Long time behaviour of a stochastic-Lagrangian particle system for the Navier-Stokes equations

Abstract. This paper is based on a formulation of the Navier-Stokes equations developed in arxiv:math.PR/0511067 (to appear in Commun. Pure Appl. Math), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take N copies of the above process (each based on independent Wiener processes), and replace the expected value with 1 N times the sum over these N copies. We prove that in two dimensions, this system has (time) global solutions with C initial data. Further, we show that as N → ∞ the system converges to the solution of Navier-Stokes equations on any finite interval [0, T ]. However for fixed N , we prove that this system retains roughly O( 1 N ) times it’s original energy t → ∞. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as t → ∞ explicitly.

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