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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)

35Q35 PDEs in connection with fluid mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
82D10 Statistical mechanics of plasmas
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