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Green’s Function and Pointwise Space-time Behaviors of the Vlasov-Poisson-Boltzmann System

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Abstract

The pointwise space-time behaviors of the Green’s function and the global solution to the Vlasov-Poisson-Boltzmann (VPB) system in spatial three dimension are studied in this paper. It is shown that, due to the influence of electrostatic potential governed by the Poisson equation, the Green’s function admits only the macroscopic nonlinear diffusive waves, the singular kinetic waves, and the remainder term decaying exponentially in time but algebraically in space. These behaviors have an essential difference from the Boltzmann equation, namely, the Huygen’s type sound wave propagation and the space-time exponential decay of remainder term for Boltzmann equation (Liu and Yu in Commun Pure Appl Math 57:1543–1608, 2004; Bull Inst Math Acad Sin (NS) 1(1): 1–78, 2006) cannot be observed for VPB system. Furthermore, we establish the pointwise space-time nonlinear diffusive behaviors of the global solution to the nonlinear VPB system in terms of the Green’s function. Some new strategies are introduced to deal with the difficulties caused by the electric fields.

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References

  1. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol. 106. Springer, New York 1994

    Book  Google Scholar 

  2. Duan, R.J., Strain, R.M.: Optimal time decay of the Vlasov-Poisson-Boltzmann system in \({{\mathbb{R}}}^3\). Arch. Ration. Mech. Anal. 199(1), 291–328, 2011

    Article  MathSciNet  Google Scholar 

  3. Duan, R.J., Yang, T.: Stability of the one-species Vlasov-Poisson-Boltzmann system. SIAM J. Math. Anal. 41, 2353–2387, 2010

    Article  MathSciNet  Google Scholar 

  4. Duan, R.J., Yang, T., Zhu, C.J.: Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete Contin. Dyn. Syst. 16, 253–277, 2006

    Article  MathSciNet  Google Scholar 

  5. Duan, R.J., Yang, T., Zhao, H.J.: The Vlasov-Poisson-Boltzmann system in the whole space: the hard potential case. J. Differ. Equ. 252, 6356–6386, 2012

    Article  ADS  MathSciNet  Google Scholar 

  6. Duan, R.J., Yang, T., Zhao, H.J.: The Vlasov-Poisson-Boltzmann system for soft potentials. Math. Models Methods Appl. Sci. 23(6), 979–1028, 2013

    Article  MathSciNet  Google Scholar 

  7. Duan, R.J., Liu, S.: Stability of the rarefaction wave of the Vlasov-Poisson-Boltzmann system. SIAM J. Math. Anal. 47(5), 3585–3647, 2015

    Article  MathSciNet  Google Scholar 

  8. Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Commun. Pure Appl. Math. 55(9), 1104–1135, 2002

    Article  MathSciNet  Google Scholar 

  9. Guo, Y.: The Vlasov-Poisson-Boltzmann system near vacuum. Commun. Math. Phys. 218(2), 293–313, 2001

    Article  ADS  MathSciNet  Google Scholar 

  10. Guo, Y., Jang, J.: Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System. Commun. Math. Phys. 299, 469–501, 2010

    Article  ADS  MathSciNet  Google Scholar 

  11. Li, H.-L., Yang, T., Zhong, M.: Spectrum analysis for the Vlasov-Poisson-Boltzmann system. Preprint, arXiv:1402.3633v1 [math.AP]

  12. Li, H.-L., Yang, T., Zhong, M.: Spectrum analysis and optimal decay rates of the bipolar Vlasov-Poisson-Boltzmann equations. Indiana Univ. Math. J. 65, 665–725, 2016

    Article  MathSciNet  Google Scholar 

  13. Li, H.-L., Wang, Y., Yang, T., Zhong, M.: Stability of nonlinear wave patterns to the bipolar Vlasov-Poisson-Boltzmann system. Arch. Rational Mech. Anal. 228, 39–127, 2018

