Abstract
It is shown that the incompressible Navier–Stokes equation can be derived from an infinite-dimensional mean-field stochastic differential equation.
Similar content being viewed by others
Notes
The input of one of the referees concerning this point is gratefully acknowledged.
It is gratefully acknowledged that one of the referees pointed out Constantin and Iyer (2008) (which was unknown to the author).
References
Ahmed, N.U., Ding, X.: A semilinear McKean–Vlasov stochastic evolution equation in Hilbert space. Stoch. Process. Appl. 60, 65–85 (1995)
Albeverio, S., Belopolskaya, Ya.: Generalized solutions of the Cauchy problem for the Navier–Stokes system and diffusion processes. Cubo 12(2), 77–96 (2010)
Chorin, A., Hughes, T., McCracken, M., Marsden, J.: Product formulas and numerical algorithms. Commun. Pure Appl. Math. 31, 205–256 (1978)
Chorin, A., Marsden, J.: Mathematical Introduction to Fluid Mechanics. Springer, New York (1993)
Cipriano, F., Cruzeiro, A.B.: Navier–Stokes equation and diffusions on the group of homeomorphisms of the torus. Commun. Math. Phys. 275(1), 255–269 (2007)
Constantin, P.: An Eulerian–Lagrangian approach for incompressible fluids: local theory. J. Am. Math. Soc. 14(2), 263–278 (2000)
Constantin, P., Iyer, G.: A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations. Commun. Pure Appl. Math. 61(3), 330–345 (2008)
Crisan, D., Flandoli, F., Holm, D.D.: Solution properties of a 3D stochastic Euler fluid equation. arXiv:1704.06989
Cruzeiro, A.B.: Hydrodynamics, probability and the geometry of the diffeomorphism group. Seminar on Stochastic Analysis, Random Fields and Applications VI pp. 83–93 (2011)
Cruzeiro, A.B., Shamarova, E.: Navier–Stokes equations and forwardbackward SDEs on the group of diffeomorphisms of a torus. Stoch. Process. Appl. 119(12), 4034–4060 (2009)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, 2nd edn. CUP, Cambridge (2014)
Del Moral, P.: Mean-field simulation for Monte-Carlo integration. Chapman Hall, London (2016)
Eyink, G.L.: Stochastic least-action principle for the incompressible Navier–Stokes equation. Phys. D 239(14), 1236–1240 (2010)
Eyink, G.L., Drivas, T.D.: A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part I. Flows with no bounding walls. J. Fluid Mech. 829, 153–189 (2017)
Ebin, D., Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92(1), 1037–1041 (1970)
Glikhlikh, Y.: Global and Stochastic Analysis with Applications to Mathematical Physics. Springer TMP, New York (2011)
Govindan, T.: Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications. Springer, New York (2016)
Iyer, G.: A stochastic Lagrangian formulation of the Navier–Stokes and related transport equations. Ph.D. Thesis, University of Chicago (2006)
Iyer, G.: A stochastic perturbation of inviscid flows. Commun. Math. Phys. 266(3), 631–645 (2006)
Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. CUP, Cambridge (2003)
Marsden, J.: On product formulas for nonlinear semigroups. J. Funct. Anal. 13(1), 51–72 (1973)
Marsden, J., Ebin, D., Fischer, A.: Diffeomorphism groups, hydrodynamics and relativity. In: Vanstone J. (ed.) Proceedings of the 13th Biennial Seminar of Canadian Mathematical Congress, pp. 135–279 (1972)
Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, New York (2005)
Rezakhanlou, F.: Stochastically symplectic maps and their applications to the Navier–Stokes equation. Annales de l’Institut Henri Poincare (C) Non Linear Anal. 33(1), 1–22 (2016)
Yasue, K.: A variational priciple for the Navier–Stokes equation. J. Funct. Anal. 51(2), 133–141 (1983)
Acknowledgements
The paper has benefited a lot from the two very insightful referee reports. Also I want to thank my colleagues at the FMA for discussing fluid mechanics over lunch time.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Charles R. Doering.
Rights and permissions
About this article
Cite this article
Hochgerner, S. Stochastic Mean-Field Approach to Fluid Dynamics. J Nonlinear Sci 28, 725–737 (2018). https://doi.org/10.1007/s00332-017-9425-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-017-9425-y