Abstract
A stochastic variational principle for the (two dimensional) Navier-Stokes equation is established. The velocity field can be considered as a generalized velocity of a diffusion process with values on the volume preserving diffeomorphism group of the underlying manifold. Navier-Stokes equation is reinterpreted as a perturbed equation of geodesics for the L 2 norm. The method described here should hold as well in higher dimensions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold V.I. (1966). Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications a l’hidrodynamique des fluides parfaits. Ann. Inst. Fourier 16: 316–361
Arnold V.I. and Khesin B.A. (1998). Topological Methods in Hydrodynamics. Springer-Verlag, New York
Constantin P. (2001). An Eulerian-Lagrangian approach to the Navier-Stokes equations. Commun. Math. Phys. 216(3): 663–686
Constantin, P., Iyer, G.: A Stochastic Lagrangian Representation of 3-Dimensional Incompressible Navier-Stokes Equations. Comm. Pure Appl. Math., in press, DOI 10.1002/cpa.20192, 2007
Fang S. (2002). Canonical Brownian motion on the diffeomorphism group of the circle. J. Funct. Anal. 196: 162–179
Fang S. (2004). Solving s.d.e.’s on Homeo S1. J. Funct. Anal. 216: 22–46
Gawedzki, K.: Simple models of turbulent transport. XIV Intern. Congress on Mathem. Physics, RiverEdge, NJ: World Scientific, 2005
Gomes D.A. (2005). A variational formulation for the Navier-Stokes equation. Commun. Math. Phys. 257(1): 227–234
Inoue A. and Funaki T. (1979). A new derivation of the Navier-Stokes equation. Commun. Math. Phys. 65: 83–90
Kunita H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge
Malliavin, P.: The canonic diffusion above the diffeomorphism group of the circle. C. R. Acad. Sci. Paris, t. 329, Série I, 325–329 (1999)
Nakagomi T., Yasue K. and Zambrini J.C. (1981). Stochastic variational derivations of the Navier-Stokes equation. Lett. Math. Phys. 160: 337–365
Shkoller S. (1998). Geometry and curvature of diffeomorphism group with H1 metric and mean hydrodynamics. J. Funct. Anal. 160(1): 337–365
Üstünel, A.S.: Stochastic analysis on Lie groups. Stoch. Anal. and Rel. Topics VI, Progr. Probab. 42, Boston: Birkhäuser 1998
Yasue K. (1983). A variational principle for the Navier-Stokes equation. J. Funct. Anal. 51(2): 133–141
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Constantin.
Rights and permissions
About this article
Cite this article
Cipriano, F., Cruzeiro, A.B. Navier-Stokes Equation and Diffusions on the Group of Homeomorphisms of the Torus. Commun. Math. Phys. 275, 255–269 (2007). https://doi.org/10.1007/s00220-007-0306-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-007-0306-3