Skip to main content
SpringerLink
Log in

Optimal Transport with Discrete Mean Field Interaction

  • Chapter
  • First Online:
Book cover 2017 MATRIX Annals

Part of the book series: MATRIX Book Series ((MXBS,volume 2))

  • 1144 Accesses

Abstract

In this note, we summarise some regularity results recently obtained for an optimal transport problem where the matter transported is either accelerated by an external force field, or self-interacting, at a given intermediate time.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 159.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    Article  MathSciNet  Google Scholar 

  2. Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44, 375–417 (1991)

    Article  MathSciNet  Google Scholar 

  3. Lee, P., McCann, R.J.: The Ma-Trudinger-Wang curvature for natural mechanical actions. Calc. Var. PDEs 41, 285–299 (2011)

    Article  MathSciNet  Google Scholar 

  4. Liu, J.: Hölder regularity of optimal mappings in optimal transportation. Calc. Var. PDEs 34, 435–451 (2009)

    Article  Google Scholar 

  5. Liu, J., Loeper, G.: Optimal transport with discrete long range mean field interactions (submitted). Available at arXiv:1809.07432

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Loeper, G.: On the regularity of solutions of optimal transportation problems. Acta Math. 202, 241–283 (2009)

    Article  MathSciNet  Google Scholar 

  8. Ma, X.-N., Trudinger, N.S., Wang, X.-J.: Regularity of potential functions of the optimal transportation problems. Arch. Ration. Mech. Anal. 177, 151–183 (2005)

    Article  MathSciNet  Google Scholar 

  9. McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997)

    Article  MathSciNet  Google Scholar 

  10. Trudinger, N.S., Wang, X.-J.: On the second boundary value problem for Monge-Ampère type equations and optimal transportation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8, 143–174 (2009)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, Australia

    Jiakun Liu

  2. School of Mathematical Sciences, Monash University, Melbourne, VIC, Australia

    Grégoire Loeper

Authors
  1. Jiakun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Grégoire Loeper
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiakun Liu .

Editor information

Editors and Affiliations

  1. The University of Melbourne, Melbourne, Australia

    Jan de Gier

  2. University of Western Australia, Perth, Australia

    Cheryl E. Praeger

  3. UCLA, Los Angeles, CA, USA

    Terence Tao

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Liu, J., Loeper, G. (2019). Optimal Transport with Discrete Mean Field Interaction. In: de Gier, J., Praeger, C., Tao, T. (eds) 2017 MATRIX Annals. MATRIX Book Series, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-04161-8_15

Download citation

Publish with us

Policies and ethics

Search

Navigation