Skip to main content
SpringerLink
Log in

A Continuous Model of Transportation Revisited

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We review two models of optimal transport where congestion effects during the transport can possibly by taken into account. The first model is Beckmann's one, where the transport activities are modeled by vector fields with given divergence. The second one is the model by Carlier et al., which is in turn a continuous reformulation of Wardrop's model on graphs. We discuss the extensions of these models to their natural functional analytic setting and show that they are indeed equivalent, by using Smirnov's decomposition theorem for normal 1-currents.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Ambrosio and B. Kirchheim, "Currents in metric spaces," Acta Math., 185, 1–80(2000).

    Article  MathSciNet  Google Scholar 

  2. M. J. Beckmann, "A continuous model of transportation, "Econometrica, 20, 643–660(1952).

    Article  MathSciNet  Google Scholar 

  3. F. Benmansour, G. Carlier, G. Peyré, and F. Santambrogio, "Numerical approximation of continuous traffic congestion equilibria," Netw. Heterog. Media, 4, 605–623(2009).

    Article  MathSciNet  Google Scholar 

  4. M. Bernot, V. Caselles, and J.-M. Morel, Optimal Transportation Networks. Models and Theory, Lect. Notes Math., 1955, Springer-Verlag, Berlin (2009).

  5. G. Bouchitté and G. Buttazzo," Characterization of optimal shapes and masses through Monge–Kantorovich equation," J. Eur. Math. Soc., 3, 139–168 (2001).

    Article  MathSciNet  Google Scholar 

  6. G. Bouchitté, G. Buttazzo, and L. De Pascale, "The Monge–Kantorovich problem for distributions and applications," J. Convex Anal., 17, 925–943 (2010).

    MathSciNet  MATH  Google Scholar 

  7. G. Bouchitté, T. Champion, and C. Jimenez, "Completion of the space of measures in the Kantorovich norm," Riv. Mat. Univ. Parma, 4,127–139(2005).

    MathSciNet  MATH  Google Scholar 

  8. L. Brasco and G. Carlier, "Congested traffic equilibria and degenerate anisotropic PDEs," to appear in Dyn. Games Appl. (2013); doi:10.1007/s13235-013-0081-z.

  9. L. Brasco and G. Carlier, "On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds," to appear in Adv. Cal. Var. (2013); doi:10.1515/av-2013-0007.

  10. L. Brasco, G. Carlier, and F. Santambrogio, "Congested traffic dynamics, weak flows and very degenerate elliptic equations," J. Math. Pures Appl., 93, 652–671 (2010).

    Article  MathSciNet  Google Scholar 

  11. H. Brezis, J. M. Coron, and E. Lieb, "Harmonic maps with defects," Comm. Math. Phys., 107, 649–705 (1986).

    Article  MathSciNet  Google Scholar 

  12. G. Carlier, C. Jimenez, and F. Santambrogio, "Optimal transportation with traffic congestion and Wardrop equilibria," SIAM J. Control Optim., 47, 1330–1350 (2008).

    Article  MathSciNet  Google Scholar 

  13. B. Dacorogna and J. Moser, "On a partial differential equation involving the Jacobian determinant," Ann. Inst. H. Poincaré Anal. Non Linéaire, 7, 1–26(1990).

    Article  MathSciNet  Google Scholar 

  14. C. Dellacherie and P.-A. Meyer, Probabilities and Potentials, North-Holland, Amsterdam–NewYork (1978).

    MATH  Google Scholar 

  15. L. De Pascale and A. Pratelli, "Regularity properties for Monge transport density and for solutions of some shape optimization problems," Cal. Var. Partial Differential Equations, 14, 249–274 (2002).

    Article  MathSciNet  Google Scholar 

  16. R. J. DiPerna and P.-L. Lions, "Ordinary differential equations, transport theory and Sobolev spaces," Invent. Math., 98, 511–547 (1989).

    Article  MathSciNet  Google Scholar 

  17. I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin(1990).

    Book  Google Scholar 

  18. L. C. Evans and W. Gangbo, "Differential equations methods for the Monge–Kantorovich mass transfer problem," Mem. Amer. Math. Soc., 653 (1999).

