Structural Stability of Supersonic Solutions to the Euler–Poisson System

Abstract

The structural stability for supersonic solutions of the Euler–Poisson system for hydrodynamical model in semiconductor devices and plasmas in two dimensional domain is established, under the perturbation of the flow velocity and the strength of electric field in the horizontal direction at the entrance of a channel. First, the Euler–Poisson system in the supersonic region is reformulated 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.