Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data
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- Méléard, S. Probab Theory Relat Fields (2001) 121: 367. doi:10.1007/s004400100154
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We are interested in proving Monte-Carlo approximations for 2d Navier-Stokes equations with initial data u0 belonging to the Lorentz space L2,∞ and such that curl u0 is a finite measure. Giga, Miyakawaand Osada  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  and Méléardv  obtained in the case of a vortex equation with bounded density initial data.