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The 3D Vlasov–Poisson–Landau System Near 1D Local Maxwellians

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Abstract

We consider the Cauchy problem on the Vlasov–Poisson–Landau system with the Coulomb interaction in the three dimensional space domain \({\mathbb {R}}\times {\mathbb {T}^2}\). Although there have been extensive studies on global existence and large time behavior of solutions near global Maxwellians either in \(\mathbb {T}^3\) or \(\mathbb {R}^3\), it is unknown whether solutions can be constructed around some non-trivial asymptotic profiles. In this paper, we obtain the global solutions near a spatially one-dimensional local Maxwellian connecting two distinct global Maxwellians at \(x_1=\pm \infty \), where the fluid components of the local Maxwellian are the smooth approximate rarefaction wave of the corresponding full compressible Euler system in \(x_1\in \mathbb {R}\). We also prove the large time asymptotics of global solutions towards such planar rarefaction waves, and establish the propagation of the high-order moments and regularity of solutions with large amplitude. As a byproduct, all these results can be carried over to the pure Landau equation in the same setting.

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Acknowledgements

Renjun Duan’s research was supported by the General Research Fund (Project No. 14302817) from RGC of Hong Kong and a Direct Grant (No. 4053334) from CUHK. Hongjun Yu’s research was supported by the GDUPS 2017 and the NNSFC Grant 11871229.

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Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Renjun Duan

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Hongjun Yu

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  1. Renjun Duan
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Duan, R., Yu, H. The 3D Vlasov–Poisson–Landau System Near 1D Local Maxwellians. J Stat Phys 182, 33 (2021). https://doi.org/10.1007/s10955-021-02704-6

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