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Paraxial approximation of the stationary Vlasov-Maxwell equations. (English) Zbl 0823.35149
Brezis, H. (ed.) et al., Nonlinear partial differential equations and their applications. Collège de France Seminar, volume XII. Lectures held at the weekly seminar on applied mathematics, Paris, France, 1991-1993. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 302, 158-171 (1994).
The time evolution of a beam of charged particles, which interact only by selfconsistent electromagnetic fields, can be modelled by the Vlasov- Maxwell system. The particles are described by a distribution function on phase space which obeys a continuity equation, the Vlasov equation. This equation is coupled to Maxwell’s equations for the fields and the resulting nonlinear system of PDEs is supplemented with boundary conditions for both the particle density and the electromagnetic fields.
Since the model is posed in 6-dimensional phase space, numerical simulations are expensive in terms of computational costs. The author presents an approximative model for the stationary case, which exploits the symmetry of such plasma beams in order to obtain an also numerically simpler model. To this end, the physical quantities are split into longitudinal components, i.e., components in the direction of the symmetry axis, and transverse components, and these are expanded in power of \(l/L\) where \(l\) and \(L\) are the transverse or longitudinal characteristic lengths respectively.
For the entire collection see [Zbl 0795.00016].
Reviewer: G.Rein (München)

35Q35 PDEs in connection with fluid mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35Q60 PDEs in connection with optics and electromagnetic theory