Skip to main content

Vlasov Approximation

  • Chapter
  • First Online:
Mathematical Models of Viscous Friction

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2135))

  • 1287 Accesses

Abstract

In this chapter the problem of viscous friction is considered when the medium is described by a gas of free particles in the mean field approximation. We give necessary conditions on the body/medium interaction to have a microscopic model of viscous friction and, conversely, we show that the runaway particle effect takes place in the case of bounded interaction. We conclude with a heuristic analysis of the case of singular interaction.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight 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

References

  1. Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the 1∕N limit of interacting classical particles. Commun. Math. Phys. 56, 101–111 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  2. Buttà, P., Caglioti, E., Marchioro, C.: On the long time behavior of infinitely extended systems of particles interacting via Kac Potentials. J. Stat. Phys. 108, 317–339 (2002)

    Article  MATH  Google Scholar 

  3. Buttà, P., Ferrari, G., Marchioro, C.: Speedy motions of a body immersed in an infinitely extended medium. J. Stat. Phys. 140, 1182–1194 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Buttà, P., Manzo, F., Marchioro, C.: A simple Hamiltonian model of runaway particle with singular interaction. Math. Models Methods Appl. Sci. 15, 753–766 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Caglioti, E., Caprino, S., Marchioro, C., Pulvirenti, M.: The Vlasov equation with infinite mass. Arch. Ration. Mech. Anal. 159, 85–108 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Caprino, S., Marchioro, C.: On the plasma-charge model. Kinet. Relat. Models 3, 241–254 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Caprino, S., Marchioro, C., Pulvirenti, M.: On the Vlasov–Helmholtz equations with infinite mass. Commun. Partial Differ. Equ. 27, 791–808 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dobrushin, R.L.: Vlasov equations. Soviet J. Funct. Anal. 13, 60–110 (1979)

    MathSciNet  Google Scholar 

  9. Hauray, M., Jabin, P.E.: N-particles approximation of the Vlasov equations with singular potential. Arch. Ration. Mech. Anal. 183, 489–524 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Horst, E.: On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I. Math. Methods Appl. Sci. 3, 229–248 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  11. Horst, E.: On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation II. Math. Methods Appl. Sci. 4, 19–32 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gruber, Ch., Piasecki, Jb.: Stationary motion of the adiabatic piston. Physica A 268, 412–423 (1999)

    Article  Google Scholar 

  13. Landau, L.D., Lifshitz, E.M.: Physical Kinetics. Course of Theoretical Physics, vol. 10. Oxford/Pergamon Press, New York/Frankfurt (1981)

    Google Scholar 

  14. Lebowitz, J.L., Piasecki, J., Sinai, Ya.: Scaling dynamics of a massive piston in an ideal gas. In: Hard Ball Systems and the Lorentz Gas. Encylopaedia of Mathematical Sciences, vol. 101, pp. 217–227. Springer, Berlin (2000)

    Google Scholar 

  15. Lions, P.-L., Perthame, B.: Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system. Invent. Math. 105, 415–430 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  16. Loeper, G.: Uniqueness of the solution to the Vlasov–Poisson system with bounded density. J. Math. Pure Appl. 86, 68–79 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Majda, A., Majda, G., Zheng, Y.: Concentrations in the one-dimensional Vlasov–Poisson equations. I Temporal development and non-unique weak solutions in the single component case. Physica D 74, 268–300 (1994)

    MATH  MathSciNet  Google Scholar 

  18. Majda, A., Majda, G., Zheng, Y.: Concentrations in the one-dimensional Vlasov–Poisson equations. II Screening and the necessity for measure-valued solutions in the two component case. Physica D 79, 41–76 (1994)

    MATH  MathSciNet  Google Scholar 

  19. Marchioro, C., Miot, E., Pulvirenti, M.: The Cauchy problem for the 3 − D Vlasov–Poisson system with point charges. Arch. Ration. Mech. Anal. 201, 1–26 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Neunzert, H.: An introduction to the nonlinear Boltzmann–Vlasov equation. In: Kinetic Theories and the Boltzmann Equation (Montecatini, 1981). Lecture Notes in Mathematics, vol. 1048, pp. 60–110. Springer, Berlin (1981)

    Google Scholar 

  21. Pankavich, S.: Global existence for a three dimensional Vlasov–Poisson system with steady spatial asymptotics. Commun. Partial Differ. Equ. 31, 349–370 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Pfaffelmoser, K.: Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data. J. Differ. Equ. 95, 281–303 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Salort, D.: Transport equations with unbounded force fields and application to the Vlasov–Poisson equation. Math. Models Methods Appl. Sci. 19, 199–228 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sganga, D.: Moto di una particella carica in un fluido di Vlasov libero. Graduate thesis, SAPIENZA Università di Roma [in Italian] (2008)

    Google Scholar 

  25. Schaeffer, J.: Global existence of smooth solutions to the Vlasov–Poisson system in three dimensions. Commun. Partial Differ. Equ. 16, 1313–1335 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  26. Schaeffer, J.: Steady spatial asymptotics for the Vlasov–Poisson system. Math. Methods Appl. Sci. 26, 273–296 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Schaeffer, J.: The Vlasov Poisson system with steady spatial asymptotics. Commun. Partial Differ. Equ. 28, 1057–108 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Schaeffer, J.: Global existence for the Vlasov–Poisson system with steady spatial asymptotic behavior. Kinet. Relat. Models 5, 129–153 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. Spohn, H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3, 445–455 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ukai, S., Okabe, T.: On classical solutions in the large in time of two-dimensional Vlasov’s equation. Osaka J. Math. 15, 245–261 (1978)

    MATH  MathSciNet  Google Scholar 

  31. Wollman, S.: Global in time solutions to the two-dimensional Vlasov–Poisson system. Commun. Pure Appl. Math. 33, 173–197 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  32. Wollman S.: Global in time solution to the three-dimensional Vlasov–Poisson system. J. Math. Anal. Appl. 176, 76–91 (1993)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Buttà, P., Cavallaro, G., Marchioro, C. (2015). Vlasov Approximation. In: Mathematical Models of Viscous Friction. Lecture Notes in Mathematics, vol 2135. Springer, Cham. https://doi.org/10.1007/978-3-319-14759-8_2

Download citation

Publish with us

Policies and ethics