Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics. (English) Zbl 0722.35091

The time evolution of a collisionless plasma consisting of various species of particles which interact by selfconsistent, electromagnetic forces is described by the relativistic Vlasov-Maxwell system.
The author investigates the behaviour of classical solutions to the corresponding initial value problem under small perturbations of the initial data. First it is shown that the solutions depend continuously on the initial data with respect to various norms.
The main result is on global solutions. A global solution whose electromagnetic field decays in a certain way for large times is shown to remain global under small perturbations of the initial data and to retain the decay behaviour of the field. Therefore, such global solutions are generic.
This global perturbation result implies the existence of global classical solutions for small and for nearly neutral data - results which were known from the earlier work by R. Glassey and J. Schaeffer [Commun. Math. Phys. 119, No.3, 353-384 (1988; Zbl 0673.35070)] and R. Glassey and W. Strauss [ibid. 113, 191-208 (1987; Zbl 0646.35072)] and a new global existence result for nearly spherically symmetric initial data.
Reviewer: G.Rein


35Q72 Other PDE from mechanics (MSC2000)
82D10 Statistical mechanics of plasmas
35B20 Perturbations in context of PDEs
35L15 Initial value problems for second-order hyperbolic equations
Full Text: DOI


[1] Bardos, C., Degond, P.: Global existence for the Vlasov Poisson equation in 3 space variables with small initial data. Ann. Inst. Henri Poincaré, Analyse non linéaire2, 101–118 (1985) · Zbl 0593.35076
[2] Batt, J.: Global symmetric solutions of the initial value problem in stellar dynamics. J. Diff. Eqns.25, 342–364 (1977) · Zbl 0366.35020
[3] Batt, J.: The nonlinear Vlasov-Poisson system of partial differential equations in stellar dynamics. Publications de L.U.E.R. Mathématiques Pures Appliquées, Année 835, 1–30 (1983)
[4] Glassey, R., Schaeffer, J.: On symmetric solutions of the relativistic Vlasov-Poisson system. Commun. Math. Phys.101, 459–473 (1985) · Zbl 0582.35110
[5] Glassey, R., Schaeffer, J.: Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data. Commun. Math. Phys.119, 353–384 (1988) · Zbl 0673.35070
[6] Glassey, R., Strauss, W.: Singularity formation in a collisionless plasma could occur only at high velocities. Arch. Rat. Mech. Anal.92, 59–90 (1986) · Zbl 0595.35072
[7] Glassey, R., Strauss, W.: High velocity particles in a collisionless plasma. Math. Meth. Appl. Sci.9, 46–52 (1987) · Zbl 0649.35079
[8] Glassey, R., Strauss, W.: Absence of shocks in an initially dilute collisionless plasma. Commun. Math. Phys.113, 191–208 (1987). · Zbl 0646.35072
[9] Horst, E.: On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation, Part I and II. Math. Meth. Appl. Sci.3, 229–248 (1981),4, 19–32 (1982) · Zbl 0463.35071
[10] Horst, E.: Symmetrical Plasmas and Their Decay. Preprint (1989)
[11] Pfaffelmoser, K.: Globale klassische Lösungen des dreidimensionalen Vlasov-Poisson Systems. Ph.D. dissertation, Munich (1989) · Zbl 0722.35090
[12] Rein, G.: Das Verhalten klassischer Lösungen des relativischen Vlasov-Maxwell-Systems bei kleinen Störungen der Anfangsdaten und Aussagen über globale Existenz. Ph.D. dissertation, Munich (1989)
[13] Rein, G.: A two species plasma in the limit of large ion mass. Math. Meth. Appl. Sci.13, 159–167 (1990) · Zbl 0708.35091
[14] Schaeffer, J.: Global existence for the Poisson-Vlasov system with nearly symmetric data. J. Diff. Eqns.69, 111–148 (1987) · Zbl 0642.35058
[15] Schaeffer, J.: The classical limit of the relativistic Vlasov-Maxwell system. Commun. Math. Phys.104, 409–421 (1986) · Zbl 0597.35109
[16] Wollman, S.: An existence and uniqueness theorem for the Vlasov-Maxwell system. Commun. Pure Appl. Math.37, 457–462 (1984) · Zbl 0592.45010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.