# A stochastic-Lagrangian particle system for the Navier-Stokes equations

Published

Journal Article

This paper is based on a formulation of the Navier-Stokes equations developed by Constantin and the first author (Commun. Pure Appl. Math. at press, arXiv:math.PR/0511067), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take N copies of the above process (each based on independent Wiener processes), and replace the expected value with 1/N times the sum over these N copies. (We note that our formulation requires one to keep track of N stochastic flows of diffeomorphisms, and not just the motion of N particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space C1,α which consists of differentiable functions whose first derivative is α Hölder continuous (see section 3 for the precise definition). Further, we show that as N → ∞ the system converges to the solution of Navier-Stokes equations on any finite interval [0, T]. However for fixed N, we prove that this system retains roughly O(1/N) times its original energy as t → ∞. Hence the limit N → ∞ and T → ∞ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as t → ∞ explicitly. © 2008 IOP Publishing Ltd and London Mathematical Society.

### Full Text

### Duke Authors

### Cited Authors

- Iyer, G; Mattingly, J

### Published Date

- November 1, 2008

### Published In

### Volume / Issue

- 21 / 11

### Start / End Page

- 2537 - 2553

### Electronic International Standard Serial Number (EISSN)

- 1361-6544

### International Standard Serial Number (ISSN)

- 0951-7715

### Digital Object Identifier (DOI)

- 10.1088/0951-7715/21/11/004

### Citation Source

- Scopus