Resetting of a particle system for the Navier-Stokes equations

This work is based on a formulation of the incompressible Navier-Stokes equations developed by P. Constantin and G. Iyer, where the velocity field of a viscous incompressible uid is written as the expected value of a stochastic process. If we take N copies of the above process (each based on independent Wiener processes), and replace the expected value with the empirical mean, then it was shown that the particle system for the Navier-Stokes equations does not dissipate all its energy as time goes to infinity. This is in contrast to the true (unforced) Navier-Stokes equations, which dissipate all of their energy as time goes to infinity. The objective of this short note is to describe a resetting procedure that removes this deficiency. We prove that if we repeat this resetting procedure often enough, then the new particle system for the Navier-Stokes equations dissipates all its energy.