, Volume 121, Issue 3, pp 367-388

Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data

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Abstract.

We are interested in proving Monte-Carlo approximations for 2d Navier-Stokes equations with initial data u 0 belonging to the Lorentz space L 2,∞ and such that curl u 0 is a finite measure. Giga, Miyakawaand Osada [7] proved that a solution u exists and that u=K* curl u, where K is the Biot-Savartkernel and v = 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 equationby a stochastic interacting particlesystem and deduce a Monte-Carlo approximation for a solution of the Navier-Stokesequation. That gives in this case a pathwise proofof the vortex algorithm introducedby Chorin and consequently generalizes the works ofMarchioro-Pulvirenti [12] and Méléardv [15] obtained in the case of a vortex equation with bounded density initial data.

Received: 6 October 1999 / Revised version: 15 September 2000 / Published online: 9 October 2001