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Asymptotic stability of a nonlinear wave for an outflow problem of the bipolar Navier–Stokes–Poisson system under large initial perturbation

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Abstract

In this paper, we study the time-asymptotic behavior of solutions to an outflow problem for the one-dimensional bipolar Navier–Stokes–Poisson system in the half space. First, we make some suitable assumptions on the boundary data and space-asymptotic states such that the time-asymptotic state of the solution is a nonlinear wave, which is the superposition of the transonic stationary solution and the 2-rarefaction wave. Next, we show the stability of this nonlinear wave under a class of large initial perturbation, provided that the strength of the transonic stationary solution is small enough, while the amplitude of the 2-rarefaction wave can be arbitrarily large. The proof is completed by a delicate energy method and a continuation argument. The key point is to derive the positive upper and lower bounds of the particle densities.

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Acknowledgements

Wu would like to express his gratitude to Prof. Lin He for his helpful suggestions. Wu and Zhu are supported in part by Science and Technology Commission of Shanghai Municipality (Grant No. 20JC1413600).

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Qiwei Wu, Xiaofeng Hou & Peicheng Zhu

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  1. Qiwei Wu
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  2. Xiaofeng Hou
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QW wrote the main manuscript text. All authors reviewed the manuscript.

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Wu, Q., Hou, X. & Zhu, P. Asymptotic stability of a nonlinear wave for an outflow problem of the bipolar Navier–Stokes–Poisson system under large initial perturbation. Z. Angew. Math. Phys. 74, 146 (2023). https://doi.org/10.1007/s00033-023-02029-2

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