Skip to main content
SpringerLink
Log in
Book cover

From Particle Systems to Partial Differential Equations III pp 185–194Cite as

Symmetries and Martingales in a Stochastic Model for the Navier-Stokes Equation

  • Conference paper
  • First Online:

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 162))

Abstract

A stochastic description of solutions of the Navier-Stokes equation is investigated. These solutions are represented by laws of finite dimensional semi-martingales and characterized by a weak Euler-Lagrange condition. A least action principle, related to the relative entropy, is provided. Within this stochastic framework, by assuming further symmetries, the corresponding invariances are expressed by martingales, stemming from a weak Noether’s theorem.

This is a preview of subscription content, 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   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
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

References

  1. Bismut, J.-M.: Mécanique aléatoire. Lecture Notes in Mathematics. Springer, New York (1981)

    Book  MATH  Google Scholar 

  2. Bossy, M., Jabir, J.F., Talay, D.: On conditional McKean Lagrangian stochastic models. Probab. Theory Relat. Fields 151, 319–351 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cruzeiro, A.B., Cipriano, F.: Navier-Stokes equation and diffusions on the group of homeomorphisms of the torus. Commun. Math. Phys. 275, 255–269 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cruzeiro, A.B., Lassalle, R.: On the least action principle for the Navier-Stokes equation. In: Stochastic Analysis and Applications 2014. Springer Proceedings in Mathematics and Statistics, vol. 100. Springer (2014)

    Google Scholar 

  5. Constantin, P.: An Eulerian-Lagrangian approach to the Navier-Stokes equations. Commun. Math. Phys. 216(3), 663–686 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Constantin, P., Iyer, G.: A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations. Commun. Pure Appl. Math. Phys. 61(3), 330–345 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cruzeiro, A.B., Lassalle, R.: Weak calculus of variations for functionals of laws of semi-martingales (2015)

    Google Scholar 

  8. Cruzeiro, A.B., Shamarova, E.: Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus. Stoch. Process. Appl. 119(12), 4034–4060 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gomes, D.A.: A variational formulation for the Navier-Stokes equation. Commun. Math. Phys. 257(1), 227–234 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North Holland, Amsterdam (Kodansha Ltd., Tokyo) (1981)

    Google Scholar 

  11. Inoue, A., Funaki, T.: A new derivation of the Navier-Stokes equation. Commun. Math. Phys. 65, 83–90 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Föllmer, H.: Random fields and diffusion processes. In: Ecole d’ Été de Saint Flour XV–XVII (1988)

    Google Scholar 

  13. Lassalle, R., Zambrini, J.C.: A weak approach to the stochastic deformation of classical mechanics. J. Geom. Mech. (2016)

    Google Scholar 

  14. Leonard, C.: A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Continu. Dyn. Syst. A 34(4), 1533 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mikulevicius, R., Rozovskii, B.L.: Stochastic Navier-Stokes equations for turbulent flows. SIAM J. Math. Anal. 35(5), 1250–1310 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mikulevicius, R., Rozovskii, B.L.: On equations of stochastic fluid mechanics. In: Hida, T., Karandikar, R., Kunita, H., Rajput, P., Watanabe, S., Xiong, J. (eds.) Stochastics in Finite and Infinite Dimensions, pp. 285–302. Birkhauser Boston, Boston, MA, (2001)

    Google Scholar 

  17. Nakagomi, T., Yasue, K., Zambrini, J.C.: Stochastic variational derivations of the Navier-Stokes equation. Lett. Math. Phys. 5(6), 545–552 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tan, X., Touzi, N.: Optimal transportation under controlled stochastic dynamics. Ann. Prob. (2012)

    Google Scholar 

  19. Thieullen, M., Zambrini, J.C.: Probability and quantum symmetries I, the theorem of Noether in Schrödinger’s Euclidean quantum mechanics. Ann. Inst. H.Poincaré, Phys. Theo. 67(3), 297 (1997)

    Google Scholar 

  20. Yasue, K.: A variational principle for the Navier-Stokes equation. J. Funct. Anal. 51(2), 133–141 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zambrini, J.C.: Stochastic mechanics according to E. Schrödinger. Phys. Rev. A. 33(3), 1532–1548 (1986)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75016, Paris, France

    Rémi Lassalle

  2. GFMUL and Departamento Matemática Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Ana Bela Cruzeiro

Authors
  1. Rémi Lassalle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ana Bela Cruzeiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ana Bela Cruzeiro .

Editor information

Editors and Affiliations

  1. University of Minho Department of Mathematics, Braga, Portugal

    Patrícia Gonçalves

  2. University of Minho Department of Mathematics, Braga, Portugal

    Ana Jacinta Soares

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Lassalle, R., Cruzeiro, A.B. (2016). Symmetries and Martingales in a Stochastic Model for the Navier-Stokes Equation. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations III. Springer Proceedings in Mathematics & Statistics, vol 162. Springer, Cham. https://doi.org/10.1007/978-3-319-32144-8_9

Download citation

Publish with us

Policies and ethics

Search

Navigation