Structural stability of cylindrical supersonic solutions to the steady Euler–Poisson system

This paper concerns the structural stability of smooth cylindrically symmetric supersonic Euler–Poisson flows in nozzles. Both three-dimensional and axisymmetric perturbations are considered. On one hand, we establish the existence and uniqueness of three-dimensional smooth supersonic solutions to the potential flow model of the steady Euler–Poisson system. On the other hand, the existence and uniqueness of smooth supersonic flows with nonzero vorticity to the steady axisymmetric Euler–Poisson system are proved. The problem is reduced to solve a nonlinear boundary value problem for a hyperbolic–elliptic mixed system. One of the key ingredients in the analysis of three-dimensional supersonic irrotational flows is the well-posedness theory for a linear second-order hyperbolic–elliptic coupled system, which is achieved by using the multiplier method and the reflection technique to derive the energy estimates. For smooth axisymmetric supersonic flows with nonzero vorticity, the deformation-curl-Poisson decomposition is utilized to reformulate the steady axisymmetric Euler–Poisson system as a deformation-curl-Poisson system together with several transport equations, so that one can design a two-layer iteration scheme to establish the nonlinear structural stability of the background supersonic flow within the class of axisymmetric rotational flows.