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Ionized Gas in an Annular Region

Walter A. Strauss, Masahiro Suzuki

TL;DR

This work studies a Townsend discharge model confined to a bounded, star-shaped domain, formulating a mixed hyperbolic-parabolic-elliptic PDE system for ion and electron densities coupled to the electrostatic potential. The authors introduce a stability index κ(λ) via the lowest eigenvalue of a Schrödinger-type operator and show a κ(λ)–dependent breakdown of the trivial state, with sparking voltage λ* marking the threshold for discharge. They establish a local bifurcation from the trivial solution at λ* and prove a global radial bifurcation curve that either drives densities unbounded or terminates at a second trivial state at λ#, using Lyapunov–Schmidt reduction and Rabinowitz-type global theory. The electron-free case is treated to reveal a stable manifold structure, while positivity constraints on the densities shape the global bifurcation diagram, yielding physically meaningful scenarios in cylindrical and spherical geometries. Overall, the paper provides a rigorous framework for existence, stability, and global continuation of stationary states in ionization models beyond planar geometries, with implications for sparking and plasma formation analyses.

Abstract

We consider a plasma that is created by a high voltage difference $λ$, which is known as a Townsend discharge. We consider it to be confined to the region $Ω$ between two concentric spheres, two concentric cylinders, or more generally between two star-shaped surfaces. We first prove that if the plasma is initially relatively dilute, then either it may remain dilute for all time or it may not, depending on a certain parameter $κ(λ, Ω)$. Secondly, we prove that there is a connected one-parameter family of steady states. This family connects the non-ionized gas to a plasma, either with a sparking voltage $λ^*$ or with very high ionization, at least in the cylindrical or spherical cases.

Ionized Gas in an Annular Region

TL;DR

This work studies a Townsend discharge model confined to a bounded, star-shaped domain, formulating a mixed hyperbolic-parabolic-elliptic PDE system for ion and electron densities coupled to the electrostatic potential. The authors introduce a stability index κ(λ) via the lowest eigenvalue of a Schrödinger-type operator and show a κ(λ)–dependent breakdown of the trivial state, with sparking voltage λ* marking the threshold for discharge. They establish a local bifurcation from the trivial solution at λ* and prove a global radial bifurcation curve that either drives densities unbounded or terminates at a second trivial state at λ#, using Lyapunov–Schmidt reduction and Rabinowitz-type global theory. The electron-free case is treated to reveal a stable manifold structure, while positivity constraints on the densities shape the global bifurcation diagram, yielding physically meaningful scenarios in cylindrical and spherical geometries. Overall, the paper provides a rigorous framework for existence, stability, and global continuation of stationary states in ionization models beyond planar geometries, with implications for sparking and plasma formation analyses.

Abstract

We consider a plasma that is created by a high voltage difference , which is known as a Townsend discharge. We consider it to be confined to the region between two concentric spheres, two concentric cylinders, or more generally between two star-shaped surfaces. We first prove that if the plasma is initially relatively dilute, then either it may remain dilute for all time or it may not, depending on a certain parameter . Secondly, we prove that there is a connected one-parameter family of steady states. This family connects the non-ionized gas to a plasma, either with a sparking voltage or with very high ionization, at least in the cylindrical or spherical cases.
Paper Structure (18 sections, 25 theorems, 192 equations, 3 figures)

This paper contains 18 sections, 25 theorems, 192 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Geometric assumption and notation
  4. Dilute plasma
  5. Sparking voltage
  6. Stationary solutions
  7. Stability analysis of the trivial solution
  8. Nonlinear stability for $\kappa>0$
  9. Nonlinear instability for $\kappa<0$
  10. Electron-free case
  11. Local bifurcation
  12. Transport equation
  13. Linearized operator
  14. The system, rewritten
  15. Proof of local bifurcation
  16. ...and 3 more sections

Key Result

Theorem 2.1

For given $\lambda$, suppose the stability index $\kappa(\lambda)$ is positive. There exists $\varepsilon>0$ such that if the initial data ${\mathcal{R}}_0 \in X$ with $\|{\mathcal{R}}_0\|_{X} < \varepsilon$ satisfies the compatibility conditions com, then the problem Mmodel has a unique solution ${ Moreover, there is a constant $C$ such that $\sup_{0\le t<\infty} \|{\mathcal{R}}(t)\|_X \le C\|{\m

Figures (3)

  • Figure 1: domain $\Omega$
  • Figure 2: alternative (i)
  • Figure 3: alternative (ii)

Theorems & Definitions (48)

  • Theorem 2.1: Stability
  • Theorem 2.2: Instability
  • Proposition 2.3: No electrons
  • Lemma 2.4
  • proof
  • Theorem 2.5: Local Bifurcation
  • Theorem 2.6: Global Bifurcation
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 38 more