Abstract
Our purpose is to extend to the framework of Electromagnetism the Monge-Kantorovich theory [9] which originally comes from Continuum Mechanics and has become very popular in the last ten years in the field of nonlinear PDEs, especially because of its connection with the Monge-Ampère equation [2],[3],[4], [5],[6],[7]…We construct, in the framework of Electromagnetism, an Action which is analogous to the Monge-Kantorovich (or Wasserstein) distance. From this Action, we derive a system of PDEs that look like non-linear Maxwell equations. Then, through suitable approximations, we formally recover not only the linear Maxwell equation, as expected, but, more surprisingly, an approximation to the complete (pressureless) Euler-Maxwell system.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
J.-D. Benamou, Y. Brenier, A Computational Fluid Mechanics solution to the MongeKantorovich mass transfer problem, Numerische Math. 84 (2000) 375–393.
Y.Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm Pure Appl. Math. 64 (1991) 375–417.
L.A. Caffarelli, Boundary regularity of maps with convex potentials. Ann. of Math. (2) 144 (1996), no. 3, 453–496.
L.C. Evans,Partial differential equations and Monge-Kantorovich mass transfer, Lecture Notes, 1998.
W. Gangbo, R.J. McCann, The geometry of optimal transportation, Acta Math. 177 (1996), no. 2, 113–161.
D. Kinderlehrer, R. Jordan, F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998), no. 1,1–17.
F. Otto, Evolution of microstructure in unstable porous media flow: a relaxational approach,. Comm. Pure Appl. Math. 52 (1999) 873–915.
J. Polchinski, String theory. Vol. I., Cambridge University Press, 1998.
S.T. Rachev, L. Röschendorf, Mass transportation problems, Vol. I and II. Probability and its Applications, Springer-Verlag, New York, 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Brenier, Y. (2001). A Monge-Kantorovich Approach to the Maxwell Equations. In: Freistühler, H., Warnecke, G. (eds) Hyperbolic Problems: Theory, Numerics, Applications. International Series of Numerical Mathematics, vol 140. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8370-2_19
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8370-2_19
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9537-8
Online ISBN: 978-3-0348-8370-2
eBook Packages: Springer Book Archive