In this paper, we review some transport models based on the continuity equation, starting with the so-called Benamou − Brenier formula, which is nothing but a fluid mechanics reformulation of the Monge − Kantorovich problem with cost c(x, y) = |x − y|2. We discuss some of its applications (gradient flows, sharp functional inequalities, etc.), as well as some variants and generalizations to dynamical transport problems, where interaction effects among mass particles are considered. Bibliography: 43 titles.
Similar content being viewed by others
References
M. Agueh, N. Ghoussoub, and X. Kang, “Geometric inequalities via a general comparison principle for interacting gases,” Geom. Funct. Anal., 14, 215-244 (2004).
L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Birkhäuser Verlag, Basel (2008).
F. Andreu, V. Caselles, and J. M. Mazón, “Some regularity results on the ‘relativistic” heat equation,” J. Differential Equations, 245, 3639-3663 (2008).
W. Beckner, “Geometric proof of Nash’s inequality,” Int. Math. Res. Not., 1998, No. 2, 67-71 (1998).
J.-D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kant0r0vich mass transfer problem,” Numer. Math., 84, 375-393 (2000).
M. Bernot, V. Caselles, and J.-M. Morel, “Traffic plans,” Publ. Mat., 49, 417-451 (2005).
M. Bernot, V. Caselles, and J.-M. Morel, “The structure of branched transportation networks,” Calc. Var. Partial Dijferential Equations, 2, 279-317 (2008).
M. Bernot, V. Caselles, and J.-M. Morel, Optimal Transportation Networks. Models and Theory, Lect. Notes Math., 1955, Springer-Verlag, Berlin (2009).
M. Bernot and A. Figalli, “Synchronized traffic plans and stability of optima,” ESAIM Control Optim. Calc. Var., 14, 864-878 (2008).
G. Bouehitté and G. Buttazzo, “New lower semicontinuity results for nonconvex functionals defined on measures,” Nonlinear Anal., 15, 679-692 (1990).
Y. Brenier, “Extended Monge-Kantorovich theory,” in: Optimal Transportation and Applications, Lect. Notes Math., 1813, Springer, Berlin (2003), pp. 91-121.
A. Brancolini, G. Buttazzo, and F. Santambrogio, “Path functionals over Wasserstein spaces,” J. Eur. Math. Soc., 8, 415-434 (2006).
L. Brasco, G. Buttazzo, and F. Santambrogio, “A Benamou-Brenier approach to branched transport,” SIAM J. Math. Anal., 43, 1023-1040 (2011).
L. Brasco and F. Santambrogio, “An equivalent path functional formulation of branched transportation problems,” Discrete Contin. Dyn. Syst., 29, 845-871 (2011).
G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Longman Scientific & Technical, Harlow (1989).
G. Buttazzo, C. Jimenez, and E. Oudet, “An optimization problem for mass transportation with congested dynamics,” SIAM J. Control Optim., 48, 1961-1976 (2009).
J. A. Carrillo, S. Lisini, G. Savaré, and D. Slepcĕv, “Nonlinear mobility continuity equations and generalized displacement convexity,” J. Funct. Anal., 258, 1273-1309 (2010).
D. Cordero-Erausquin, B. Nazaret, and C. Villani, “A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities,” Adv. Math., 182, 307-332 (2004).
C. Dellacherie and P.-A. Meyer, Probabilités et potentiel, Chapitres I à IV, Hermann, Paris (1975).
M. Del Pino and J. Dolbeault, “Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions,” J. Math. Pures Appl., 81, 847-875 (2002).
J. Dolbeault, B. Nazaret, and G. Savare, “A new class of transport distances between measures,” Calc. Var. Partial Differential Equations, 34, 193-231 (2009).
E. N. Gilbert, “Minimum cost communication networks,” Bell System Tech. J., 46, 2209-2227 (1967).
L. Gross, “Logarithmic Sobolev inequalities,” Amer. J. Math., 97, 1061-1083 (1975).
T. Hillen and K. J. Painter, “A user’s guide to PDE models for chemotaxis,” J. Math. Biol., 58, 183-217 (2009).
R. Jordan, D. Kinderlehrer, and F. Otto, “The variational formulation of the Fokker Planck equation,” SIAM J. Math. Anal., 29, 1-17 (1998).
J. M. Lasry and P.-L. Lions, “Mean field games,” Japan J. Math., 2, 229-260 (2007).
S. Lisini, “Characterization of absolutely continuous curves in Wasserstein spaces,” Calc. Var. Partial Differential Equations, 28, 85-120 (2007).
S. Lisini and A. Marigonda, “On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals,” Manuscripta Math., 133, 197-224 (2010).
F. Maddalena, J. M. Morel, and S. Solimini, “A variational model of irrigation patterns,” Interfaces Free Bound., 5, 391-415 (2003).
F. Maggi and C. Villani, “Balls have the worst best Sobolev inequalities. II. Variants and extensions,” Calc. Var. Partial Differential Equations, 31, 47-74 (2008).
F. Maggi and C. Villani, “Balls have the worst best Sobolev inequalities,” J. Geom. Anal., 15, 83-121 (2005).
R. McCann, “A convexity principle for interacting gases,” Adv. Math., 128, 153-179 (1997).
R. McCann and M. Puel, “Constructing a relativistic heat flow by transport time steps,” Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 2539-2580 (2009).
J.-M. Morel and F. Santambrogio, “Comparison of distances between measures,” Appl. Math. Lett., 20, 427-432 (2007).
B. Nazaret. “Best constant in Sobolev trace inequalities on the half space,” Nonlinear Anal., 65, 1977-1985 (2006).
F. Otto, “The geometry of dissipative evolution equation: the porous medium equation,” Comm. Partial Differential Equations, 26, 101-174 (2001).
F. Otto, “Dynamics of labyrinthine pattern formation in magnetic fluids: a mean—field theory,” Arch. Rational Mech. Anal., 141, 63-103 (1998).
F. Otto and C. Villani, “Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality,” J. Funct. Anal., 173, 361-400 (2000).
E. Paolini and E. Stepanov, “Optimal transportation networks as flat chains,” Interfaces Free Bound., 8, 393-436 (2006).
G. Talenti, “Best constant in Sobolev inequality,” Ann. Mat. Pura Appl., 110, 353-372 (1976).
C. Villani, Optimal Transport. Old and New, Springer-Verlag, Berlin (2009).
C. Villani, “Optimal transportation, dissipative PDE’s and functional inequalities,” in: Optimal Transportation and Applications, Lect. Notes Math., 1813, Springer, Berlin (2003), pp. 53-89.
Q. Xia, “Optimal paths related to transport problems,” Commun. Contemp. Math., 5, 251-279 (2003).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 390, 2011, pp. 5 − 51.
Rights and permissions
About this article
Cite this article
Brasco, L. A Survey on dynamical transport distances. J Math Sci 181, 755–781 (2012). https://doi.org/10.1007/s10958-012-0713-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-012-0713-7