Communications in Mathematical Sciences

Volume 16 (2018)

Number 3

Resetting of a particle system for the Navier–Stokes equations

Pages: 879 – 886

(Fast Communication)

DOI: http://dx.doi.org/10.4310/CMS.2018.v16.n3.a13

Authors

Alexei Novikov (Department of Mathematics, Pennsylvania State University, University Park, Pa., U.S.A.)

Karim Shikh-Khalil (Pennsylvania State University, University Park, Pa., U.S.A.)

Abstract

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 fluid 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.

Keywords

stochastic Lagrangian formulation, particle system, resetting

2010 Mathematics Subject Classification

60F10, 60J75, 62P10, 92C37

Full Text (PDF format)

Received 5 September 2017

Received revised 28 January 2018

Accepted 28 January 2018