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

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.