Article

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

Nonlinearity (Impact Factor: 1.6). 03/2008; DOI: 10.1088/0951-7715/21/11/004

Source: arXiv

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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.ESAIM Mathematical Modelling and Numerical Analysis 09/2010; · 1.03 Impact Factor - [Show abstract] [Hide abstract]

**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.Communications in Mathematical Physics 03/2011; · 1.97 Impact Factor - [Show abstract] [Hide abstract]

**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.The Annals of Probability 11/2008; · 1.38 Impact Factor

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