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 $r=0$

Supersonic flows of the Euler-Poisson system in three-dimensional cylinders

In this paper, we prove the unique existence of three-dimensional supersonic solutions to the steady Euler-Poisson system in cylindrical nozzles when prescribing the velocity, entropy, and the strength of electric field at the entrance. We first establish the unique existence of irrotational supersonic solutions in a cylindrical nozzle with an arbitrary cross section by extending the results of \cite{bae2021three} with an aid of weighted Sobolev norms. Then, we establish the unique existence of three-dimensional axisymmetric supersonic solutions to the Euler-Poisson system with nonzero vorticity in a circular cylinder. In particular, we construct a three-dimensional solution with a nonzero angular momentum density (or equivalently a nonzero swirl). Therefore this is truly a three dimensional flow in the sense that the Euler-Poisson system cannot be reduced to a two dimensional system via a stream function formulation. The main idea is to reformulate the system into a second order hyperbolic-elliptic coupled system and two transport equations via the method of Helmholtz decomposition, and to employ the method of iterations. Several technical issues, including the issue of singularities on the axis of symmetry and the issue of corner singularities in a Lipschitz domain, are carefully addressed.Comment: 67 pages, 1 figur

Structural stability of Supersonic solutions to the Euler-Poisson system

The well-posedness for the supersonic solutions of the Euler-Poisson system for hydrodynamical model in semiconductor devices and plasmas is studied in this paper. We first reformulate the Euler-Poisson system in the supersonic region into a second order hyperbolic-elliptic coupled system together with several transport equations. One of the key ingredients of the analysis is to obtain the well-posedness of the boundary value problem for the associated linearized hyperbolic-elliptic coupled system, which is achieved via a delicate choice of multiplier to gain energy estimate. The nonlinear structural stability of supersonic solution in the general situation is established by combining the iteration method with the estimate for hyperbolic-elliptic system and the transport equations together.Comment: The paper was revised substantially in this new version. In particular, we constructed the new multiplier under general conditions on the background solution