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

  • Rémi Lassalle
  • Ana Bela Cruzeiro
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (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.

Keywords

Stochastic analysis Stochastic control Navier-Stokes equation 

Mathematics Subject Classification:

93E20 60H30 

References

  1. 1.
    Bismut, J.-M.: Mécanique aléatoire. Lecture Notes in Mathematics. Springer, New York (1981)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bossy, M., Jabir, J.F., Talay, D.: On conditional McKean Lagrangian stochastic models. Probab. Theory Relat. Fields 151, 319–351 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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. 5.
    Constantin, P.: An Eulerian-Lagrangian approach to the Navier-Stokes equations. Commun. Math. Phys. 216(3), 663–686 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cruzeiro, A.B., Lassalle, R.: Weak calculus of variations for functionals of laws of semi-martingales (2015)Google Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gomes, D.A.: A variational formulation for the Navier-Stokes equation. Commun. Math. Phys. 257(1), 227–234 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North Holland, Amsterdam (Kodansha Ltd., Tokyo) (1981)Google Scholar
  11. 11.
    Inoue, A., Funaki, T.: A new derivation of the Navier-Stokes equation. Commun. Math. Phys. 65, 83–90 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Föllmer, H.: Random fields and diffusion processes. In: Ecole d’ Été de Saint Flour XV–XVII (1988)Google Scholar
  13. 13.
    Lassalle, R., Zambrini, J.C.: A weak approach to the stochastic deformation of classical mechanics. J. Geom. Mech. (2016)Google Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mikulevicius, R., Rozovskii, B.L.: Stochastic Navier-Stokes equations for turbulent flows. SIAM J. Math. Anal. 35(5), 1250–1310 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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. 17.
    Nakagomi, T., Yasue, K., Zambrini, J.C.: Stochastic variational derivations of the Navier-Stokes equation. Lett. Math. Phys. 5(6), 545–552 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tan, X., Touzi, N.: Optimal transportation under controlled stochastic dynamics. Ann. Prob. (2012)Google Scholar
  19. 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. 20.
    Yasue, K.: A variational principle for the Navier-Stokes equation. J. Funct. Anal. 51(2), 133–141 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zambrini, J.C.: Stochastic mechanics according to E. Schrödinger. Phys. Rev. A. 33(3), 1532–1548 (1986)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.CEREMADE, Université Paris-DauphineParisFrance
  2. 2.GFMUL and Departamento Matemática Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal

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