The Annals of Applied Probability

Stochastic vortex method for forced three-dimensional Navier–Stokes equations and pathwise convergence rate

J. Fontbona

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Abstract

We develop a McKean–Vlasov interpretation of Navier–Stokes equations with external force field in the whole space, by associating with local mild Lp-solutions of the 3d-vortex equation a generalized nonlinear diffusion with random space–time birth that probabilistically describes creation of rotation in the fluid due to nonconservativeness of the force. We establish a local well-posedness result for this process and a stochastic representation formula for the vorticity in terms of a vector-weighted version of its law after its birth instant. Then we introduce a stochastic system of 3d vortices with mollified interaction and random space–time births, and prove the propagation of chaos property, with the nonlinear process as limit, at an explicit pathwise convergence rate. Convergence rates for stochastic approximation schemes of the velocity and the vorticity fields are also obtained. We thus extend and refine previous results on the probabilistic interpretation and stochastic approximation methods for the nonforced equation, generalizing also a recently introduced random space–time-birth particle method for the 2d-Navier–Stokes equation with force.

Article information

Source
Ann. Appl. Probab., Volume 20, Number 5 (2010), 1761-1800.

Dates
First available in Project Euclid: 25 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1282747400

Digital Object Identifier
doi:10.1214/09-AAP672

Mathematical Reviews number (MathSciNet)
MR2724420

Zentralblatt MATH identifier
1222.60077

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 65C35: Stochastic particle methods [See also 82C80] 76M23: Vortex methods 76D17: Viscous vortex flows
Secondary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Keywords
3d-Navier–Stokes equation with external force McKean–Vlasov model with random space–time birth stochastic vortex method propagation of chaos convergence rate

Citation

Fontbona, J. Stochastic vortex method for forced three-dimensional Navier–Stokes equations and pathwise convergence rate. Ann. Appl. Probab. 20 (2010), no. 5, 1761--1800. doi:10.1214/09-AAP672. https://projecteuclid.org/euclid.aoap/1282747400


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