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Nonresonant velocity averaging and the Vlasov-Maxwell system

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Transport Phenomena and Kinetic Theory

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Abstract

The Vlasov equation governs the number density in single-particle phase space of a large particle system (typically a rarefied ionized gas or plasma), subject to some external force field (for instance the Lorentz force acting on charged particles). Most importantly, collisions between particles are neglected in the Vlasov equation, unlike the case of the Boltzmann equation. Hence the only possible source of nonlinearity in the Vlasov equation for charged particles is the self-consistent electromagnetic field created by charges in motion: each particle is subject to the electromagnetic field created by all the particles other than itself.

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References

  1. Ambrosio, L., Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158 (2004), 227–260.

    Article  MATH  MathSciNet  Google Scholar 

  2. Bouchut, F., Golse, F., Pallard, C., Nonresonant smoothing for coupled wave+transport equations and the Vlasov-Maxwell system, Revistà Mat. Iberoam. 20 (2004), 865–892.

    MATH  MathSciNet  Google Scholar 

  3. Bouchut, F., Golse, F., Pallard, C., Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. for Rational Mech. and Anal. 170 (2003), 1–15.

    Article  MATH  MathSciNet  Google Scholar 

  4. DiPerna, R.J., Lions, P.-L., Global weak solutions of the Vlasov-Maxwell system, Comm. on Pure and Appl. Math. 42 (1989), 729–757.

    Article  MATH  MathSciNet  Google Scholar 

  5. DiPerna, R.J., Lions, P.-L., Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547.

    Article  MATH  MathSciNet  Google Scholar 

  6. Gel’fand, I.M., Shilov, G.E., Generalized functions. Vol. 1. Properties and operations. Academic Press, New York-London, 1964.

    Google Scholar 

  7. Glassey, R., Strauss, W., High velocity particles in a collisionless plasma. Math. Methods Appl. Sci. 9 (1987), 46–52.

    Article  MATH  MathSciNet  Google Scholar 

  8. Glassey, R., Strauss, W., Absence of shocks in an initially dilute collisionless plasma. Comm. Math. Phys. 113 (1987), 191–208.

    Article  MATH  MathSciNet  Google Scholar 

  9. Glassey, R., Strauss, W., Large velocities in the relativistic Vlasov-Maxwell equations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), 615–627.

    MATH  MathSciNet  Google Scholar 

  10. Glassey, R.T., Schaeffer, J.W., Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data. Comm. Math. Phys. 119 (1988), 353–384.

    Article  MATH  MathSciNet  Google Scholar 

  11. Glassey, R.T., Schaeffer, J.W., The “two and one-half-dimensional” relativistic Vlasov Maxwell system. Comm. Math. Phys. 185 (1997), 257–284.

    Article  MATH  MathSciNet  Google Scholar 

  12. Glassey, R.T., Schaeffer, J.W., The relativistic Vlasov-Maxwell system in two space dimensions. I, II. Arch. for Rational Mech. and Anal. 141 (1998), 331–354 & 355–374.

    Article  MATH  MathSciNet  Google Scholar 

  13. Glassey, R.T., Strauss, W.A., Singularity formation in a collisionless plasma could occur only at high velocities. Arch. for Rational Mech. and Anal. 92 (1986), 59–90.

    MATH  MathSciNet  Google Scholar 

  14. Jackson, J.D., Classical Electrodynamics Wiley, New York, 1975.

    MATH  Google Scholar 

  15. Klainerman, S., Staffilani, G., A new approach to study the Vlasov-Maxwell system, Comm. on Pure and Appl. Anal. 1 (2002), 103–125.

    MATH  MathSciNet  Google Scholar 

  16. Landau, L.D., Lifshitz, E.M., Cours de Physique Théorique. Vol. 2: Théorie des champs, Editions Mir, Moscow, 1970.

    Google Scholar 

  17. Pallard, C., Nonresonant smoothing in Sobolev spaces for coupled wave+transport equations, Bull. Sci. Math. 127 (2003), 705–718.

    Article  MATH  MathSciNet  Google Scholar 

  18. Peral, J., L p estimates for the wave equation, J. Funct. Anal. 36 (1980), 114–145.

    Article  MATH  MathSciNet  Google Scholar 

  19. Seeger, A.; Sogge, C.D.; Stein, E.M. Regularity properties of Fourier integral operators. Ann. of Math. (2) 134 (1991), 231–251.

    Article  MathSciNet  Google Scholar 

  20. Stein, E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Université Paris 7 & Laboratoire Jacques-Louis Lions, Boîte courrier 187, F75252, Paris cedex 05

    François Golse

Authors
  1. François Golse
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Editors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Torino, 32 Piazza Leonardo da Vinci, 20133, Milano, Italy

    Carlo Cercignani

  2. Dipartimento di Matematica “F. Castorati”, Università degli Studi di Pavia, 1 via Ferrata, 27100, Pavia, Italy

    Ester Gabetta

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© 2007 Birkhäuser Boston

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Golse, F. (2007). Nonresonant velocity averaging and the Vlasov-Maxwell system. In: Cercignani, C., Gabetta, E. (eds) Transport Phenomena and Kinetic Theory. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4554-0_3

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  • DOI: https://doi.org/10.1007/978-0-8176-4554-0_3

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-4489-5

  • Online ISBN: 978-0-8176-4554-0

  • eBook Packages: EngineeringEngineering (R0)

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