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A stochastic-Lagrangian particle system for the Navier-Stokes equations

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Date
2008-11-01
Authors
Iyer, Gautam
Mattingly, Jonathan
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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 article
Subject
Science & Technology
Physical Sciences
Mathematics, Applied
Physics, Mathematical
Mathematics
Physics
Permalink
https://hdl.handle.net/10161/21353
Published Version (Please cite this version)
10.1088/0951-7715/21/11/004
Publication 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.
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Scholars@Duke

Mattingly

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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