A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations
Article first published online: 4 JUN 2007
Copyright © 2007 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Volume 61, Issue 3, pages 330–345, March 2008
How to Cite
Constantin, P. and Iyer, G. (2008), A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations. Comm. Pure Appl. Math., 61: 330–345. doi: 10.1002/cpa.20192
- Issue published online: 20 DEC 2007
- Article first published online: 4 JUN 2007
- Manuscript Received: APR 2006
- National Science Foundation Grant. Grant Number: DMS-0504213
In this paper we derive a probabilistic representation of the deterministic three-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 Lagrangian-averaged Navier-Stokes alpha models. © 2007 Wiley Periodicals, Inc.