A stochastic-Lagrangian particle system for the Navier-Stokes equations
Abstract
This paper is based on a formulation of the Navier-Stokes equations developed by Constantin
and the first author (Commun. Pure Appl. Math. at press, arXiv:math.PR/0511067), 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 note that our formulation requires one to keep track
of N stochastic flows of diffeomorphisms, and not just the motion of N particles.)
We prove that in two dimensions, this system of interacting diffeomorphisms has (time)
global solutions with initial data in the space C1,α which consists of differentiable
functions whose first derivative is α Hölder continuous (see section 3 for the precise
definition). 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 its original energy as t → ∞. Hence
the limit N → ∞ and T → ∞ do not commute. 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. © 2008 IOP Publishing Ltd and London Mathematical Society.
Type
Journal articleSubject
Science & TechnologyPhysical Sciences
Mathematics, Applied
Physics, Mathematical
Mathematics
Physics
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https://hdl.handle.net/10161/21353Published Version (Please cite this version)
10.1088/0951-7715/21/11/004Publication Info
Iyer, Gautam; & Mattingly, Jonathan (2008). A stochastic-Lagrangian particle system for the Navier-Stokes equations. Nonlinearity, 21(11). pp. 2537-2553. 10.1088/0951-7715/21/11/004. Retrieved from https://hdl.handle.net/10161/21353.This is constructed from limited available data and may be imprecise. To cite this
article, please review & use the official citation provided by the journal.
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Show full item recordScholars@Duke
Jonathan Christopher Mattingly
Kimberly J. Jenkins Distinguished University Professor of New Technologies
Jonathan Christopher Mattingly grew up in Charlotte, NC where he attended Irwin Ave
elementary and Charlotte Country Day. He graduated from the NC School of Science
and Mathematics and received a BS is Applied Mathematics with a concentration in physics
from Yale University. After two years abroad with a year spent at ENS Lyon studying
nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to
attend Princeton University where he obtained a PhD in Applied and

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