# Nonresonant velocity averaging and the Vlasov-Maxwell system

• François Golse
Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

## 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.

## Keywords

Cauchy Problem Fundamental Solution Vlasov Equation Collisionless Plasma Maxwell System
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.

## References

1. [1]
Ambrosio, L., Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158 (2004), 227–260.
2. [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.
3. [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.
4. [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.
5. [5]
DiPerna, R.J., Lions, P.-L., Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547.
6. [6]
Gel’fand, I.M., Shilov, G.E., Generalized functions. Vol. 1. Properties and operations. Academic Press, New York-London, 1964.Google Scholar
7. [7]
Glassey, R., Strauss, W., High velocity particles in a collisionless plasma. Math. Methods Appl. Sci. 9 (1987), 46–52.
8. [8]
Glassey, R., Strauss, W., Absence of shocks in an initially dilute collisionless plasma. Comm. Math. Phys. 113 (1987), 191–208.
9. [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.
10. [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.
11. [11]
Glassey, R.T., Schaeffer, J.W., The “two and one-half-dimensional” relativistic Vlasov Maxwell system. Comm. Math. Phys. 185 (1997), 257–284.
12. [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.
13. [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.
14. [14]
Jackson, J.D., Classical Electrodynamics Wiley, New York, 1975.
15. [15]
Klainerman, S., Staffilani, G., A new approach to study the Vlasov-Maxwell system, Comm. on Pure and Appl. Anal. 1 (2002), 103–125.
16. [16]
Landau, L.D., Lifshitz, E.M., Cours de Physique Théorique. Vol. 2: Théorie des champs, Editions Mir, Moscow, 1970.Google Scholar
17. [17]
Pallard, C., Nonresonant smoothing in Sobolev spaces for coupled wave+transport equations, Bull. Sci. Math. 127 (2003), 705–718.
18. [18]
Peral, J., L p estimates for the wave equation, J. Funct. Anal. 36 (1980), 114–145.
19. [19]
Seeger, A.; Sogge, C.D.; Stein, E.M. Regularity properties of Fourier integral operators. Ann. of Math. (2) 134 (1991), 231–251.
20. [20]
Stein, E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970.