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Classical solutions to a mixed-type PDE with a Keldysh-type degeneracy and accelerating transonic solutions to the Euler-Poisson system

Myoungjean Bae, Ben Duan, Chunjing Xie

TL;DR

This work develops a rigorous framework for classical solutions to a class of mixed-type PDEs with Keldysh-type degeneracy using singular perturbations and $H^m$-regularity, generalizing prior Kuzmin-type results. It then applies the linear theory to the steady Euler–Poisson system to establish the structural stability of a one-dimensional smooth transonic state and to construct a two-dimensional classical solution whose sonic interface is a regular graph, across which all flow variables remain $C^1$. The key methods combine vanishing-viscosity approximations, extension arguments for global regularity, and a carefully designed linearization with an iterative scheme to handle the coupling between elliptic and Keldysh-type equations. The results provide a rigorous existence and stability theory for smooth transonic flows with an accelerating transition, highlighting the electric field’s role in modifying transonic behavior and offering a foundation for further analysis of multidimensional plasma and semiconductor models.

Abstract

In this paper, we first prove the existence of classical solutions to a class of Keldysh-type equations. Next, we apply this existence result to prove the structural stability of one-dimensional smooth transonic solutions to the steady Euler-Poisson system. Most importantly, the solutions constructed in this paper are classical solutions to the Euler-Poisson system, thus their sonic interfaces are not weak discontinuities in the sense that all the flow variables, such as density, velocity and pressure, are at least $C^1$ across the interfaces.

Classical solutions to a mixed-type PDE with a Keldysh-type degeneracy and accelerating transonic solutions to the Euler-Poisson system

TL;DR

This work develops a rigorous framework for classical solutions to a class of mixed-type PDEs with Keldysh-type degeneracy using singular perturbations and -regularity, generalizing prior Kuzmin-type results. It then applies the linear theory to the steady Euler–Poisson system to establish the structural stability of a one-dimensional smooth transonic state and to construct a two-dimensional classical solution whose sonic interface is a regular graph, across which all flow variables remain . The key methods combine vanishing-viscosity approximations, extension arguments for global regularity, and a carefully designed linearization with an iterative scheme to handle the coupling between elliptic and Keldysh-type equations. The results provide a rigorous existence and stability theory for smooth transonic flows with an accelerating transition, highlighting the electric field’s role in modifying transonic behavior and offering a foundation for further analysis of multidimensional plasma and semiconductor models.

Abstract

In this paper, we first prove the existence of classical solutions to a class of Keldysh-type equations. Next, we apply this existence result to prove the structural stability of one-dimensional smooth transonic solutions to the steady Euler-Poisson system. Most importantly, the solutions constructed in this paper are classical solutions to the Euler-Poisson system, thus their sonic interfaces are not weak discontinuities in the sense that all the flow variables, such as density, velocity and pressure, are at least across the interfaces.

Paper Structure

This paper contains 19 sections, 20 theorems, 467 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Well-posedness for a boundary value problem of linear Keldysh equation
  3. Main theorem
  4. Boundary value problem with a singular perturbation
  5. A priori $H^1$-estimate of $v^{\varepsilon}$
  6. Local $H^2$-estimate of $v^{\varepsilon}$
  7. Extension of boundary value problem \ref{['bvp-aux']}
  8. Proof of Theorem \ref{['theorem-1-full']}
  9. Accelerating transonic solution to the steady Euler-Poisson system
  10. Preliminary
  11. Restatement of Theorem \ref{['theorem-2-pre']}
  12. Proof of Theorem \ref{['theorem-HD']}
  13. Linearized boundary value problem for iteration
  14. Approximated sonic interfaces
  15. Well-posedness of \ref{['lbvp main']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $m\ge 4$ be fixed in $\mathbb{N}$. Assume that $a_{ij}, a_1\in W^{1,\infty}(\Omega)\cap H^{m-1}(\Omega)$ and $f\in H^{m-1}(\Omega)$. Suppose that there exists a constant $\lambda>0$ to satisfy In addition, if it holds that and then the boundary value problem has a unique solution $u\in H^m(\Omega)$.

Figures (1)

  • Figure 1: The critical trajectory

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 39 more