Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Analysis of the relativistic Vlasov-Maxwell model in an interval


Authors: Francis Filbet, Yan Guo and Chi-Wang Shu
Journal: Quart. Appl. Math. 63 (2005), 691-714
MSC (2000): Primary 35Q72, 76X05, 82D99
DOI: https://doi.org/10.1090/S0033-569X-05-00977-9
Published electronically: August 17, 2005
MathSciNet review: 2187927
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Abstract: The dynamics of a collisionless electron gas under the influence of the self-consistent electromagnetic field is studied in an interval, where electrons are emitted at one end of the interval and absorbed at the other. The electron distribution $f$ depends on one space variable $x$ and two dimensional velocity variables $(v_{1},v_{2})$, whereas the electromagnetic field satisfies the Maxwell system. It is shown that the electromagnetic field is smooth enough to define the trajectories of particles. On the other hand, due to the absorbing boundary condition, $f$ is in general not smooth, and only has bounded total variations. Such $BV$ regularity is sufficient to guarantee the uniqueness of the nonlinear electron dynamics. Finally, numerical simulations are performed to demonstrate the formation of a singularity from the boundary for the electron distribution.


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Additional Information

Francis Filbet
Affiliation: Mathématiques pour l’Industrie et la Physique, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse cedex 04, France
Email: filbet@mip.ups-tlse.fr

Yan Guo
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: guoy@dam.brown.edu

Chi-Wang Shu
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: shu@dam.brown.edu

DOI: https://doi.org/10.1090/S0033-569X-05-00977-9
Keywords: Vlasov-Maxwell system, boundary problem, BV estimate
Received by editor(s): February 20, 2005
Published electronically: August 17, 2005
Additional Notes: The research of Y. Guo was supported in part by NSF grant DMS-0305161 and by a Salomon award from Brown University. The research of C.-W. Shu was supported in part by ARO grant W911NF-04-1-0291, NSF grant DMS-0207451 and AFOSR grant F49620-02-1-0113.
Article copyright: © Copyright 2005 Brown University

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