Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Steady states of the Vlasov-Maxwell system

Author: Jack Schaeffer
Journal: Quart. Appl. Math. 63 (2005), 619-643
MSC (2000): Primary 35Q60; Secondary 86A25
DOI: https://doi.org/10.1090/S0033-569X-05-00984-5
Published electronically: September 22, 2005
MathSciNet review: 2187923
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Abstract: The Vlasov-Maxwell system models collisionless plasma. Solutions are considered that depend on one spatial variable, $x$, and two velocity variables, $v_1$ and $v_2$. As $x\rightarrow - \infty$ it is required that the phase space densities of particles approach a prescribed function, $F\left(v_1,v_2\right)$, and all field components approach zero. It is assumed that $F\left(v_1,v_2\right) = 0$ if $v_1 \leq W_1$, where $W_1$ is a positive constant. An external magnetic field is prescribed and taken small enough so that no particle is reflected ($v_1$ remains positive).

The main issue is to identify the large-time behavior; is a steady state approached and, if so, can it be identified from the time independent Vlasov-Maxwell system? The time-dependent problem is solved numerically using a particle method, and it is observed that a steady state is approached (on a bounded $x$ interval) for large time. For this steady state, one component of the electric field is zero at all points, the other oscillates without decay for $x$ large; in contrast the magnetic field tends to zero for large $x$. Then it is proven analytically that if the external magnetic field is sufficiently small, then (a reformulation of) the steady problem has a unique solution with $B \rightarrow 0$ as $x \rightarrow +\infty$. Thus the ``downstream'' condition, $B \rightarrow 0$ as $x\rightarrow + \infty$, is used to identify the large time limit of the system.

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Additional Information

Jack Schaeffer
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email: js5m@andrew.cmu.edu

DOI: https://doi.org/10.1090/S0033-569X-05-00984-5
Received by editor(s): October 13, 2004
Published electronically: September 22, 2005
Article copyright: © Copyright 2005 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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