Navier-Stokes Equation and Diffusions on the Group of Homeomorphisms of the Torus

Cruzeiro, A. B.; Cipriano, F.

doi:10.1007/s00220-007-0306-3

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Navier-Stokes Equation and Diffusions on the Group of Homeomorphisms of the Torus

Author(s): F. Cipriano , A. B. Cruzeiro

Publication date (Print): July 25 2007

Journal: Communications in Mathematical Physics

Publisher: Springer Nature

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Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits

Vladimir Arnold (1966)

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A stochastic Lagrangian representation of the 3-dimensional incompressible Navier-Stokes equations

Peter Constantin, Gautam Iyer (2005)

In this paper we derive a representation of the deterministic 3-dimensional Navier-Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for the Euler equations of ideal fluids is used to recover the velocity field. This method admits a self-contained proof of local existence for the nonlinear stochastic system, and can be extended to formulate stochastic representations of related hydrodynamic-type equations, including viscous Burgers equations and LANS-alpha models.