Stability of stationary solutions to the compressible bipolar Euler-Poisson equations

Hong Cai; Zhong Tan

doi:10.3934/dcds.2017201

Article Contents

2017, Volume 37, Issue 9: 4677-4696. Doi: 10.3934/dcds.2017201

This issue Previous Article Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach Next Article Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media

Stability of stationary solutions to the compressible bipolar Euler-Poisson equations

Hong Cai^1, and
Zhong Tan^2, ,

1.
School of Mathematical Sciences, Xiamen University, Fujian, Xiamen 361005, China,
2.
School of Mathematical Sciences and Fujian Provincial Key Laboratory, on Mathematical Modeling and Scientific Computing, Xiamen University, Fujian, Xiamen 361005, China

* Corresponding author: Zhong Tan, tan85@xmu.edu.cn
* Corresponding author: Zhong Tan, tan85@xmu.edu.cn

Received: March 31, 2016

Revised: March 31, 2017

Published: October 2017

The authors are supported by National Natural Science Foundation of China-NSAF (No.11271305, 11531010)

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Abstract

In this paper, we study the compressible bipolar Euler-Poisson equations with a non-flat doping profile in three-dimensional space. The existence and uniqueness of the non-constant stationary solutions are established under the smallness assumption on the gradient of the doping profile. Then we show the global existence of smooth solutions to the Cauchy problem near the stationary state provided the $H^3$ norms of the initial density and velocity are small, but the higher derivatives can be arbitrarily large.

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Mathematics Subject Classification: Primary: 35M10, 35Q35; Secondary: 35Q60.

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