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Journal of Statistical Physics

, Volume 136, Issue 2, pp 297–330 | Cite as

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

  • Roberta Bosi
  • Maria J. Cáceres
Article

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.

Keywords

BGK model Boltzmann equation External force Long time behaviour Maxwellian steady states 

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References

  1. 1.
    Andries, P., Aoki, K., Perthame, B.: A consistent BGK-type model for gas mixtures. J. Stat. Phys. 106(5–6), 993–1018 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bhatnagar, P.L., Gross, E.P., Krook, M.: A model of collision processes in gases. Phys. Rev. 94, 511 (1954) zbMATHCrossRefADSGoogle Scholar
  3. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bouchut, F.: Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94(4), 623–672 (2003) zbMATHMathSciNetGoogle Scholar
  5. 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) MathSciNetGoogle Scholar
  6. 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) zbMATHCrossRefADSMathSciNetGoogle Scholar
  7. 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) zbMATHCrossRefGoogle Scholar
  8. 8.
    Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, Berlin (1988) zbMATHGoogle Scholar
  9. 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) zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Desvillettes, L.: Convergence to equilibrium in large time for Boltzmann and BGK equations. Arch. Ration. Mech. Anal. 110(1), 73–91 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 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) zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    DiPerna, R., Lions, P.-L.: On the Cauchy problem for Boltzmann equation: global existence and weak stability. Ann. Math. 130, 321–366 (1989) CrossRefMathSciNetGoogle Scholar
  13. 13.
    DiPerna, R., Lions, P.-L.: Global weak solution of Vlasov-Maxwell systems. Commun. Pure Appl. Math. 42, 729–757 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 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) zbMATHMathSciNetGoogle Scholar
  16. 16.
    Escobedo, M., Mischler, S.: On a quantum Boltzmann equation for a gas of photons. J. Math. Pures Appl. 80(5), 471–515 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gasser, I., Markowich, P.A., Perthame, B.: Dispersion and moment lemmas revised. J. Differ. Equ. 156, 254–281 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 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) CrossRefMathSciNetGoogle Scholar
  19. 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) CrossRefADSGoogle Scholar
  20. 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) zbMATHMathSciNetGoogle Scholar
  21. 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. 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) zbMATHMathSciNetGoogle Scholar
  23. 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) zbMATHMathSciNetGoogle Scholar
  24. 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) zbMATHMathSciNetGoogle Scholar
  25. 25.
    Perthame, B.: Global existence to the BGK model of Boltzmann equation. J. Differ. Equ. 82, 191–205 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Perthame, B.: Kinetic Formulation of Conservation Laws. Oxford University Press, Oxford (2002) zbMATHGoogle Scholar
  27. 27.
    Perthame, B.: Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27(6), 1405–1421 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Perthame, B., Pulvirenti, M.: Weighted L bounds and uniqueness for the Boltzmann BGK model. Arch. Ration. Mech. Anal. 125, 289–295 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Saint-Raymond, L.: From the BGK model to the Navier-Stokes equations. Ann. Sci. Ec. Norm. Super., Sér. 4 36, 271–317 (2003) zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK
  2. 2.Departamento de Matemática AplicadaUniversidad de GranadaGranadaSpain

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