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

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

2005-11-03

arxiv.org/abs/math/0511067v4

Comm. Pure Appl. Math. 61 (2008), no. 3, 330--345

Mathematics

Probability

v4: Minor corrections to bibliography, and final version that will apear in CPAM. v3: Minor corrections to the algebra in the

Scientific paper

10.1002/cpa.20192

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.

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