# Radial transonic shock solutions of Euler-Poisson system in convergent nozzles

## TL;DR

This paper proves the existence and uniqueness of a transonic shock solution for the Euler-Poisson system in a convergent nozzle under specific boundary conditions and electric field strength, advancing understanding of shock phenomena in electrohydrodynamics.

## Contribution

It establishes the first rigorous proof of a unique transonic shock solution for the Euler-Poisson system in a convergent nozzle with specified boundary data.

## Key findings

- Existence of a transonic shock solution under large electric field conditions.
- Uniqueness of the shock solution in the given setting.
- Dependence of solution properties on electric field strength and nozzle length.

## Abstract

Given constant data of density $\rho_0$, velocity $-u_0{\bf e}_r$, pressure $p_0$ and electric force $-E_0{\bf e}_r$ for supersonic flow at the entrance, and constant pressure $p_{\rm ex}$ for subsonic flow at the exit, we prove that Euler-Poisson system admits a unique transonic shock solution in a two dimensional convergent nozzle, provided that $u_0>0$, $E_0>0$, and that $E_0$ is sufficiently large depending on $(\rho_0, u_0, p_0)$ and the length of the nozzle.