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Three-dimensional Supersonic flows for the steady Euler-Poisson system in divergent nozzles

Hyangdong Park

TL;DR

This work establishes the existence and stability of axisymmetric supersonic flows for the steady Euler-Poisson system in a three-dimensional divergent nozzle with nonzero vorticity and angular momentum. It introduces a Helmholtz decomposition-based reformulation and constructs a robust multi-layer fixed-point scheme to obtain a unique solution under small entrance-data perturbations, providing precise Sobolev-norm estimates. A key feature is the careful handling of axis singularities without employing weighted norms, enabling standard Sobolev theory to yield regularity from the reformulated elliptic system. The results lay a rigorous foundation for multi-dimensional transonic-flow analysis and potential transonic-shock studies in Euler-Poisson settings.

Abstract

We are concerned with the unique existence of an axisymmetric supersonic solution with nonzero vorticity and nonzero angular momentum density for the steady Euler-Poisson system in three-dimensional divergent nozzles when prescribing the velocity, strength of electric field, and the entropy at the entrance. We first reformulate the problem via the method of the Helmholtz decomposition for three-dimensional axisymmetric flows and obtain a solution to the reformulated problem by the iteration method. Furthermore, we deal carefully with singularity issues related to the polar angle on the axis of the divergent nozzle.

Three-dimensional Supersonic flows for the steady Euler-Poisson system in divergent nozzles

TL;DR

This work establishes the existence and stability of axisymmetric supersonic flows for the steady Euler-Poisson system in a three-dimensional divergent nozzle with nonzero vorticity and angular momentum. It introduces a Helmholtz decomposition-based reformulation and constructs a robust multi-layer fixed-point scheme to obtain a unique solution under small entrance-data perturbations, providing precise Sobolev-norm estimates. A key feature is the careful handling of axis singularities without employing weighted norms, enabling standard Sobolev theory to yield regularity from the reformulated elliptic system. The results lay a rigorous foundation for multi-dimensional transonic-flow analysis and potential transonic-shock studies in Euler-Poisson settings.

Abstract

We are concerned with the unique existence of an axisymmetric supersonic solution with nonzero vorticity and nonzero angular momentum density for the steady Euler-Poisson system in three-dimensional divergent nozzles when prescribing the velocity, strength of electric field, and the entropy at the entrance. We first reformulate the problem via the method of the Helmholtz decomposition for three-dimensional axisymmetric flows and obtain a solution to the reformulated problem by the iteration method. Furthermore, we deal carefully with singularity issues related to the polar angle on the axis of the divergent nozzle.

Paper Structure

This paper contains 7 sections, 12 theorems, 184 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problems and Main Theorems
  3. One-dimensional radial solutions
  4. Problems and Theorems
  5. Reformulation Problems via Helmholtz decomposition
  6. Proof of Theorem \ref{['M-Thm-super']}
  7. Proof of Lemma \ref{['lema-po']}

Key Result

Lemma 2.1

For given $\gamma>1$, $r_{\rm en}>1$, $m_0>0$, and $S_0>0$, suppose that constants $\rho_0$, $E_0$, and $M_0$ satisfy ini-con-super-1. For a given constant $\bar{\delta}>0$ sufficiently small, there exists a constant $r_{\ast}\in(r_{\rm en},\infty)$ depending on $r_{\rm en}$, $\gamma$, $\bar{b}$, $m for $\rho_s$ in ini-con-super-1.

Figures (1)

  • Figure 1.1: Flows in a divergent nozzle $\Omega$

Theorems & Definitions (22)

  • Definition 1.1
  • Lemma 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • Remark 2.5
  • Theorem 3.1
  • Proposition 4.1
  • proof : Proof of Proposition \ref{['su-lemma-1']}
  • ...and 12 more