  19. H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin–Heidelberg–New York (1969).

    MATH  Google Scholar 

  20. M. Feldman and R. McCann, "Uniqueness and transport density in Monges's mass transportation problem," Calc. Var. Partial Differential Equations, 15, 81–113 (2004).

    Article  MathSciNet  Google Scholar 

  21. M. Giaquinta, G. Modia, and J. Souček, Cartesian Currents in the Calculus of Variations I, SpringerVerlag, Berlin (1998).

    Book  Google Scholar 

  22. L. Hanin, "Duality for general Lipschitz classes and applications," Proc. London Math. Soc.,75, 134–156 (1997).

    Article  MathSciNet  Google Scholar 

  23. R. Hardt and T. Rivière, "Connecting topological Hopf singularities," Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5, 287–344 (2003).

    MathSciNet  MATH  Google Scholar 

  24. L. Kantorovich, "On the translocation of masses," Dokl. Akad. Nauk. SSSR, 37, 227–229 (1942).

    MathSciNet  Google Scholar 

  25. J. Malý, "Non-absolutely convergent integrals with respect to distributions," to appear in Ann. Mat. Pura Appl. (2013); doi:10.1007/s10231-013-0338-6.

  26. J. Moser, "On the volume elements on a manifold," Trans. Amer. Math. Soc., 120, 286–294 (1965).

    Article  MathSciNet  Google Scholar 

  27. E. Paolini and E. Stepanov, "Decomposition of acylic normal currents in a metric space," J. Funt. Anal., 263, 3358–3390 (2012).

    Article  Google Scholar 

  28. E. Paolini and E. Stepanov, "Structure of metric cycles and normal one-dimensional currents," J. Funt. Anal., 264, 1269–1295 (2013).

    Article  MathSciNet  Google Scholar 

  29. E. Paolini and E. Stepanov, "Optimal transportation networks as flat chains," Interfaces Free Bound.,8, 393–436 (2006).

    Article  MathSciNet  Google Scholar 

  30. M. Petrache, "Notes on a slice distance for singular LP-bundles," preprint (2012), http://cvgmt.sns.it/paper/1752/.

  31. M. Petrache, "Interior partial regularity for minimal LP-vector felds with integer fluxes," preprint (2012), http://cvgmt.sns.it/paper/1751/.

  32. M. Petrache and T. Rivière, "Weak closure of singular abelian LP-bundles in 3 dimensions," Geom. Funt. Anal., 21, 1419–1442 (2011).

    Article  Google Scholar 

  33. A. C. Ponce, "On the distributions of the form ∑iδpi − δni," J. Funct. Anal., 210, 391–435 (2004).

  34. T. Rivière, "Lines vortices in the U(1)-Higgs model," ESAIM Control Optim. Calc. Var.,1, 77–167 (1996).

    Article  MathSciNet  Google Scholar 

  35. E. Sandier, "Ginzburg–Landau minimizers from ℝn+1 to ℝn and minimal connetions," Indiana Univ. Math. J., 50, 1807–1844 (2001).

  36. S. K. Smirnov, "Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents," St. Petersburg Math. J., 5, 841–867 (1994).

    MathSciNet  Google Scholar 

  37. G. Strang, "L1and L∞ approximation of vector fields in the plane," Lect. Notes Numer. Appl. Anal., 5, 273–288 (1982).

    Google Scholar 

  38. T. Valkonen, "Optimal transportation networks and stations," Interfaces Free Bound.,11, 569–597 (2009).

    Article  MathSciNet  Google Scholar 

  39. C. Villani, Topics in Optimal Transportation, Amer. Math. Soc., Providence, Rhode Island (2003).

  40. J. G. Wardrop, "Some theoretical aspects of road traffic research," Proc. Inst. Civ. Eng., 2, 325–378 (1952).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire dˈAnalyse, Topologie, Probabilités, Aix-Marseille Université, Marseille, France

    L. Brasco

  2. ETH, Departement Mathematik, Zürich, Switzerland

    M. Petrache

Authors
  1. L. Brasco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Petrache
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Brasco.

Additional information

Published in Zapiski Nauhnykh Seminarov POMI, Vol. 411, 2013, pp. 5–37.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brasco, L., Petrache, M. A Continuous Model of Transportation Revisited. J Math Sci 196, 119–137 (2014). https://doi.org/10.1007/s10958-013-1644-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1644-7

Keywords

Search

Navigation