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We develop a McKean-Vlasov interpretation of Navier-Stokes equations withexternal force field in the whole space, by associating with local mild$L^p$-solutions of the 3d-vortex equation a generalized nonlinear diffusionwith random space-time birth that probabilistically describes creation ofrotation in the fluid due to nonconservativeness of the force. We establish alocal well-posedness result for this process and a stochastic representationformula for the vorticity in terms of a vector-weighted version of its lawafter its birth instant. Then we introduce a stochastic system of 3d vorticeswith mollified interaction and random space-time births, and prove thepropagation of chaos property, with the nonlinear process as limit, at anexplicit pathwise convergence rate. Convergence rates for stochasticapproximation schemes of the velocity and the vorticity fields are alsoobtained. We thus extend and refine previous results on the probabilisticinterpretation and stochastic approximation methods for the nonforced equation,generalizing also a recently introduced random space-time-birth particle methodfor the 2d-Navier-Stokes equation with force.

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