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

TL;DR

This paper proves the existence of classical solutions to a Keldysh-type PDE and applies this to establish the stability of smooth transonic solutions in the Euler-Poisson system, ensuring regularity across sonic interfaces.

Contribution

It introduces new classical solutions to a mixed-type PDE with degeneracy and demonstrates their application to stable, smooth transonic flows in the Euler-Poisson system.

Findings

Existence of classical solutions to Keldysh-type equations.

Structural stability of smooth transonic solutions.

Solutions have $C^1$ regularity across sonic interfaces.

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.