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

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.