Mattingly J.: A stochastic-Lagrangian particle system for the Navier-Stokes equations. Nolinearity 21, 2537-2553

Nonlinearity (Impact Factor: 1.21). 03/2008; 21(11). DOI: 10.1088/0951-7715/21/11/004
Source: arXiv


This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (\texttt{arxiv:math.PR/0511067}, to appear), 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 $\frac{1}{N}$ times the sum over these $N$ copies. (We remark 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 $\holderspace{1}{\alpha}$ which consists of differentiable functions whose first derivative is $\alpha$ H\"older continuous (see Section \ref{sGexist} for the precise definition). Further, we show that as $N \to \infty$ 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 $O(\frac{1}{N})$ times its original energy as $t \to \infty$. Hence the limit $N \to \infty$ and $T\to \infty$ 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 $t \to \infty$ explicitly. Comment: v3: Typo fixes, and a few stylistic changes. 17 pages, 2 figures

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Available from: Jonathan C. Mattingly, Dec 24, 2013
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    • "Lastly, we mention that other stochastic approaches for incompressible Navier-Stokes equations can be found in the references [16] [4] [5] [8] [9], etc.; and compared with the analytic arguments , one of the main advantages of representation (1.4) is that it is convenient for numerical simulations (cf. [17] [14]). This is in fact our main motivation for studying the stochastic representation of fractal Navier-Stokes equation ( "
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    ABSTRACT: In this article we study the fractal Navier-Stokes equations by using stochastic Lagrangian particle path approach in Constantin and Iyer \cite{Co-Iy}. More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by L\'evy processes. Basing on this representation, a self-contained proof for the existence of local unique solution for the fractal Navier-Stokes equation with initial data in $\mW^{1,p}$ is provided, and in the case of two dimensions or large viscosity, the existence of global solution is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for L\'evy processes with time dependent and discontinuous drifts is proved.
    Preview · Article · Mar 2011 · Communications in Mathematical Physics
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    • "Such a representation will be proved in Section 2. We emphasize that representations (3) and (5) are useful in numerical computations (cf. [19] [15]). Direct calculations shows that the second equation in (5) is equivalent to "
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    ABSTRACT: By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer's forward formulations in {Co-Iy}. Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for the incompressible Navier-Stokes equation in the whole space. In two dimensions, an alternative proof to global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero.
    Preview · Article · Nov 2008 · Probability Theory and Related Fields
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    • "Proposition 6.1 can be proved using a standard Picard's iteration. A proof of the analogous result for the Navier-Stokes equations appeared in the appendix of [11] (see also [9] [10]). The proof of 6.1 is very similar, and we do not provide it here. "
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    ABSTRACT: We study the dissipation mechanism of a stochastic particle system for the Burgers equation. The velocity field of the viscous Burgers and Navier-Stokes equations can be expressed as an expected value of a stochastic process based on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3 (2008) 330-345]. In this paper we study a particle system for the viscous Burgers equations using a Monte-Carlo version of the above; we consider N copies of the above stochastic flow, each driven by independent Wiener processes, and replace the expected value with $\frac{1}{N}$ times the sum over these copies. A similar construction for the Navier-Stokes equations was studied by Mattingly and the first author of this paper [Iyer and Mattingly Nonlinearity 21 (2008) 2537-2553]. Surprisingly, for any finite N, the particle system for the Burgers equations shocks almost surely in finite time. In contrast to the full expected value, the empirical mean $\frac{1}{N}\sum_1^N$ does not regularize the system enough to ensure a time global solution. To avoid these shocks, we consider a resetting procedure, which at first sight should have no regularizing effect at all. However, we prove that this procedure prevents the formation of shocks for any $N\geq2$, and consequently as $N\to\infty$ we get convergence to the solution of the viscous Burgers equation on long time intervals.
    Preview · Article · Nov 2008 · The Annals of Probability
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