Subsonic solutions for steady Euler-Poisson system in two dimensional nozzles

TL;DR

This paper proves the existence and stability of subsonic steady flows in a 2D nozzle governed by the Euler-Poisson system, using Helmholtz decomposition and stream function methods to handle the nonlinear coupled equations.

Contribution

It introduces a novel approach combining Helmholtz decomposition and stream function formulation to analyze subsonic Euler-Poisson flows in nozzles, establishing existence and stability results.

Findings

Existence of subsonic solutions in a 2D nozzle.

Stability of these solutions under boundary conditions.

Development of estimates for the flow map ensuring uniqueness.

Abstract

In this paper, we prove the existence and stability of subsonic flows for steady full Euler-Poisson system in a two dimensional nozzle of finite length when imposing the electric potential difference on non-insulated boundary from a fixed point at the entrance, and prescribing the pressure at the exit of the nozzle. The Euler-Poisson system for subsonic flow is a hyperbolic-elliptic coupled nonlinear system. One of the crucial ingredient of this work is the combination of Helmholtz decomposition for the velocity field and stream function formulation together. In terms of the Helmholtz decomposition, the Euler-Poisson system is rewritten as a second order nonlinear elliptic system of three equations and transport equations for entropy and pseudo-Bernoulli's invariant. The associated elliptic system in a Lipschitz domain with nonlinear boundary conditions is solved with the help of the estimates developed in [2] based on its nice structure. The transport equations are resolved via the flow map induced by the stream function formulation. Furthermore, the delicate estimates for the flow map give the uniqueness of the solutions.