Journal of Statistical Physics

, Volume 174, Issue 3, pp 562–578 | Cite as

Mean Field Limit of Interacting Filaments for 3D Euler Equations

  • Hakima BessaihEmail author
  • Michele Coghi
  • Franco Flandoli


The 3D Euler equations, precisely local smooth solutions of class \(H^s\) with \(s>5/2\) are obtained as a mean field limit of finite families of interacting curves, the so called vortex filaments, described by means of the concept of 1-currents. This work is a continuation of a previous paper, where a preliminary result in this direction was obtained, with the true Euler equations replaced by a vector valued non linear PDE with a mollified Biot–Savart relation.


3D Euler equations Vortex filaments Currents Mean field theory 

Mathematics Subject Classification

Primary 35Q31 70F45 Secondary 37C10 76B47 49Q15 



We would like to thank the anonymous referees for their careful reading and valuable remarks which helped improving this paper. Hakima Bessaih’s research was partially supported by NSF Grant DMS-1418838. Franco Flandoli was partially supported by the PRIN 2015 project “Deterministic and stochastic evolution equations”.


  1. 1.
    Berselli, L.C., Bessaih, H.: Some results for the line vortex equation. Nonlinearity 15(6), 1729–1746 (2002)ADSMathSciNetzbMATHGoogle Scholar
  2. 2.
    Berselli, L.C., Gubinelli, M.: On the global evolution of vortex filaments, blobs, and small loops in 3D ideal flows. Commun. Math. Phys. 269(3), 693–713 (2007)ADSMathSciNetzbMATHGoogle Scholar
  3. 3.
    Bessaih, H., Flandoli, F.: Limit behaviour of a dense collection of vortex filaments. Math. Models Methods Appl. Sci. 14(2), 189–215 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bessaih, H., Wijeratne, C.: Fractional Brownian motion and an application to fluids. In: Stochastic Equations for Complex Systems: Theoretical and Computational Topics, pp. 37–52. Springer (2015)Google Scholar
  5. 5.
    Bessaih, H., Gubinelli, M., Russo, F.: The evolution of a random vortex filament. Ann. Probab. 33(5), 1825–1855 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bessaih, H., Coghi, M., Flandoli, F.: Mean field limit of interacting filaments and vector valued non-linear PDEs. J. Stat. Phys. 166(5), 1276–1309 (2017)ADSMathSciNetzbMATHGoogle Scholar
  7. 7.
    Bourguignon, J.P., Brezis, H.: Remarks on the Euler equations. J. Funct. Anal. 15, 341–363 (1974)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Brzezniak, Z., Gubinelli, M., Neklyudov, M.: Global solutions of the random vortex filament equation. Nonlinearity 26(9), 2499–2514 (2013)ADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Chorin, A.J.: The evolution of a turbulent vortex. Commun. Math. Phys. 83(4), 517–535 (1982)ADSMathSciNetzbMATHGoogle Scholar
  10. 10.
    Dobrushin, R.L.: Vlasov equation. Funct. Anal. Appl. 13, 115–123 (1979)zbMATHGoogle Scholar
  11. 11.
    Flandoli, F.: A probabilistic description of small scale structures in 3D fluids. Ann. Inst. Henri Poincaré, Probab. Stat. 38(2), 207–228 (2002)ADSMathSciNetzbMATHGoogle Scholar
  12. 12.
    Flandoli, F., Gubinelli, M.: The Gibbs ensembles of vortex filaments. Probab. Theory Relat. Fields 122(3), 317–340 (2002)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Giaquinta, M., Modica, G., Soucek, J.: Cartesian Currents in the Calculus of Variations I: Cartesian Currents. Springer, Berlin (1998)zbMATHGoogle Scholar
  14. 14.
    Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Krantz, S.G., Parks, H.R.: Geometric Integration Theory. Springer, Berlin (2008)zbMATHGoogle Scholar
  16. 16.
    Leonard, A.: Computing three-dimensional incompressible flows with vortex elements. Ann. Rev. Fluid Mech. 17, 523–559 (1985)ADSGoogle Scholar
  17. 17.
    Lions, P.L.: Mathematical Topics in Fluid Mechanics. Incompressible Models, vol. 1. Oxford Univ Press, Oxford (1996)Google Scholar
  18. 18.
    Lions, P.L.: On Euler equations and statistical physics, Cattedra Galileiana [Galileo Chair]. Scuola Normale Superiore, Classe di Scienze, Pisa (1998)Google Scholar
  19. 19.
    Lions, P.L., Majda, A.: Equilibrium statistical theory for nearly parallel vortex filaments. Commun. Pure Appl. Math. 53, 76–142 (2000)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Majda, A.J., Bertozzi, A.: Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)Google Scholar
  21. 21.
    Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Noviscous Fluids. Springer, Berlin (1994)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Hakima Bessaih
    • 1
    Email author
  • Michele Coghi
    • 2
  • Franco Flandoli
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of WyomingLaramieUnited States
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  3. 3.Scuola Normale SuperiorePisaItaly

Personalised recommendations