Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges

The Euler–Poisson equations with varying background charges in finitely long flat nozzles are investigated, for which two and only two transonic shock solutions are constructed. In $\href{https://dx.doi.org/10.4310/CMS.2012.v10.n2.a1}{[\textrm{T. Luo and Z.P. Xin, Commun. Math. Sci., 10:419–462, 2012}]}$, Luo and Xin established the wellposedness of steady Euler–Poisson equations for the constant background charge. Motivated by their pioneering work and combined with the special physical character of semiconductor devices, we propose the transonic shock problem in which the density of the background charge is a piecewise constant function and its discontinuity is determined only by shock fronts. The existence and non-uniqueness of transonic shock solutions are obtained via the method of shock matching.

Keywords

Euler–Poisson equations, transonic shock, non-uniqueness, varying background charges

2010 Mathematics Subject Classification

35L65, 35L67, 35R35, 76H05

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The research of Ben Duan is supported by NSFC grants 12271205 and 12171498.

The research of Haoran Zheng is partly supported by the National Key Research and Development Program of China grant 2020YFA0713602, by NSFC grants 12171199 and 11971198, and by the Jilin Provincial Department of Science and Technology grant 20210201015GX.

Received 30 November 2022

Accepted 1 September 2023

Published 4 March 2024