Skip to main content

Plasma kinetic models: the Fokker-Planck-Landau equation

  • Chapter
Modeling and Computational Methods for Kinetic Equations

Abstract

In this work, we present an approach for the Landau equation based on the relationship between entropy and entropy dissipation. Thanks to the same estimate, we recover on one hand an explicit bound on the long time behavior of the spatially homogeneous equation, and on the other hand the strong L 1 compactness of the solutions of the spatially inhomogeneous equation.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Alexandre and C. Villani. On the Landau approximation in plasma physics. To appear in Ann. I.H.P. An. non line’aire.

    Google Scholar 

  2. D. Bakry and M. Emery. Diffusions hypercontractives. in Sent. Proba. XIX, n. 1123 in Lecture Notes in Math., Springer, 177–206,1985.

    Google Scholar 

  3. H. Brdzis. Analyse Fonctionnelle. Masson, Paris, 1983.

    Google Scholar 

  4. E. A. Carlen and M. C. Carvalho. Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation. J. Stat. Phys., 67 (3–4): 575–608,1992.

    Article  MathSciNet  MATH  Google Scholar 

  5. E. A. Carlen and M. C. Carvalho. Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J. Stat. Phys., 74 (3–4): 743–782, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Cercignani. The Boltzmann equation and its applications. Springer, New York, 1988.

    Book  MATH  Google Scholar 

  7. S. Chapman and T.G. Cowling. The mathematical theory of non-uniform gases. Cambridge Univ. Press., London, 1952.

    Google Scholar 

  8. I. Csiszar. Information-type measures of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung., 2: 299–318,1967.

    MathSciNet  MATH  Google Scholar 

  9. P. Degond and M. Lemou. Dispersion relations for the linearized Fokker-Planck equation. Arch. Rat. Mech. Anal, 138: 137–167, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Degond, and B. Lucquin-Desreux. The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Mod. Meth. in Appl. Sci., 2: 2 (1992), 167–182.

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Desvillettes. On asymptotics of the Boltzmann equation when the collisions become grazing. Transp. Th. Stat. Phys., 21: 3 (1992), 259–276.

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Desvillettes and C. Villani. On the spatially homogeneous Landau equation for hard potentials. Part I: Existence, uniqueness and smoothness. Comm. Partial Differential Equations 25, 1/2 (2000), 179–259.

    Article  MathSciNet  Google Scholar 

  13. L. Desvillettes and C. Villani. On the spatially homogeneous Landau equation for hard potentials. Part II: H-theorem and applications. Comm. Partial Differential Equations 25, 1/2 (2000), 261–298.

    Article  MathSciNet  Google Scholar 

  14. R. DiPerna and P.-L. Lions. On the Cauchy problem for the Boltzmann equation: Global existence and weak stability. Ann. Math., 130: 312–366,1989.

    Article  MathSciNet  Google Scholar 

  15. R. DiPerna, P.-L. Lions, Y. Meyer. L p regularity of velocity averages. Ann. I.H.P, Analyse non-lineaire, 8: 271–287,1991.

    MathSciNet  MATH  Google Scholar 

  16. F. Golse, B. Perthame and R. Sentis, Un résultat de compacité pour l’équation de transport et application au calcul de la valeur propre principale d’un opérateur de transport, C. R. Acad. Se, 301, 341–344,1985.

    MathSciNet  MATH  Google Scholar 

  17. F. Golse, P.-L. Lions, B. Perthame, R. Sentis, Regularity of the moments of the solution of a transport equation, J. Fund. Anal., 76,110–125,1988.

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math., 97: 1061–1083,1975.

    Article  MathSciNet  Google Scholar 

  19. L. Gross. Logarithmic Sobolev inequalities and contractive properties of semigroups. In Dirichlet Forms, Lect. Notes Maths, 1563, Springer-Verlag, Berlin, 54–88,1992.

    Google Scholar 

  20. S. Kullback. A lower bound for discrimination information in terms of variation. IEEE Trans. Inf. The., 4: 126–127, 1967.

    Article  Google Scholar 

  21. E.M. Lifshitz and L.P. Pitaevskii. Physical kinetics. Perg. Press., Oxford, 1981.

    Google Scholar 

  22. P.-L. Lions. On Boltzmann and Landau equations. Phil. Trans. R. Soc. Lond., A, 346: 191–204, 1994.

    Article  MATH  Google Scholar 

  23. M.M. Rao., Z.D. Ren, Theory ofOrlicz Spaces Pure and Appl. Math., 146, N.-Y, Marcel Dekker Inc.

    Google Scholar 

  24. G. Toscani. Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation. Quart. Appl. Math., 57: 521–541, 1999.

    MathSciNet  MATH  Google Scholar 

  25. G. Toscani., C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Comm. Math. Phys., 203:667–706,1999.

    Article  MathSciNet  Google Scholar 

  26. G. Toscani., C. Villani, On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Statist. Phys., 98: 5/6,1279–1309, 2000.

    Article  MathSciNet  Google Scholar 

  27. C. Villani. On the Cauchy problem for Landau equation: sequential stability, global existence. Adv. Diff. Eq., 1: 793–816, 1996.

    MathSciNet  MATH  Google Scholar 

  28. C. Villani. On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Mod. Meth. Appl. Sci., 8: 957–983,1998.

    Article  MathSciNet  MATH  Google Scholar 

  29. C. Villani. Cercignani’s conjecture is sometimes true and always almost true. To appear in Comm. Math. Phys.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this chapter

Cite this chapter

Desvillettes, L. (2004). Plasma kinetic models: the Fokker-Planck-Landau equation. In: Degond, P., Pareschi, L., Russo, G. (eds) Modeling and Computational Methods for Kinetic Equations. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8200-2_6

Download citation

  • DOI: https://doi.org/10.1007/978-0-8176-8200-2_6

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6487-3

  • Online ISBN: 978-0-8176-8200-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics