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

TL;DR

This paper introduces a stochastic vortex method for the 3D Navier-Stokes equations with external forces, providing a probabilistic interpretation, a well-posedness result, and explicit convergence rates for stochastic approximations.

Contribution

It develops a McKean-Vlasov framework for forced 3D Navier-Stokes, establishing well-posedness, a stochastic representation, and propagation of chaos with explicit convergence rates.

Findings

Established local well-posedness for the stochastic vortex process.

Proved propagation of chaos with explicit pathwise convergence rate.

Derived convergence rates for stochastic schemes of velocity and vorticity fields.

Abstract

We develop a McKean-Vlasov interpretation of Navier-Stokes equations with external force field in the whole space, by associating with local mild $L^p$-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.