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.
Publié le : 2000-11-14
Classification:
Two-dimensional Navier-Stokes equation,
vortex method,
interacting particle systems,
propogation of chaos,
60K35,
76D05
@article{1019487613,
author = {M\'el\'eard, Sylvie},
title = {A trajectorial proof of the vortex method for the two-dimensional
Navier-Stokes equation},
journal = {Ann. Appl. Probab.},
volume = {10},
number = {2},
year = {2000},
pages = { 1197-1211},
language = {en},
url = {http://dml.mathdoc.fr/item/1019487613}
}
Méléard, Sylvie. A trajectorial proof of the vortex method for the two-dimensional
Navier-Stokes equation. Ann. Appl. Probab., Tome 10 (2000) no. 2, pp. 1197-1211. http://gdmltest.u-ga.fr/item/1019487613/