Supersonic flows of the Euler-Poisson system in three-dimensional cylinders

TL;DR

This paper proves the existence and uniqueness of three-dimensional supersonic solutions to the Euler-Poisson system in cylindrical nozzles, including cases with nonzero vorticity and swirl, using advanced mathematical techniques.

Contribution

It extends previous results to three-dimensional flows with nonzero vorticity and swirl, employing a reformulation into hyperbolic-elliptic coupled systems and addressing singularities.

Findings

Existence and uniqueness of irrotational supersonic solutions in cylindrical nozzles.

Existence and uniqueness of axisymmetric solutions with nonzero vorticity and swirl.

Development of methods to handle singularities on the axis and domain corners.

Abstract

In this paper, we prove the unique existence of three-dimensional supersonic solutions to the steady Euler-Poisson system in cylindrical nozzles when prescribing the velocity, entropy, and the strength of electric field at the entrance. We first establish the unique existence of irrotational supersonic solutions in a cylindrical nozzle with an arbitrary cross section by extending the results of \cite{bae2021three} with an aid of weighted Sobolev norms. Then, we establish the unique existence of three-dimensional axisymmetric supersonic solutions to the Euler-Poisson system with nonzero vorticity in a circular cylinder. In particular, we construct a three-dimensional solution with a nonzero angular momentum density (or equivalently a nonzero swirl). Therefore this is truly a three dimensional flow in the sense that the Euler-Poisson system cannot be reduced to a two dimensional system via a stream function formulation. The main idea is to reformulate the system into a second order hyperbolic-elliptic coupled system and two transport equations via the method of Helmholtz decomposition, and to employ the method of iterations. Several technical issues, including the issue of singularities on the axis of symmetry and the issue of corner singularities in a Lipschitz domain, are carefully addressed.