Optimal multiphase transportation with prescribed momentum
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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: 10.1051/cocv:2002024
Classification: 65K10,  35Q
Keywords: 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://archive.numdam.org/articles/10.1051/cocv:2002024/}
}
TY  - JOUR
AU  - Brenier, Yann
AU  - Puel, Marjolaine
TI  - Optimal multiphase transportation with prescribed momentum
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2002
DA  - 2002///
SP  - 287
EP  - 343
VL  - 8
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2002024/
UR  - https://zbmath.org/?q=an%3A1091.49034
UR  - https://www.ams.org/mathscinet-getitem?mr=1932954
UR  - https://doi.org/10.1051/cocv:2002024
DO  - 10.1051/cocv:2002024
LA  - en
ID  - COCV_2002__8__287_0
ER  - 
%0 Journal Article
%A Brenier, Yann
%A Puel, Marjolaine
%T Optimal multiphase transportation with prescribed momentum
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2002
%P 287-343
%V 8
%I EDP-Sciences
%U https://doi.org/10.1051/cocv:2002024
%R 10.1051/cocv:2002024
%G en
%F COCV_2002__8__287_0
Brenier, Yann; Puel, Marjolaine. Optimal multiphase transportation with prescribed momentum. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), pp. 287-343. doi : 10.1051/cocv:2002024. http://archive.numdam.org/articles/10.1051/cocv:2002024/

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

[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

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

[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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

[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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

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

Cited by Sources: