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

dc.contributor.author Iyer, Gautam
dc.contributor.author Mattingly, Jonathan
dc.date.accessioned 2020-08-29T16:35:04Z
dc.date.available 2020-08-29T16:35:04Z
dc.date.issued 2008-11-01
dc.identifier.issn 0951-7715
dc.identifier.issn 1361-6544
dc.identifier.uri https://hdl.handle.net/10161/21353
dc.description.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.
dc.language English
dc.publisher IOP Publishing
dc.relation.ispartof Nonlinearity
dc.relation.isversionof 10.1088/0951-7715/21/11/004
dc.subject Science & Technology
dc.subject Physical Sciences
dc.subject Mathematics, Applied
dc.subject Physics, Mathematical
dc.subject Mathematics
dc.subject Physics
dc.title A stochastic-Lagrangian particle system for the Navier-Stokes equations
dc.type Journal article
duke.contributor.id Mattingly, Jonathan|0297691
dc.date.updated 2020-08-29T16:35:03Z
pubs.begin-page 2537
pubs.end-page 2553
pubs.issue 11
pubs.organisational-group Trinity College of Arts & Sciences
pubs.organisational-group Mathematics
pubs.organisational-group Duke
pubs.publication-status Published
pubs.volume 21
duke.contributor.orcid Mattingly, Jonathan|0000-0002-1819-729X


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