Skip to main content
SpringerLink
Log in

Global Existence of Classical Solutions to the Vlasov-Poisson-Boltzmann System

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The time evolution of the distribution function for the charged particles in a dilute gas is governed by the Vlasov–Poisson–Boltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this paper, we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global Maxwellian. Moreover, the convergence rate in time to the global Maxwellian is also obtained through the energy method. The proof is based on the theory of compressible Navier–Stokes equations with forcing and the decomposition of the solutions to the Boltzmann equation with respect to the local Maxwellian introduced in [23] and elaborated in [31].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Boltzmann L. (1964) (translated by Stephen G. Brush): Lectures on Gas Theory. New York, Dover Publications Inc

  2. Carleman T. (1932) Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60, 91–142

    Article  MathSciNet  Google Scholar 

  3. Cercignani C. (1988) The Boltzmann equation and its applications Applied Mathematical Sciences 67. New York, Springer-Verlag

    Google Scholar 

  4. Cercignani C., Illner R., Pulvirenti M. (1994) The Mathematical Theory of Dilute Gases Applied Mathematical Sciences 106. New York, Springer-Verlag

    Google Scholar 

  5. Deckelnick K. (1992) Decay estimates for the compressible Navier–Stokes equations in unbounded domains. Math. Z. 209, 115–130

    MATH  MathSciNet  Google Scholar 

  6. Desvillettes L., Dolbeault J. (1991) On long time asymptotics of the Vlasov–Poisson–Boltzmann equation. Comm. Partial Differ. Eqs. 16 (2–3): 451–489

    MATH  MathSciNet  Google Scholar 

  7. Duan R., Yang T., Zhu C.-J. (2005) Boltzmann equation with external force in infinite vacuum. J. Math. Phys. 46(5): 053307

    Article  MathSciNet  ADS  Google Scholar 

  8. Glassey R.: The Cauchy Problem in Kinetic Theory. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1996

  9. Glassey R., Strauss W.-A. (1999) Decay of the linearized Boltzmann–Vlasov system. Transport Theory Statist. Phys. 28(2): 135–156

    MATH  MathSciNet  ADS  Google Scholar 

  10. Glassey R., Strauss W.-A. (1999) Perturbation of essential spectra of evolution operators and the Vlasov–Poisson–Boltzmann system. Discrete Contin. Dynam. Systems 5(3): 457–472

    Article  MATH  MathSciNet  Google Scholar 

  11. Glassey R., Schaeffer J., Zheng Y.-X. (1996) Steady states of the Vlasov–Poisson–Fokker–Planck system. J. Math. Anal. Appl. 202(3): 1058–1075

    Article  MATH  MathSciNet  Google Scholar 

  12. Golse F., Perthame B., Sulem C. (1986) On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Rational Mech. Anal. 103, 81–96

    Article  MathSciNet  ADS  Google Scholar 

  13. Grad H.: Asymptotic Theory of the Boltzmann Equation II, Rarefied Gas Dynamics. J. A. Laurmann, ed., Vol. 1, New York: Academic Press 1963, pp. 26–59

  14. Guo Y. (2002) The Vlasov–Poisson–Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55(9): 1104–1135

    Article  MATH  MathSciNet  Google Scholar 

  15. Guo Y. (2003) The Vlasov–Maxwell–Boltzmann system near Maxwellians. Invent. Math. 153(3): 593–630

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  ADS  Google Scholar 

  17. Guo Y. (2004) The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53(4): 1081–1094

    Article  MATH  MathSciNet  Google Scholar 

  18. Hilbert D., (1953) Grundzuge einer Allgemeinen Theorie der Linearen Integralgleichungen (in German). New York N.-Y, Chelsea Publishing Company

    Google Scholar 

  19. Huang F.-M., Xin Z.-P., Yang T.: Contact discontinuity with general perturbations for gas motions. Preprint 2004, available at http://www.cityu.edu.hk/rcms/publication/preprint18.pdf

  20. Kawashima S., Matsumura A. (1985) Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101(1): 97–127

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Li H.-L., Matsumura A.: Asymptotic behavior of compressible Navier–Stokes–Poisson system. Preprint

  22. Lions P.-L.: On kinetic equations. In: Proceedings of the International Congress of Mathematicians, Kyoto: Math. Soc. Japan, 1991, pp.1173–1185

  23. Liu T.-P., Yang T., Yu S.-H. (2004) Energy method for the Boltzmann equation. Physica D 188(3-4): 178–192

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Liu T.-P., Yang T., Yu S.-H., Zhao H.-J. (2006) Nonlinear stability of rarefaction waves for the Boltzmann equation. Arch. Rational Mech. Anal. 181(2): 333–371

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Liu T.-P., Yu S.-H. (2004) Boltzmann equation: Micro-macro decompositions and positivity of shock profiles. Commun. Math. Phys. 246(1): 133–179

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Matsumura A., Nishida T. (1983) Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89(4): 445–464

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Mischler S. (2000) On the initial boundary value problem for the Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 210(2): 447–466

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Ukai S., Yang T., Zhao H.-J. (2005) Global solutions to the Boltzmann equation with external forces. Analysis and Applications 3(2): 157–193

    Article  MATH  MathSciNet  Google Scholar 

  29. Ukai S., Yang T., Zhao H.-J. (2006) Convergence rate to stationary solutions for Boltzmann equation with external force. To appear in Chin. Ann. Math. Ser. B. 27(4): 363–378

    Article  MATH  MathSciNet  Google Scholar 

  30. Yang T., Yu H.-J., Zhao H.-J.: Cauchy problem for the Vlasov–Poisson–Boltzmann system. Arch. Rational Mech. Anal. 182(3), (2006)

  31. Yang T., Zhao H.-J. (2006) A new energy method for the Boltzmann equation. J. Math. Phys. 47(5): 053301

    Article  MathSciNet  ADS  Google Scholar 

  32. Yang T., Zhao H.-J. (2005) A half-space problem for the Boltzmann equation with specular reflection boundary condition. Commun. Math. Phys. 255(3): 683–726

    Article  MATH  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

    Tong Yang

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Huijiang Zhao

Authors
  1. Tong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huijiang Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tong Yang.

Additional information

Communicated by H.-T. Yau.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, T., Zhao, H. Global Existence of Classical Solutions to the Vlasov-Poisson-Boltzmann System. Commun. Math. Phys. 268, 569–605 (2006). https://doi.org/10.1007/s00220-006-0103-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-006-0103-4

Keywords

Search

Navigation