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 the convergence of Monte-Carlo approximations for
vortex equations in bounded domains of $R^2$ with Neumann's condition on the
boundary. This work is the first step to justify theorically some
numerical algorithms for Navier-Stokes equations in bounded domains
with no-slip conditions.
We prove that the vortex equation has a unique solution in an
appropriate space and
can be interpreted in a
probabilistic point of view through a nonlinear reflected process with
space-time random
births on the boundary of the domain.
Next, we approximate the solution $w$ of this vortex equation
by the weighted empirical measure of interacting diffusive particles with normal reflecting boundary
conditions and
space-time random births on the boundary. The weights are related to
the initial data and to the Neumann condition. We can deduce from
this result a simple stochastic particle
algorithm to approximate $w$.

**Mots Clés:** *Vortex equation on a bounded domain ; Monte-Carlo approximation ;
Interacting particle systems with reflection ; space-time random births
*

**Date:** 2002-06-27

**Prépublication numéro:** *PMA-742*

**Pdf file : **PMA-742.pdf