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52 (1999) pp. 411-452. [13] Brenier, Yann; Derivation of the Euler equations from a caricature of Coulomb interaction, Commun. Math. Physics, 212 (2000) pp. 93-104. [14] Brenier, Yann and Marjolaine Puel; Optimal multiphase transportation with prescribed momentum, ESAIM Control Optim. Calc. , 8 (2002) pp. 287-343. [15] Caffarelli, Luis; The regularity of mappings with a convex potential, J. Amer. Math. , 5 (1992) pp. 99-104. [16] Caffarelli, Luis; Boundary regularity of maps with convex potentials.

Sci. Paris Ser. , 305 (1987) pp. 805-808. [10] Brenier, Yann; Polar factorization and monotone rearrangement of vectorvalued functions, Comm. Pure Appl. , 44 (1991) pp. 375-417. [11] Brenier, Yann; A homogenized model for vortex sheets, Arch. Rational Mech. , 138 (1997) pp. 319-343. [12] Brenier, Yann; Minimal geodesies on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. , 52 (1999) pp. 411-452. [13] Brenier, Yann; Derivation of the Euler equations from a caricature of Coulomb interaction, Commun.

Indeed Notice, at this point, that our model of springs with optimal assignment can be seen as a caricature of charged particles subject to Coulomb interactions. Finally, letting e go to zero, we find which is nothing but the Euler equations for an inviscid incompressible fluid with unit density. Comment The Euler equations can be related to the theory of optimal transportation in a different way through the concept of multiphase (or generalized) flow, as developed in [11, 12, 14]. 2 Vibrating strings and Born-lnfeld nonlinear electromagnetism Our goal is twofold1: 1.

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