Radial solutions of the hydrodynamic model of semiconductors with sonic boundary

TL;DR

This paper investigates the existence and uniqueness of radial subsonic and supersonic solutions for the steady hydrodynamic model of semiconductors with sonic boundary conditions, extending previous one-dimensional results to higher dimensions.

Contribution

It introduces new methods to establish solutions in higher dimensions, especially for supersonic cases, under general doping profile conditions.

Findings

Proved existence and uniqueness of radial subsonic solutions.

Established existence of radial supersonic solutions.

Extended previous one-dimensional results to higher dimensions.

Abstract

The purpose of this paper is to study radial solutions for steady hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary. The existence and uniqueness of radial subsonic solution, and the existence of radial supersonic solutions are derived by using the energy method and the compactness method, but under a general condition of the doping profile. In particular, for radial supersonic solutions, it is more difficult to get the related estimates by the effect of high dimensional space and the sonic boundary, so we apply a special iteration to complete the proofs. The results obtained essentially improve and develop the previous studies in the one-dimensional case.