Structural stability of supersonic spiral flows with large angular velocity for the Euler-Poisson system

TL;DR

This paper proves the existence, uniqueness, and stability of smooth supersonic spiral flows with large angular velocity in the Euler-Poisson system, addressing the hyperbolic-elliptic mixed structure in a cylindrical setting.

Contribution

It introduces a novel approach for establishing the stability of large angular velocity supersonic flows within the Euler-Poisson framework, including both cylindrical and axisymmetric cases.

Findings

Existence and uniqueness of smooth supersonic spiral flows with large angular velocity.

Structural stability of these flows under small perturbations.

Development of energy estimates for hyperbolic-elliptic coupled systems.

Abstract

This paper concerns the structural stability of smooth cylindrically symmetric supersonic spiral flows with large angular velocity for the steady Euler-Poisson system in a concentric cylinder. We establish the existence and uniqueness of some smooth supersonic Euler-Poisson flows with nonzero angular velocity and vorticity including both cylindrical spiral flows and axisymmetric spiral flows. The deformation-curl-Poisson decomposition for the steady Euler-Poisson system is utilized to deal with the hyperbolic-elliptic mixed structure in the supersonic region. For smooth cylindrical supersonic spiral flows, the key point lies on the well-posedness of a boundary value problem for a linear second order hyperbolic-elliptic coupled system, which is achieved by finding an appropriate multiplier to obtain the important basic energy estimates. The nonlinear structural stability is established by designing a two-layer iteration and combining the estimates for the hyperbolic-elliptic system and the transport equations. For smooth axisymmetric supersonic spiral flows, we use the special structure of the steady Euler-Poisson system to derive a priori estimates of the linearized second order elliptic system, which enable us to establish the structural stability of the background supersonic flow within the class of axisymmetric flows.