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Optimal Multiphase Transportation with prescribed momentum

Published online by Cambridge University Press:  15 August 2002

Yann Brenier  and
Marjolaine Puel
Yann Brenier
Affiliation:
Laboratoire J.A Dieudonné, Parc Valrose, 06100 Nice, France; brenier@math.unice.fr.
Marjolaine Puel
Affiliation:
Laboratoire d'Analyse Numérique, Université Paris 6, BC. 187, 75252 Paris Cedex 05, France.
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Abstract

A multiphase generalization of the Monge–Kantorovich optimal transportation problem is addressed. Existence of optimal solutions is established. The optimality equations are related to classical Electrodynamics.

Keywords

Optimal transportationmultiphase flowelectrodynamics.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 287 - 343
Copyright
© EDP Sciences, SMAI, 2002

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References

Barthe, F., Optimal Young's inequality and its converse: A simple proof. Geom. Funct. Anal. 8 (1998) 234-242. CrossRef
Benamou, J.-D. and Brenier, Y., A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375-393. CrossRef
Born, M. and Infeld, L., Foundations of the new field theory. Proc. Roy. Soc. London A 144 (1934) 425-451. CrossRef
Bouchitté, G. and Buttazzo, G., Characterization of optimal shapes and masses through Monge-Kantorovich equation. J. Eur. Math. Soc. (JEMS) 3 (2001) 139-168.
Y. Brenier, A combinatorial algorithm for the Euler equations of incompressible flows, in Proc. of the Eighth International Conference on Computing Methods in Applied Sciences and Engineering. Versailles (1987). Comput. Methods Appl. Mech. Engrg. 75 (1989) 325-332.
Brenier, Y., Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 805-808.
Brenier, Y., Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 64 (1991) 375-417. CrossRef
Brenier, Y., A homogenized model for vortex sheets. Arch. Rational Mech. Anal. 138 (1997) 319-353. CrossRef
Brenier, Y., Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Comm. Pure Appl. Math. 52 (1999) 411-452. 3.0.CO;2-3>CrossRef
Y. Brenier, Extension of the Monge-Kantorovich theory to classical electrodynamics. Summer School on mass transportation methods in kinetic theory and hydrodynamics. Ponta Delgada, Azores, Portugal (2000).
H. Brézis, Analyse fonctionnelle. Masson, Paris (1974).
Caffarelli, L.A., Boundary regularity of maps with convex potentials. Ann. of Math. (2) 144 (1996) 453-496. CrossRef
Cullen, M.J. and Purser, R.J., An extended Lagrangian theory of semigeostrophic frontogenesis. J. Atmos. Sci. 41 (1984) 1477-1497. 2.0.CO;2>CrossRef
L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999).
Gangbo, W. and McCann, R.J., The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. CrossRef
Kinderlehrer, D., Jordan, R. and Otto, F., The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1-17.
Kantorovich, L.V., On a problem of Monge. Uspekhi Mat. Nauk. 3 (1948) 225-226.
McCann, R.J., A convexity principle for interacting gases. Adv. Math. 128 (1997) 153-179. CrossRef
Otto, F., Viscous fingering: An optimal bound on the growth rate of the mixing zone. SIAM J. Appl. Math. 57 (1997) 982-990. CrossRef
Otto, F., The geometry of dissipative evolution equations: The porous medium equation. Comm. Partial Differential Equations 26 (2001) 101-174 CrossRef
Otto, F. and Villani, C., Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000) 361-400. CrossRef
A.V. Pogorelov, The Minkowski multidimensional problem. John Wiley, New York-Toronto-London, Scr. Ser. in Math. (1978).
S.T. Rachev and L. Rüschendorf, Mass transportation problems, Vols. I and II. Probability and its Applications. Springer-Verlag.
G. Strang, Introduction to applied mathematics. Wellesley-Cambridge Press, Wellesley, MA (1986).
Sudakov, V.N., Geometric problems in the theory of infinite-dimensional probability distributions. Proc. Steklov Inst. 141 (1979) 1-178.