Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60K35 Interacting random processes; statistical mechanics type models; percolation theory, See also {82B43, 82C43}
- 76D05 Navier-Stokes equations, See also {35Q30}

**Résumé:** We are interested in proving stochastic approximations for
a 2d Navier-Stokes equation with an initial data $u_0$ belonging to the
Lorentz space
$L^{2,\infty}$ and such that curl $u_0$ is a finite measure. Giga, Miyakawa
and Osada \cite{Giga:88} proved that
a solution $u$ exists and that $u=K*{\rm curl}\ u$,
where $K$ is the Biot-Savart
kernel and $v={\rm curl}\ u$ is solution of a nonlinear equation in
dimension one, called the
vortex equation.
In this paper, we approximate a solution $v$ of this vortex equation
by a stochastic interacting particle
system and deduce a Monte-Carlo approximation for a solution of the
Navier-Stokes
equation. That gives in this case a pathwise proof
of the vortex algorithm introduced
by Chorin and consequently generalizes the works of
Marchioro-Pulvirenti
\cite{Marchioro:82} and M\'el\'eard
\cite{Meleard:98} obtained in the case of an initial measure with bounded
density.

**Mots Clés:** *Vortex equation ; Monte-Carlo approximation ; Probabilistic interpretation*

**Date:** 1999-07-20

**Prépublication numéro:** *PMA-520*