TY - JOUR

T1 - Resetting of a particle system for the Navier-Stokes equations

AU - Novikov, Alexei

AU - Shikh-Khalil, Karim

N1 - Funding Information:
Acknowledgements. The first author was partially supported by the NSF grant DMS-1515187. The authors would like to thank the reviewer for the valuable comments and remarks.
Funding Information:
The first author was partially supported by the NSF grant DMS-1515187. The authors would like to thank the reviewer for the valuable comments and remarks.
Publisher Copyright:
© 2018 International Press.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

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U2 - 10.4310/CMS.2018.v16.n3.a12

DO - 10.4310/CMS.2018.v16.n3.a12

M3 - Article

AN - SCOPUS:85053269032

SN - 1539-6746

VL - 16

SP - 879

EP - 886

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

IS - 3

ER -