Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berselli, L.C., Bessaih, H.: Some results for the line vortex equation. Nonlinearity 15(6), 1729–1746 (2002)
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)
Bessaih, H., Flandoli, F.: Limit behaviour of a dense collection of vortex filaments. Math. Models Methods Appl. Sci. 14(2), 189–215 (2004)
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)
Bessaih, H., Gubinelli, M., Russo, F.: The evolution of a random vortex filament. Ann. Probab. 33(5), 1825–1855 (2005)
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)
Bourguignon, J.P., Brezis, H.: Remarks on the Euler equations. J. Funct. Anal. 15, 341–363 (1974)
Brzezniak, Z., Gubinelli, M., Neklyudov, M.: Global solutions of the random vortex filament equation. Nonlinearity 26(9), 2499–2514 (2013)
Chorin, A.J.: The evolution of a turbulent vortex. Commun. Math. Phys. 83(4), 517–535 (1982)
Dobrushin, R.L.: Vlasov equation. Funct. Anal. Appl. 13, 115–123 (1979)
Flandoli, F.: A probabilistic description of small scale structures in 3D fluids. Ann. Inst. Henri Poincaré, Probab. Stat. 38(2), 207–228 (2002)
Flandoli, F., Gubinelli, M.: The Gibbs ensembles of vortex filaments. Probab. Theory Relat. Fields 122(3), 317–340 (2002)
Giaquinta, M., Modica, G., Soucek, J.: Cartesian Currents in the Calculus of Variations I: Cartesian Currents. Springer, Berlin (1998)
Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)
Krantz, S.G., Parks, H.R.: Geometric Integration Theory. Springer, Berlin (2008)
Leonard, A.: Computing three-dimensional incompressible flows with vortex elements. Ann. Rev. Fluid Mech. 17, 523–559 (1985)
Lions, P.L.: Mathematical Topics in Fluid Mechanics. Incompressible Models, vol. 1. Oxford Univ Press, Oxford (1996)
Lions, P.L.: On Euler equations and statistical physics, Cattedra Galileiana [Galileo Chair]. Scuola Normale Superiore, Classe di Scienze, Pisa (1998)
Lions, P.L., Majda, A.: Equilibrium statistical theory for nearly parallel vortex filaments. Commun. Pure Appl. Math. 53, 76–142 (2000)
Majda, A.J., Bertozzi, A.: Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)
Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Noviscous Fluids. Springer, Berlin (1994)
Acknowledgements
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”.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bessaih, H., Coghi, M. & Flandoli, F. Mean Field Limit of Interacting Filaments for 3D Euler Equations. J Stat Phys 174, 562–578 (2019). https://doi.org/10.1007/s10955-018-2189-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-018-2189-4