Open Access
December 2004 Order preserving vibrating strings and applications to electrodynamics and magnetohydrodynamics
Yann Brenier
Methods Appl. Anal. 11(4): 515-532 (December 2004).


The motion of a collection of vertical strings subject to horizontal linear vibrations in the plane can be described by a system of first order nonlinear conservations laws. This system -that we call the Chaplygin-Born-Infeld (CBI) system- is related to Magnetohydrodynamics and more specifically to its shallow water version. Then, each vibrating string can be interpreted as a magnetic line. The CBI system is also related to the Born-Infeld theory for the electromagnetic field, a nonlinear correction to the classical Maxwell’s equations.

Due to the linearity of vibrations, there is a priori no mechanism to prevent the strings to cross each other, at least for sufficiently large initial impulse. These crossings generate concentration sin- gularities in the CBI system. A numerical scheme is introduced to maintain order preserving strings beyond singularities. This order preserving scheme is shown to be convergent to a distinguished limit, which can be interpreted, through maximal monotone operator theory, as a vanishing viscosity limit of the CBI system. Finally, models of pressureless gas with sticky particles are revisited and a new formulation is provided.


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Yann Brenier. "Order preserving vibrating strings and applications to electrodynamics and magnetohydrodynamics." Methods Appl. Anal. 11 (4) 515 - 532, December 2004.


Published: December 2004
First available in Project Euclid: 13 April 2006

zbMATH: 1107.74028
MathSciNet: MR2195368

Primary: 74K05
Secondary: 35L70 , 74H45 , 76W05 , 78A25

Rights: Copyright © 2004 International Press of Boston

Vol.11 • No. 4 • December 2004
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