonic potential flow for steady Euler-Poisson system in a multidimensional nozzle of a finite length when prescribing the electric potential difference on 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 $C^{1,\alp}$ estimates of solutions to a quasi-linear second order elliptic system with mixed boundary conditions in a multidimensional domain with Lipschitz continuous boundary. Particularly, we discovered a special structure of the Euler-Poisson system which enables us to obtain $C^{1,\alp}$ estimates of velocity potential and electric potential functions, and this leads us to establish structural stability of subsonic flows for the Euler-Poisson system under perturbations of various data.