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
Published electronically: September 22, 2005
MathSciNet review: 2187923
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35Q60, 86A25

Retrieve articles in all journals with MSC (2000): 35Q60, 86A25

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.

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2016 Brown University
Comments: qam-query@ams.org
AMS Website