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

• ##### Gautam Iyer
Nonlinearity (Impact Factor: 1.6). 03/2008; DOI: 10.1088/0951-7715/21/11/004
Source: arXiv

ABSTRACT 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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##### Article: A Stochastic Representation for Backward Incompressible Navier-Stokes Equations
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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.
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##### Article: Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations
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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.
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##### Article: Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities
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ABSTRACT: To filter perturbed local measurements on a random medium, a dynamic model jointly with an observation transfer equation are needed. Some media given by PDE could have a local probabilistic representation by a Lagrangian stochastic process with mean-field interactions. In this case, we define the acquisition process of locally homogeneous medium along a random path by a Lagrangian Markov process conditioned to be in a domain following the path and conditioned to the observations. The nonlinear filtering for the mobile signal is therefore those of an acquisition process contaminated by random errors. This will provide a Feynman-Kac distribution flow for the conditional laws and a N particle approximation with a O( 1 p N ) asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described. Resume. Pour filtrer les mesures locales et perturbees d'un milieu aleatoire, un modele de dynamique et uneequation de transfert sont necessaires. Certains milieux decrits par une EDP peuvent avoir une representation probabiliste locale l'aide d'un processus stochastique Lagrangien en interaction avec un champ moyen. Dans ce cas, nous definissons le processus d'acquisition d'un milieu localement homogene le long d'un chemin aleatoire par un processus de Markov Lagrangien conditionnea vivre dans un domaine suivant le chemin et conditionne aux observations. Le filtrage non-lineaire du signal local est alors celui du processus d'acquisition bruite par des erreurs aleatoires. Ceci nous donne un flot de distribution de Feynman-Kac et une convergence asymptotique de l'approximation particulaires avec une erreur en O( 1 p N ). Une application au filtrage non-lineaire de fluides atmospherique 3D turbulent sera decrite.
http://dx.doi.org/10.1051/m2an/2010047. 01/2010;