Skip to main content
SpringerLink
Log in

The BGK Model with External Confining Potential: Existence, Long-Time Behaviour and Time-Periodic Maxwellian Equilibria

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We study global existence and long time behaviour for the inhomogeneous nonlinear BGK model for the Boltzmann equation with an external confining potential. For an initial datum f 0≥0 with bounded mass, entropy and total energy we prove existence and strong convergence in L 1 to a Maxwellian equilibrium state, by compactness arguments and multipliers techniques. Of particular interest is the case with an isotropic harmonic potential, in which Boltzmann himself found infinitely many time-periodic Maxwellian steady states. This behaviour is shared with the Boltzmann equation and other kinetic models. For all these systems we study the multistability of the time-periodic Maxwellians and provide necessary conditions on f 0 to identify the equilibrium state, both in L 1 and in Lyapunov sense. Under further assumptions on f, these conditions become also sufficient for the identification of the equilibrium in L 1.

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. Andries, P., Aoki, K., Perthame, B.: A consistent BGK-type model for gas mixtures. J. Stat. Phys. 106(5–6), 993–1018 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model of collision processes in gases. Phys. Rev. 94, 511 (1954)

    Article  MATH  ADS  Google Scholar 

  3. Bouchut, F.: Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Stat. Phys. 95(1–2), 113–170 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bouchut, F.: Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94(4), 623–672 (2003)

    MATH  MathSciNet  Google Scholar 

  5. Bouchut, F., Dolbeault, J.: On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with coulombic and newtonian potentials. Differ. Integral Equ. 8(3), 487–514 (1995)

    MathSciNet  Google Scholar 

  6. Bouchut, F., Perthame, B.: A BGK model for small Prandtl number in the Navier-Stokes approximation. J. Stat. Phys. 71(1–2), 191–207 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Cáceres, M.J., Carrillo, J.A., Goudon, T.: Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles. Commun. PDE 28(5–6), 969–989 (2003)

    Article  MATH  Google Scholar 

  8. Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, Berlin (1988)

    MATH  Google Scholar 

  9. Crouseilles, N., Degond, P., Lemou, M.: A hybrid kinetic-fluid model for solving the Vlasov-BGK equation. J. Comput. Phys. 203(2), 572–601 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Desvillettes, L.: Convergence to equilibrium in large time for Boltzmann and BGK equations. Arch. Ration. Mech. Anal. 110(1), 73–91 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. Desvillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. DiPerna, R., Lions, P.-L.: On the Cauchy problem for Boltzmann equation: global existence and weak stability. Ann. Math. 130, 321–366 (1989)

    Article  MathSciNet  Google Scholar 

  13. DiPerna, R., Lions, P.-L.: Global weak solution of Vlasov-Maxwell systems. Commun. Pure Appl. Math. 42, 729–757 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. DiPerna, R., Lions, P.-L.: Global solutions of Boltzmann’s equation and the entropy inequality. Arch. Ration. Mech. Anal. 114(1), 47–55 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  15. DiPerna, R., Lions, P.-L., Meyer, Y.: L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8(34), 271–287 (1991)

    MATH  MathSciNet  Google Scholar 

  16. Escobedo, M., Mischler, S.: On a quantum Boltzmann equation for a gas of photons. J. Math. Pures Appl. 80(5), 471–515 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gasser, I., Markowich, P.A., Perthame, B.: Dispersion and moment lemmas revised. J. Differ. Equ. 156, 254–281 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Golse, F., Lions, P.-L., Perthame, B., Sentis, R.: Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76, 434–460 (1988)

    Article  MathSciNet  Google Scholar 

  19. Guery-Odelin, D., Zambelli, F., Dalibard, J., Stringari, S.: Collective oscillations of a classical gas confined in harmonic traps. Phys. Rev. A 60(6), 4856–4851 (1999)

    Article  ADS  Google Scholar 

  20. Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications I. J. Math. Kyoto Univ. 34(2), 391–427 (1994)

    MATH  MathSciNet  Google Scholar 

  21. Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications II. J. Math. Kyoto Univ. 34(2), 429–461 (1994)

    Google Scholar 

  22. Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications III. J. Math. Kyoto Univ. 34(3), 539–584 (1994)

    MATH  MathSciNet  Google Scholar 

  23. Lions, P.-L., Perthame, B.: Lemme de moments, de moyenne et de dispersion. C. R. Acad. Sci. Paris Ser. I, Math. 314, 801–806 (1992)

    MATH  MathSciNet  Google Scholar 

  24. Mischler, S.: Uniqueness for the BGK-equation in R N and rate of convergence for a semi-discrete scheme. Differ. Integral Equ. 9(5), 1119–1138 (1996)

    MATH  MathSciNet  Google Scholar 

  25. Perthame, B.: Global existence to the BGK model of Boltzmann equation. J. Differ. Equ. 82, 191–205 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  26. Perthame, B.: Kinetic Formulation of Conservation Laws. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  27. Perthame, B.: Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27(6), 1405–1421 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  28. Perthame, B., Pulvirenti, M.: Weighted L ∞ bounds and uniqueness for the Boltzmann BGK model. Arch. Ration. Mech. Anal. 125, 289–295 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  29. Saint-Raymond, L.: Du modèle BGK de l’équation de Boltzmann aux équations d’Euler des fluides incompressibles. Bull. Sci. Math. 126(6), 493–506 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Saint-Raymond, L.: From the BGK model to the Navier-Stokes equations. Ann. Sci. Ec. Norm. Super., Sér. 4 36, 271–317 (2003)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University College London, Gower Street, WC1E 6BT, London, UK

    Roberta Bosi

  2. Departamento de Matemática Aplicada, Universidad de Granada, 18071, Granada, Spain

    Maria J. Cáceres

Authors
  1. Roberta Bosi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria J. Cáceres
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberta Bosi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bosi, R., Cáceres, M.J. The BGK Model with External Confining Potential: Existence, Long-Time Behaviour and Time-Periodic Maxwellian Equilibria. J Stat Phys 136, 297–330 (2009). https://doi.org/10.1007/s10955-009-9782-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-009-9782-5

Keywords

Search

Navigation