Archive for Rational Mechanics and Analysis

, Volume 194, Issue 2, pp 531–584

Vlasov–Maxwell–Boltzmann Diffusive Limit

Article

Abstract

Inspired by the work (Bastea et al. in J Stat Phys 1011087–1136, 2000) for binary fluids, we study the diffusive expansion for solutions around Maxwellian equilibrium and in a periodic box to the Vlasov–Maxwell–Boltzmann system, the most fundamental model for an ensemble of charged particles. Such an expansion yields a set of dissipative new macroscopic PDEs, the incompressible Vlasov–Navier–Stokes–Fourier system and its higher order corrections for describing a charged fluid, where the self-consistent electromagnetic field is present. The uniform estimate on the remainders is established via a unified nonlinear energy method and it guarantees the global in time validity of such an expansion up to any order.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bardos C., Golse F., Levermore D.: Fluid dynamic limits of kinetic equations. I Formal derivations. J. Statist. Phys. 63, 323–344 (1991)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Bardos C., Golse F., Levermore D.: Fluid dynamic limits of kinetic equations. II convergence proofs for the Boltzmann equation. Comm. Pure appl. Math. 46, 667–753 (1993)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bardos C., Golse F., Levermore D.: The acoustic limit for the Boltzmann equation. Arch. Rational. Mech. Anal. 153, 177–204 (2000)MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Bastea S., Esposito R., Lebowitz J.L., Marra R.: Binary fluids with long range segregating interaction. I: Derivation of kinetic and hydrodynamic equations. J. Statist. Phys. 101, 1087–1136 (2000)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    DiPerna R., Lions P.-L.: On the Cauchy problem for the Boltzmann equations: global existence and weak stability. Ann. Math. 130, 321–366 (1989)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Golse F., Levermore D.: Stokes–Fourier and Acoustic limits for the Boltzmann equation: Convergence proofs. Comm. Pure Appl. Math. 55, 336–393 (2002)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Golse F., Saint-Raymond L.: The Vlasov–Poisson system with strong magnetic field. J. Math. Pures Appl. 78, 791–817 (1999)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Golse F., Saint-Raymond L.: The Vlasov-Poisson system with strong magnetic field in quasineutral regime. Math. Models Methods Appl. Sci. 13, 661–714 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Golse F., Saint-Raymond L.: The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155, 81–161 (2004)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Guo Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153, 593–630 (2003)MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Guo Y.: Boltzmann diffusive limit beyond the Navier–Stokes approximation. Comm. Pure Appl. Math. 59, 626–687 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Levermore, D., Masmoudi, N.: From the Boltzamnn equation to an incompressible Navier–Stokes-Fourier system (preprint)Google Scholar
  13. 13.
    Lions P.-L., Masmoudi N.: From Boltzmann equations to incompressible fluid mechanics equation. I. Arch. Rational. Mech. Anal. 158, 173–193 (2001)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lions P.-L., Masmoudi N.: From Boltzmann equations to incompressible fluid mechanics equation. II. Arch. Rational. Mech. Anal. 158, 195–211 (2001)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Masmoudi N.: Hydrodynamic limits of the Boltzmann equation: Recent developments. Bal. Soc. Esp. Mat. Apl. 26, 57–78 (2003)MathSciNetGoogle Scholar
  16. 16.
    Masmoudi N., Saint-Raymond L.: From the Boltzmann equation to Stokes-Fourier system in a bounded domain. Comm. Pure Appl. Math. 56, 1263–1293 (2003)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Masmoudi N., Tayeb M.L.: Diffusion limit of a semiconductor Boltzmann–Poisson system. SIAM J. Math. Anal. 38, 1788–1807 (2007)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Saint-Raymond L.: From the BGK model to the Navier–Stokes equations. Ann. Sci. Ecole. Norm. Sup. 36, 271–317 (2003)MATHMathSciNetGoogle Scholar
  19. 19.
    Strain R.: The Vlasov–Maxwell–Boltzmann system in the whole space. Comm. Math. Phys. 268, 543–567 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Strain R., Guo Y.: Almost exponential decay near Maxwellians. Comm. Partial Diff. Equ. 31, 417–429 (2006)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ukai S., Asano K.: The Euler limit and initial layer of the nonlinear Boltzmann equation. Hokkido Math. J. 12, 311–332 (1983)MATHMathSciNetGoogle Scholar
  22. 22.
    Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of Mathematical Fluid Mechanics, vol. I, pp. 71–305, 2002Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institute for Advanced studyPrincetonUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations