×

zbMATH — the first resource for mathematics

On the one and one-half dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity. (English) Zbl 0911.35091
The relativistic Vlasov-Maxwell system describes the time evolution of a collisionless, hot plasma. The electrons or ions are given by a phase-space density which satisfies a first order partial differential equation, the Vlasov equation, and this equation is self-consistently coupled to the Maxwell equations for the electromagnetic field. Including a linearized approximation of collisional effects like diffusion and viscosity into the Vlasov equation results in the relativistic Vlasov-Fokker-Planck-Maxwell system. The terminology “relativistic” seems questionable since the system does not seem to be Lorentz invariant. Since the question of global existence of classical solutions to such systems is as yet unresolved in dimension 3, lower dimensional cases are considered. The lowest dimension in which the hyperbolic structure of the Maxwell equations is retained and the magnetic field couples to the Vlasov equation is the case of one space and two momentum variables.
The author proves global existence and uniqueness of classical solutions to the initial-value problem for this “one and one-half dimensional” Vlasov-Fokker-Planck-Maxwell system. This extends previous work of the author where the same problem was considered for the system with vanishing viscosity.
Reviewer: G.Rein (München)

MSC:
35Q35 PDEs in connection with fluid mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
82D10 Statistical mechanics of plasmas
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bouchut, J. Funct. Anal. 111 pp 239– (1993)
[2] Bouchut, Differential Integral Equations 8 pp 487– (1995)
[3] Braasch, Math. Meth. Appl. Sci. 20 pp 667– (1997)
[4] Carrillo, Appl. Math. Lett. 10 pp 45– (1997)
[5] Carrillo, J. Math. Anal. Appl. 207 pp 475– (1997)
[6] Carrillo, J. Funct. Anal. 141 pp 99– (1996)
[7] Degond, Ann. Sci. Éc. Norm. Sup., 4eSér. 19 pp 519– (1986)
[8] DiPerna, Comm. Pure Appl. Math. 42 pp 729– (1989)
[9] Glassey, Comm. Math. Phys. 119 pp 353– (1988)
[10] Glassey, Math. Meth. Appl. Sci. 13 pp 169– (1990)
[11] Glassey, J. Math. Anal., Appl. 202 pp 1058– (1996)
[12] Glassey, Comm. Math. Phys. 113 pp 191– (1987)
[13] Illner, Math. Meth. Appl. Sci. 19 pp 1409– (1996)
[14] Lai, Math. Meth. Appl. Sci. 18 pp 1013– (1995)
[15] Rein, Comm. Math. Phys. 135 pp 41– (1990)
[16] Triolo, Math. Meth. Appl. Sci. 10 pp 487– (1988)
[17] Wollman, Comm. Pure. Appl. Math. 37 pp 457– (1984)
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.