Skip to main content

Gradient flows

  • Chapter

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE,volume 87)

Abstract

In this chapter we present one of the most spectacular applications of optimal transport and Wasserstein distances to PDEs. We will see that several evolution PDEs can be interpreted as a steepest descent movement in the space \(\mathbb{W}_{2}\). This includes the Heat equation, the Fokker-Planck equation, and many others. We will present the main ideas, provide a rigorous analysis of the Fokker-Planck case, and, possible extension in the discussion section. The discussion also presents complementary topics about the theoretical framework of gradient flows in metric spaces, and about other models in evolutionary PDEs which are connected to optimal transport but are not gradient flows.

Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    We avoid here the distinction between minimizing movements and generalized minimizing movements, which is not crucial in our analysis.

  2. 2.

    Unfortunately, some knowledge of French is required (even if not forbidden, English is unusual in the Bourbaki seminar, since “Nicolas Bourbaki a une préférence pour le français”).

  3. 3.

    The original ideas which led to this theory come from [198] and [246] through what is known as “Otto’s formal calculus”, and have later been reconsidered in [15].

  4. 4.

    Note that a similar argument, based on doubling the variables, is also often performed for the differentiation of W 2 2 along curves in \(\mathbb{W}_{2}\), which we did differently in Section 5.3.5.

References

  1. L. Ambrosio, Movimenti minimizzanti. Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Natur. 113, 191–246 (1995)

    MathSciNet  Google Scholar 

  2. L. Ambrosio, N. Gigli, A user’s guide to optimal transport, in Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics (2013), Springer Berlin Heidelberg, pp. 1–155

    Google Scholar 

  3. L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Spaces of Probability Measures. Lectures in Mathematics, ETH Zurich (Birkhäuser, Basel, 2005)

    MATH  Google Scholar 

  4. L. Ambrosio, M. Colombo, G. De Philippis, A. Figalli, Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case Commun. Part. Differ. Equat. 37(12), 2209–2227 (2012)

    Article  MATH  Google Scholar 

  5. L. Ambrosio, M. Colombo, G. De Philippis, A. Figalli, A global existence result for the semigeostrophic equations in three dimensional convex domains. Discr. Contin. Dyn. Syst. 34(4), 1251–1268 (2013)

    Article  Google Scholar 

  6. L. Ambrosio, N. Gigli, G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below Inv. Math. 195(2), 289–391 (2014)

    MATH  MathSciNet  Google Scholar 

  7. V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. (French) Ann. Inst. Fourier (Grenoble) 16(1), 319–361 (1996)

    Google Scholar 

  8. J.-D. Benamou, Y. Brenier, Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem. SIAM J. Appl. Math. 58, 1450–1461 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. J.-D. Benamou, Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. J.-D. Benamou, G. Carlier, Q. Mérigot, É. Oudet, Discretization of functionals involving the Monge-Ampère operator. (2014)

    Google Scholar 

  11. F. Bernardeau, S. Colombi, E. Gaztanaga, R. Scoccimarro, Large-Scale Structure of the Universe and Cosmological Perturbation Theory. Phys. Rep. 367, 1–248 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Bernot, A. Figalli, F. Santambrogio, Generalized solutions for the Euler equations in one and two dimensions. J. Math. Pures et Appl. 91(2), 137–155 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Blanchet, V. Calvez, J.A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model. SIAM J. Numer. Anal. 46(2), 691–721 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Blanchet, J.-A. Carrillo, D. Kinderlehrer, M. Kowalczyk, P. Laurençot, S. Lisini, A hybrid variational principle for the Keller-Segel system in \(\mathbb{R}^{2}\). ESAIM M2AN (2015).

    Google Scholar 

  15. N. Bonnotte, Unidimensional and evolution methods for optimal transportation. Ph.D. Thesis, Université Paris-Sud, 2013

    Google Scholar 

  16. G. Bouchitté, G. Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation. J. Eur. Math. Soc. 3(2), 139–168 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. G. Bouchitté, G. Buttazzo, P. Seppecher, Shape optimization solutions via Monge-Kantorovich equation. C. R. Acad. Sci. Paris Sér. I Math. 324(10), 1185–1191 (1997)

    Article  MATH  Google Scholar 

  18. Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Am. Mat. Soc. 2, 225–255 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  19. Y. Brenier, A modified least action principle allowing mass concentrations for the early universe reconstruction problem. Confluentes Mathematici 3(3), 361–385 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Y. Brenier, Rearrangement, convection, convexity and entropy. Philos. Trans. R. Soc. A 371, 20120343 (2013)

    Article  MathSciNet  Google Scholar 

  21. Y. Brenier, U. Frisch, M. Hénon, G. Loeper, S. Matarrese, R. Mohayaee, A. Sobolevskii, Reconstruction of the early Universe as a convex optimization problem. Mon. Not. R. Astron. Soc. 346, 501–524 (2003)

