Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

Transonic shock solutions to the Euler–Poisson system in quasi-one-dimensional nozzles

Pages: 1023 – 1047



Ben Duan (Department of Mathematics, Dalian University of Technology, Dalian, China; and School of Business Informatics and Mathematics, University of Mannheim, Germany)

Zhen Luo (School of Mathematical Sciences, Xiamen University, Xiamen, China)

Jingjing Xiao (Institute of Mathematical Sciences, Chinese University of Hong Kong, Shatin, Hong Kong)


In this paper, we study the transonic shock solutions to the Euler–Poisson system in quasi-one-dimensional nozzles. For a given supersonic flow at the entrance of the nozzle, under some proper assumptions on the data and nozzle length we first obtain a class of steady transonic shock solutions for the exit pressure lying in a suitable range. The shock position is monotonically determined by the exit pressure. More importantly, by the estimates on the coupled effects of the electric field and the geometry of the nozzle, we prove the dynamic stability of the transonic shock solutions under suitable physical conditions. As a consequence, there indeed exist dynamically stable transonic shock solutions for the Euler–Poisson system in convergent nozzles, which is not true for the Euler system [T.-P. Liu, Commun. Math. Phys., 83, 243–260, 1982].


Euler–Poisson system, transonic shock, dynamic stability

2010 Mathematics Subject Classification

35B35, 35L65, 35L67, 76H05, 82D37

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