Math @ Duke

Publications [#243867] of Jonathan C. Mattingly
Papers Published
 with Iyer, G; Mattingly, J, A stochasticLagrangian particle system for the NavierStokes equations,
Nonlinearity, vol. 21 no. 11
(2008),
pp. 25372553, ISSN 09517715 [MR2448230 (2009h:76060)], [arXiv:0803.1222], [doi]
(last updated on 2018/07/20)
Abstract: This paper is based on a formulation of the NavierStokes 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 NavierStokes 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.


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