Structural stability of Supersonic solutions to the Euler-Poisson system

TL;DR

This paper investigates the stability of supersonic solutions to the Euler-Poisson system, reformulating it into a coupled hyperbolic-elliptic system and proving nonlinear structural stability using energy estimates and iterative methods.

Contribution

It introduces a new approach to analyze the well-posedness and stability of supersonic solutions in the Euler-Poisson system, including reformulation and energy estimate techniques.

Findings

Established well-posedness of the boundary value problem.

Proved nonlinear structural stability of supersonic solutions.

Developed energy estimate methods for hyperbolic-elliptic systems.

Abstract

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.