Global solutions and Relaxation Limit to the Cauchy Problem of a Hydrodynamic Model for Semiconductors

TL;DR

This paper proves the existence of global entropy solutions for a hydrodynamic semiconductor model using vanishing artificial viscosity and analyzes the relaxation limit as parameters tend to zero.

Contribution

It introduces a new approach to establish global solutions for the model and extends the relaxation limit analysis to general pressure functions.

Findings

Existence of global entropy solutions is established.

The relaxation limit is demonstrated for general pressure functions.

The vanishing artificial viscosity method is effectively applied.

Abstract

It is well-known that due to the lack of a technique to obtain the a-priori $L^{\infty}$ estimate of the artificial viscosity solutions of the Cauchy problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for semiconductors, where the energy equation is replaced by a pressure-density relation, over the past three decades, all solutions of this model were obtained by using the Lax-Friedrichs, Godounov schemes and Glimm scheme for both the initial-boundary value problem \cite{Zh1,Li} and the Cauchy problem \cite{MN1,PRV,HLY}; or by using the vanishing artificial viscosity method for the initial-boundary value problem \cite{Jo,HLYY}. In this paper, the existence of global entropy solutions, for the Cauchy problem of this model, is proved by using the vanishing artificial viscosity method. As a by-product, the known compactness framework \cite{MN2,JR} is applied to show the relaxation limit, as the relation time $ \tau$ and $\E,\delta$ go to zero, for general pressure $P(\rho)$.