Mild solutions for the relativistic Vlasov-Maxwell system for laser-plasma interaction
Author:
Mihai Bostan
Journal:
Quart. Appl. Math. 65 (2007), 163-187
MSC (2000):
Primary 35A05, 35B35; Secondary 82D10
DOI:
https://doi.org/10.1090/S0033-569X-07-01047-4
Published electronically:
February 12, 2007
MathSciNet review:
2313155
Full-text PDF Free Access
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Additional Information
Abstract: We study a reduced $1$D Vlasov-Maxwell system which describes the laser-plasma interaction. The unknowns of this system are the distribution function of charged particles, satisfying a Vlasov equation, the electrostatic field, verifying a Poisson equation and a vector potential term, solving a nonlinear wave equation. The nonlinearity in the wave equation is due to the coupling with the Vlasov equation through the charge density. We prove here the existence and uniqueness of the mild solution (i.e., solution by characteristics) in the relativistic case by using the iteration method.
References
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References
- M. Bostan, Existence and uniqueness of the mild solution for the $1$D Vlasov-Poisson initial-boundary value problem, SIAM J. Math. Anal. 37 (2005), 156-188. MR 2176927
- F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal. 170 (2003), 1-15. MR 2012645 (2004i:82062)
- S. Calogero and G. Rein, On classical solutions of the Nordstrom-Vlasov system, Comm. Partial Differential Equations 28 (2003), 1863-1885. MR 2015405 (2005c:35282)
- S. Calogero and G. Rein, Global weak solutions to the Nordstrom-Vlasov system, J. Differential Equations 204 (2004), 323-338. MR 2085540 (2005h:83164)
- J.A. Carrillo and S. Labrunie, Global solutions for the one-dimensional Vlasov-Maxwell system for laser-plasma interaction, Math. Models Methods Appl. Sci. 16 (2006), 19-57. MR 2194980
- J. Cooper and A. Klimas, Boundary-value problem for the Vlasov-Maxwell equation in one dimension, J. Math. Anal. Appl. 75 (1980), 306-329. MR 581821 (81h:78004)
- R. J. Diperna and P.-L. Lions, Global weak solutions of the Vlasov-Maxwell system, Comm. Pure Appl. Math. XVII (1989), 729-757. MR 1003433 (90i:35236)
- F. Filbet, Y. Guo and C.-W. Shu, Analysis of the relativistic Vlasov-Maxwell model in an interval, Quart. Appl. Math. 63 (2005), 691-714. MR 2187927 (2006h:35267)
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Additional Information
Mihai Bostan
Affiliation:
Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16 route de Gray 25030 Besançon France
Email:
mbostan@math.univ-fcomte.fr
Keywords:
Kinetic equations,
Vlasov-Maxwell system,
weak/mild solution,
characteristics
Received by editor(s):
July 21, 2006
Published electronically:
February 12, 2007
Article copyright:
© Copyright 2007
Brown University