Quantitative approximation of the Vlasov(-Fokker-Planck)-Navier-Stokes system by stochastic particle systems

TL;DR

This paper establishes a convergence rate for stochastic particle systems approximating the coupled Vlasov(-Fokker-Planck)-Navier-Stokes equations in 2D and 3D, relevant for modeling aerosols and sprays.

Contribution

It provides the first quantitative convergence analysis of particle systems towards the coupled fluid-particle PDE system, including the case with vanishing noise.

Findings

Empirical measures of particles converge to the Vlasov(-Fokker-Planck) component.

Fluid velocity converges to the Navier-Stokes component.

Convergence rates are established as the number of particles N tends to infinity.

Abstract

This paper is concerned with a fluid-particle system given by the incompressible Navier-Stokes equations coupled with the Vlasov(-Fokker-Planck) equation through a drag force. Such a model arises naturally in the study of aerosols, sprays, and more generally two-phase flows. In dimensions $d\in \{2,3\}$, we establish a rate of convergence for a system of $N$ interacting stochastic particles coupled with a fluid, towards the Vlasov(-Fokker-Planck)-Navier-Stokes system, as $N\to \infty$. The case of particles with a noise that vanishes as $N\to \infty$ is considered and leads specifically to the Vlasov-Navier-Stokes system. More precisely, we prove that the empirical measure associated with the particle system converges to the Vlasov(-Fokker-Planck) component, while the fluid velocity converges to the Navier-Stokes component of the coupled system. The proofs combine stochastic calculus and PDE techniques to establish energy estimates and commutator estimates for both the discrete and continuous systems.