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Inventiones mathematicae

, Volume 153, Issue 3, pp 593–630 | Cite as

The Vlasov-Maxwell-Boltzmann system near Maxwellians

Article

Abstract

Perhaps the most fundamental model for dynamics of dilute charged particles is described by the Vlasov-Maxwell-Boltzmann system, in which particles interact with themselves through collisions and with their self-consistent electromagnetic field. Despite its importance, no global in time solutions, weak or strong, have been constructed so far. It is shown in this article that any initially smooth, periodic small perturbation of a given global Maxwellian, which preserves the same mass, total momentum and reduced total energy (22), leads to a unique global in time classical solution for such a master system. The construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation: the linear collision operator L, not its time integration, is positive definite for any solutionf(t,x,v) with small amplitude to the Vlasov-Maxwell-Boltzmann system (8) and (12). As a by-product, such an estimate also yields an exponential decay for the simpler Vlasov-Poisson-Boltzmann system (24).

Keywords

Total Energy Charged Particle Electromagnetic Field Time Integration Exponential Decay 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Cercignani, C.: The Boltzmann Equation and Its Application. Springer-Verlag 1988 Google Scholar
  2. 2.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer-Verlag 1994 Google Scholar
  3. 3.
    Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases. Cambridge 1952 Google Scholar
  4. 4.
    Desvilletes, L., Dolbeault, J.: On long time asymptotics of the Vlasov-Poisson-Boltzmann equation. Commun. Partial Differ. Equations 16, 451–489 (1991) Google Scholar
  5. 5.
    Diperna, R., Lions, P.-L.: On the Cauchy problem for the Boltzmann equation: global existence and weak stability. Ann. Math. (2) 130, 321–366 (1989) Google Scholar
  6. 6.
    Diperna, R., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Lions, P.-L.: Global solutions of kinetic models and related equations. Nonequilibrium problems in many-particle systems. (Montecatini, 1992), 58–86. Lect. Notes Math. 1551. Berlin: Springer 1993 Google Scholar
  8. 8.
    Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Commun. Pure Appl. Math., Vol. LV, 1104–1135 (2002) Google Scholar
  9. 9.
    Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231, 391–434 (2002) CrossRefzbMATHGoogle Scholar
  10. 10.
    Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Rational Mech. Anal, in press (2003) Google Scholar
  11. 11.
    Glassey, R.: The Cauchy Problems in Kinetic Theory. SIAM 1996 Google Scholar
  12. 12.
    Glassey, R., Strauss, W.: Decay of the linearized Boltzmann-Vlasov system. Trans. Theory Stat. Phys. 28, 135–156 (1999) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Glassey, R., Strauss, W.: Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete Contin. Dyn. Syst. 5, 457–472 (1999) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mischler, S.: On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 210, 447–466 (2000) CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Ukai, S.: On the existence of global solutions of a mixed problem for the nonlinear Boltzmann equation. Proc. Japan Acad., Ser. A 53, 179–184 (1974)Google Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  • Yan Guo
    • 1
  1. 1.Lefshetz Center for Dynamical Systems, Division of Applied MathematicsBrown UniversityProvidenceUSA

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