Radial transonic shock solutions of Euler-Poisson system in convergent nozzles

Given constant data of density $\rho_0$, velocity $-u_0{\bf e}_r$, pressure $p_0$ and electric force $-E_0{\bf e}_r$ for supersonic flow at the entrance, and constant pressure $p_{\rm ex}$ for subsonic flow at the exit, we prove that Euler-Poisson system admits a unique transonic shock solution in a two dimensional convergent nozzle, provided that $u_0>0$, $E_0>0$, and that $E_0$ is sufficiently large depending on $(\rho_0, u_0, p_0)$ and the length of the nozzle

3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system

We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler-Poisson system in a cylinder supplemented with non small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl(=angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler-Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the vorticity in its cylindrical representation, and in analysis of streamlines near the axis $r=0$

Smooth Subsonic and Transonic Flows with Nonzero Angular Velocity and Vorticity to steady Euler-Poisson system in a Concentric Cylinder

In this paper, both smooth subsonic and transonic flows to steady Euler-Poisson system in a concentric cylinder are studied. We first establish the existence of cylindrically symmetric smooth subsonic and transonic flows to steady Euler-Poisson system in a concentric cylinder. On one hand, we investigate the structural stability of smooth cylindrically symmetric subsonic flows under three-dimensional perturbations on the inner and outer cylinders. On the other hand, the structural stability of smooth transonic flows under the axi-symmetric perturbations are examined. There is no any restrictions on the background subsonic and transonic solutions. A deformation-curl-Poisson decomposition to the steady Euler-Poisson system is utilized in our work to deal with the hyperbolic-elliptic mixed structure in subsonic region. It should be emphasized that there is a special structure of the steady Euler-Poisson system which yields a priori estimates and uniqueness of a second order elliptic system for the velocity potential and the electrostatic potential

Stability of Transonic Shock Solutions for One-Dimensional Euler-Poisson Equations

In this paper, both structural and dynamical stabilities of steady transonic shock solutions for one-dimensional Euler-Poission system are investigated. First, a steady transonic shock solution with supersonic backgroumd charge is shown to be structurally stable with respect to small perturbations of the background charge, provided that the electric field is positive at the shock location. Second, any steady transonic shock solution with the supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbation of the initial data, provided the electric field is not too negative at the shock location. The proof of the first stability result relies on a monotonicity argument for the shock position and the downstream density, and a stability analysis for subsonic and supersonic solutions. The dynamical stability of the steady transonic shock for the Euler-Poisson equations can be transformed to the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions. The analysis for the associated linearized problem plays an essential role