    Article  Google Scholar 

  22. G. Buttazzo, É. Oudet, E. Stepanov, Optimal transportation problems with free Dirichlet regions, in Variational Methods for Discontinuous Structures. PNLDE, vol. 51 (Birkhäuser, Basel, 2002), pp. 41–65

    Google Scholar 

  23. G. Buttazzo, C. Jimenez, É. Oudet, An optimization problem for mass transportation with congested dynamics. SIAM J. Control Optim. 48, 1961–1976 (2010)

    Article  MathSciNet  Google Scholar 

  24. P. Cardaliaguet, Notes on mean field games (from P.-L. Lions’ lectures at Collège de France). (2013) Available at https://www.ceremade.dauphine.fr/~cardalia/

  25. J.-A. Carrillo, R.J. McCann, C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Math. Iberoam. 19, 1–48 (2003)

    MathSciNet  Google Scholar 

  26. J.-A. Carrillo, R.J. McCann, C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Ann. 179, 217–263 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. J.-A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156(2), 229–271 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  28. M.J.P. Cullen, A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow (Imperial College Press, London, 2006)

    Book  Google Scholar 

  29. M. J. P. Cullen, W. Gangbo, A variational approach for the 2-dimensional semi-geostrophic shallow water equations. Arch. Ration. Mech. Ann. 156(3), 241–273 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. S. Daneri, G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math. Ann. 40, 1104–1122 (2008)

    Article  MATH  Google Scholar 

  31. E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications, ed. by C. Baiocchi, J.L. Lions (Masson, Paris, 1993), pp. 81–98

    Google Scholar 

  32. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics (American Mathematical Society, Providence, 2010)

    Google Scholar 

  33. A. Figalli, N. Gigli, A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions. J. Math. Pures et Appl. 94(2), 107–130 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. U. Frisch, S. Matarrese, R. Mohayaee, A. Sobolevski, A reconstruction of the initial conditions of the Universe by optimal mass transportation. Nature 417, 260–262 (2002)

    Article  Google Scholar 

  35. N. Gigli, K. Kuwada, S. Ohta, Heat flow on Alexandrov spaces. Commun. Pure Appl. Math. 66(3), 307–33 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  36. R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  37. E.F. Keller, L.A. Segel, Initiation of slide mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)

    Article  MATH  Google Scholar 

  38. E.F. Keller, L.A. Segel, Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)

    Article  MATH  Google Scholar 

  39. J.-M. Lasry, P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  40. J.-M. Lasry, P.-L. Lions, Mean-field games. Jpn. J. Math. 2, 229–260 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  41. G. Loeper, The reconstruction problem for the Euler-Poisson system in cosmology. Arch. Ration. Mech. Anal. 179(2), 153–216 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  42. B. Maury, J. Venel, Handling of contacts in crowd motion simulations. Traffic Granular Flow 07, 171–180 (2007)

    Google Scholar 

  43. B. Maury, J. Venel, A discrete contact model for crowd motion. ESAIM: M2AN 45(1), 145–168 (2011)

    Google Scholar 

  44. B. Maury, A. Roudneff-Chupin, F. Santambrogio, A macroscopic crowd motion model of gradient flow type. Math. Models Methods Appl. Sci. 20(10), 1787–1821 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  45. B. Maury, A. Roudneff-Chupin, F. Santambrogio, J. Venel, Handling congestion in crowd motion modeling. Net. Het. Media 6(3), 485–519 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  46. F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Commun. Part. Differ. Equat. 26, 101–174 (2011)

    Article  Google Scholar 

  47. A. Roudneff-Chupin, Modélisation macroscopique de mouvements de foule. Ph.D. Thesis, Université Paris-Sud (2011). Available at www.math.u-psud.fr/roudneff/Images/ these_roudneff.pdf

  48. F. Santambrogio, Flots de gradient dans les espaces métriques et leurs applications (d’après Ambrosio-Gigli-Savaré), in Proceedings of the Bourbaki Seminar, 2013 (in French)

    Google Scholar 

  49. A.I. Shnirelman, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid. (Russian) Mat. Sb. (N.S.) 128(170), 82–109 (1985)

    Google Scholar 

  50. A.I. Shnirelman, Generalized fluid flows, their approximation and applications. Geom. Funct. Anal. 4(5), 586–620 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  51. G.J. Shutts, M.J.P. Cullen, Parcel stability and its relation to semigeostrophic theory. J. Atmos. Sci. 44, 1318–1330 (1987)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Santambrogio, F. (2015). Gradient flows. In: Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications, vol 87. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20828-2_8

Download citation

Publish with us

Policies and ethics