The Annals of Applied Probability

A trajectorial proof of the vortex method for the two-dimensional Navier-Stokes equation

Sylvie Méléard

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Abstract

We consider the Navier–Stokes equation in dimension 2 and more precisely the vortex equation satisfied by the curl of the velocity field.We show the relation between this equation and a nonlinear stochastic differential equation. Next we use this probabilistic interpretation to construct approximating interacting particle systems which satisfy a propagation of chaos property: the laws of the empirical measures tend, as the number of particles tends to $\infty$, to a deterministic law for which marginals are solutions of the vortex equation.This pathwise result justifies completely the vortex method introduced by Chorin to simulate the solutions of the vortex equation.Our approach is inspired by Marchioro and Pulvirenti and we improve their results in a pathwise sense.

Article information

Source
Ann. Appl. Probab. Volume 10, Number 4 (2000), 1197-1211.

Dates
First available in Project Euclid: 22 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1019487613

Mathematical Reviews number (MathSciNet)
MR1810871

Digital Object Identifier
doi:10.1214/aoap/1019487613

Zentralblatt MATH identifier
01906259

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 76D05: Navier-Stokes equations [See also 35Q30]

Keywords
Two-dimensional Navier-Stokes equation vortex method interacting particle systems propogation of chaos

Citation

Méléard, Sylvie. A trajectorial proof of the vortex method for the two-dimensional Navier-Stokes equation. Ann. Appl. Probab. 10 (2000), no. 4, 1197--1211. doi:10.1214/aoap/1019487613. http://projecteuclid.org/euclid.aoap/1019487613.


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References

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