    Article  ADS  MathSciNet  Google Scholar 

  14. Li, H.-L., Wang, T., Wang, Y.: Stability of the superposition of a viscous contact wave with two rarefaction waves to the bipolar Vlasov-Poisson-Boltzmann system. SIAM J. Math. Anal. 228, 39–127, 2018

    MathSciNet  MATH  Google Scholar 

  15. Liu, T.-P., Yu, S.-H.: The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation. Commun. Pure Appl. Math. 57, 1543–1608, 2004

    Article  MathSciNet  Google Scholar 

  16. Liu, T.-P., Yu, S.-H.: The Green’s function of Boltzmann equation, 3D waves. Bull. Inst. Math. Acad. Sin. (N. S.)1(1), 1–78, 2006

    MathSciNet  Google Scholar 

  17. Kaup, L., Kaup, B.: Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory. Walter de Gruyter, Berlin, New York 1983

    Book  Google Scholar 

  18. Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations. Springer, Vienna 1990

    Book  Google Scholar 

  19. Mischler, S.: On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 210, 447–466, 2000

    Article  ADS  MathSciNet  Google Scholar 

  20. Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japan Acad. 50, 179–184, 1974

    Article  MathSciNet  Google Scholar 

  21. Ukai, S., Yang, T.: Mathematical Theory of Boltzmann Equation. Lecture Notes Series-No. 8, Hong Kong: Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, March 2006

  22. Wang, W., Zhu, Z.: Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions. J. Differ. Equ. 248, 1617–1636, 2010

    Article  ADS  MathSciNet  Google Scholar 

  23. Wang, Y.: Decay of the two-species Vlasov-Poisson-Boltzmann system. J. Differ. Equ. 254(5), 2304–2340, 2013

    Article  ADS  MathSciNet  Google Scholar 

  24. Yang, T., Yu, H.J., Zhao, H.J.: Cauchy problem for the Vlasov-Poisson-Boltzmann system. Arch. Rational Mech. Anal. 182, 415–470, 2006

    Article  ADS  MathSciNet  Google Scholar 

  25. Yang, T., Zhao, H.J.: Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 268, 569–605, 2006

    Article  ADS  MathSciNet  Google Scholar 

  26. Yang, T., Yu, H.J.: Optimal convergence rates of classical solutions for Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 301, 319–355, 2011

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for the helpful comments. The research of the first author was supported partially by the National Science Fund for Distinguished Young Scholars No. 11225102, the National Natural Science Foundation of China (Nos. 11871047, 11671384 and 11461161007), the project supported by Beijing Advanced Innovation Center for Imaging Theory and Technology No. 00719530012008, and the Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds 00719530050166. The research of the second author was supported by the General Research Fund of Hong Kong, CityU 11302215 and the National Natural Science Foundation of China No. 11731008. The research of the third author is supported by National Natural Science Foundation of China Nos. 11301094 and 11671100, and Guangxi Natural Science Foundation No. 2018GXNSFAA138210.

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Authors and Affiliations

  1. School of Mathematical Sciences and Academy for Multidisciplinary Studies, Capital Normal University, Beijing, China

    Hai-Liang Li

  2. Department of Mathematics, City University of Hong Kong, Hong Kong, China

    Tong Yang

  3. College of Mathematics and Statistics, Chongqing University, Chongqing, China

    Tong Yang

  4. Department of Mathematics and Information Sciences, Guangxi University, Nanning, China

    Mingying Zhong

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  1. Hai-Liang Li
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  2. Tong Yang
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  3. Mingying Zhong
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Correspondence to Hai-Liang Li.

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Communicated by T.P. Liu.

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Li, HL., Yang, T. & Zhong, M. Green’s Function and Pointwise Space-time Behaviors of the Vlasov-Poisson-Boltzmann System. Arch Rational Mech Anal 235, 1011–1057 (2020). https://doi.org/10.1007/s00205-019-01438-w

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