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Structural stability of cylindrical supersonic solutions to the steady Euler-Poisson system

Chunpeng Wang, Zihao Zhang

TL;DR

This work establishes the structural stability of smooth cylindrically symmetric supersonic Euler-Poisson flows in a nozzle for both three-dimensional irrotational and axisymmetric rotational regimes. It develops two complementary analytical frameworks: a linear-hyperbolic/elliptic energy method with a reflection-based a priori estimate and a nonlinear Galerkin-contraction argument for 3D irrotational flows, and a deformation-curl-Poisson decomposition together with a two-layer iteration for axisymmetric rotational flows. The results prove existence and uniqueness of smooth solutions under small boundary perturbations, with careful control of regularity and boundary compatibility, thereby advancing the mathematical theory of multi-dimensional Euler-Poisson systems in nozzle geometries. The findings have implications for plasma physics and semiconductor-device modeling by validating robust, high-speed flow configurations under perturbations.

Abstract

This paper concerns the structural stability of smooth cylindrically symmetric supersonic Euler-Poisson flows in nozzles. Both three-dimensional and axisymmetric perturbations are considered. On one hand, we establish the existence and uniqueness of three-dimensional smooth supersonic solutions to the potential flow model of the steady Euler-Poisson system. On the other hand, the existence and uniqueness of smooth supersonic flows with nonzero vorticity to the steady axisymmetric Euler-Poisson system are proved. The problem is reduced to solve a nonlinear boundary value problem for a hyperbolic-elliptic mixed system. One of the key ingredients in the analysis of three-dimensional supersonic irrotational flows is the well-posedness theory for a linear second order hyperbolic-elliptic coupled system, which is achieved by using the multiplier method and the reflection technique to derive the energy estimates. For smooth axisymmetric supersonic flows with nonzero vorticity, the deformation-curl-Poisson decomposition is utilized to reformulate the steady axisymmetric Euler-Poisson system as a deformation-curl-Poisson system together with several transport equations, so that one can design a two-layer iteration scheme to establish the nonlinear structural stability of the background supersonic flow within the class of axisymmetric rotational flows.

Structural stability of cylindrical supersonic solutions to the steady Euler-Poisson system

TL;DR

This work establishes the structural stability of smooth cylindrically symmetric supersonic Euler-Poisson flows in a nozzle for both three-dimensional irrotational and axisymmetric rotational regimes. It develops two complementary analytical frameworks: a linear-hyperbolic/elliptic energy method with a reflection-based a priori estimate and a nonlinear Galerkin-contraction argument for 3D irrotational flows, and a deformation-curl-Poisson decomposition together with a two-layer iteration for axisymmetric rotational flows. The results prove existence and uniqueness of smooth solutions under small boundary perturbations, with careful control of regularity and boundary compatibility, thereby advancing the mathematical theory of multi-dimensional Euler-Poisson systems in nozzle geometries. The findings have implications for plasma physics and semiconductor-device modeling by validating robust, high-speed flow configurations under perturbations.

Abstract

This paper concerns the structural stability of smooth cylindrically symmetric supersonic Euler-Poisson flows in nozzles. Both three-dimensional and axisymmetric perturbations are considered. On one hand, we establish the existence and uniqueness of three-dimensional smooth supersonic solutions to the potential flow model of the steady Euler-Poisson system. On the other hand, the existence and uniqueness of smooth supersonic flows with nonzero vorticity to the steady axisymmetric Euler-Poisson system are proved. The problem is reduced to solve a nonlinear boundary value problem for a hyperbolic-elliptic mixed system. One of the key ingredients in the analysis of three-dimensional supersonic irrotational flows is the well-posedness theory for a linear second order hyperbolic-elliptic coupled system, which is achieved by using the multiplier method and the reflection technique to derive the energy estimates. For smooth axisymmetric supersonic flows with nonzero vorticity, the deformation-curl-Poisson decomposition is utilized to reformulate the steady axisymmetric Euler-Poisson system as a deformation-curl-Poisson system together with several transport equations, so that one can design a two-layer iteration scheme to establish the nonlinear structural stability of the background supersonic flow within the class of axisymmetric rotational flows.

Paper Structure

This paper contains 15 sections, 12 theorems, 286 equations.

Table of Contents

  1. Introduction
  2. The nonlinear problem and main results
  3. Smooth supersonic irrotational flows
  4. Smooth axisymmetric supersonic rotational flows
  5. The stability analysis within irrotational flows
  6. The linearized problem
  7. Energy estimates for the linearized problem
  8. The well-posedness of the linearized problem
  9. Proof of Theorem 2.4
  10. The stability analysis within axisymmetric rotational flows
  11. The deformation-curl-Poisson decomposition
  12. The linearized problem
  13. A second-order linear hyperbolic-elliptic coupled system
  14. The well-posedness of the linearized problem
  15. Proof of Theorem 2.6

Key Result

Proposition 2.1

Fix $b_0> 0$. Given constants $\rho_0>0$, $U_{0}>0$, $S_0>0$ and $E_0$ satisfying where Then there exists a positive constant $R_0$ depending only the data $(\gamma,r_0,b_0,\rho_0,U_0,S_0, E_0)$ such that for any $r_1\in(r_0,r_0+R_0)$, the initial value problem 1-4 and 1-4-b has a unique smooth solution $(\bar{\rho},\bar{U}, \bar{S}, \bar{E})$ satisfying

Theorems & Definitions (18)

  • Proposition 2.1
  • Definition 2.2
  • Theorem 2.4
  • Theorem 2.6
  • Remark 2.7
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 8 more