Optimal multiphase transportation with prescribed momentum
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 287-343.

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.

DOI : https://doi.org/10.1051/cocv:2002024
Classification : 65K10,  35Q
Mots clés : optimal transportation, multiphase flow, electrodynamics 
@article{COCV_2002__8__287_0,
     author = {Brenier, Yann and Puel, Marjolaine},
     title = {Optimal multiphase transportation with prescribed momentum},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {287--343},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002024},
     zbl = {1091.49034},
     mrnumber = {1932954},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2002__8__287_0/}
}
Brenier, Yann; Puel, Marjolaine. Optimal multiphase transportation with prescribed momentum. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 287-343. doi : 10.1051/cocv:2002024. http://www.numdam.org/item/COCV_2002__8__287_0/

[1] F. Barthe, Optimal Young's inequality and its converse: A simple proof. Geom. Funct. Anal. 8 (1998) 234-242. | Zbl 0902.26009

[2] J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375-393. | Zbl 0968.76069

[3] M. Born and L. Infeld, Foundations of the new field theory. Proc. Roy. Soc. London A 144 (1934) 425-451. | Zbl 0008.42203

[4] G. Bouchitté and G. Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation. J. Eur. Math. Soc. (JEMS) 3 (2001) 139-168. | Zbl 0982.49025

[5] 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. | MR 1035755 | Zbl 0687.76016

[6] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 805-808. | MR 923203 | Zbl 0652.26017

[7] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 64 (1991) 375-417. | MR 1100809 | Zbl 0738.46011

[8] Y. Brenier, A homogenized model for vortex sheets. Arch. Rational Mech. Anal. 138 (1997) 319-353. | MR 1467558 | Zbl 0962.35140

[9] Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Comm. Pure Appl. Math. 52 (1999) 411-452. | MR 1658919 | Zbl 0910.35098

[10] 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).

[11] H. Brézis, Analyse fonctionnelle. Masson, Paris (1974). | MR 697382 | Zbl 0511.46001

[12] L.A. Caffarelli, Boundary regularity of maps with convex potentials. Ann. of Math. (2) 144 (1996) 453-496. | MR 1426885 | Zbl 0916.35016

[13] M.J. Cullen and R.J. Purser, An extended Lagrangian theory of semigeostrophic frontogenesis. J. Atmos. Sci. 41 (1984) 1477-1497. | MR 881109

[14] L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999). | Zbl 0920.49004

[15] W. Gangbo and R.J. Mccann, The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. | MR 1440931 | Zbl 0887.49017

[16] D. Kinderlehrer, R. Jordan and F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1-17. | Zbl 0915.35120

[17] L.V. Kantorovich, On a problem of Monge. Uspekhi Mat. Nauk. 3 (1948) 225-226.

[18] R.J. Mccann, A convexity principle for interacting gases. Adv. Math. 128 (1997) 153-179. | MR 1451422 | Zbl 0901.49012

[19] F. Otto, Viscous fingering: An optimal bound on the growth rate of the mixing zone. SIAM J. Appl. Math. 57 (1997) 982-990. | MR 1462048 | Zbl 0901.76082

[20] F. Otto, The geometry of dissipative evolution equations: The porous medium equation. Comm. Partial Differential Equations 26 (2001) 101-174 | MR 1842429 | Zbl 0984.35089

[21] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000) 361-400. | MR 1760620 | Zbl 0985.58019

[22] A.V. Pogorelov, The Minkowski multidimensional problem. John Wiley, New York-Toronto-London, Scr. Ser. in Math. (1978). | MR 478079 | Zbl 0387.53023

[23] S.T. Rachev and L. Rüschendorf, Mass transportation problems, Vols. I and II. Probability and its Applications. Springer-Verlag. | MR 1619170 | Zbl 0990.60500

[24] G. Strang, Introduction to applied mathematics. Wellesley-Cambridge Press, Wellesley, MA (1986). | MR 870634 | Zbl 0618.00015

[25] V.N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions. Proc. Steklov Inst. 141 (1979) 1-178. | MR 530375 | Zbl 0409.60005