Communications in Mathematical Sciences

Volume 18 (2020)

Number 7

Subsonic and supersonic steady-states of bipolar hydrodynamic model of semiconductors with sonic boundary

Pages: 2005 – 2038



Pengcheng Mu (School of Mathematics and Statistics, Northeast Normal University, Changchun, China)

Mei Ming (Department of Mathematics, Champlain College Saint-Lambert, Quebec, Canada; and Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)

Kaijun Zhang (School of Mathematics and Statistics, Northeast Normal University, Changchun, China)


In this paper, we investigate the well-posedness/ill-posedness of the stationary solutions to the isothermal bipolar hydrodynamic model of semiconductors driven by Euler–Poisson equations. Here, the density of electrons is proposed with sonic boundary and considered in interiorly subsonic case or interiorly supersonic case, while the density of holes is considered in fully subsonic case or fully supersonic case. With the developed technique based on the topological degree method, the following four kinds of stationary solutions under some conditions are proved to exist: the interiorly-subsonic-vs-fully-subsonic solution, the interiorly-supersonic-vs-fully-subsonic solution, the interiorly-subsonic-vs-fully-supersonic solution, and the interiorly-supersonic-vs-fully-supersonic solution. The non-existence of the above four kinds of solutions under some conditions is also technically proved. For the existence of these physical solutions, different from the previous studies, where traditional fixed-point argument via energy estimates is used, we recognize that such an approach fails for our cases, due to that the effect of boundary degeneracy for the electrons causes difficulty in estimating the upper and lower bounds for the holes. Instead of it, we use the topological degree method to prove the existence of physical solutions.


bipolar hydrodynamic model of semiconductors, Euler–Poisson equations, sonic boundary, steady-states, subsonic/supersonic solutions, topological degree method

2010 Mathematics Subject Classification

35J70, 35Q35, 76N10

M. Mei was supported in part by NSERC Grant RGPIN 354724-2016, and FRQNT Grant No. 256440. K.J Zhang was supported by the NSFC Grant No. 11771071.

Received 10 October 2019

Accepted 12 May 2020

Published 11 December 2020