Derivation of the Monokinetic Vlasov-Stokes equations

TL;DR

This paper derives the monokinetic Vlasov-Stokes equations from a microscopic particle model with inertia, establishing a non-perturbative mean-field limit without requiring vanishing inertia, for particles with initial velocities following a Lipschitz profile.

Contribution

It provides a non-perturbative derivation of the monokinetic Vlasov-Stokes equations from particle dynamics with inertia, extending previous results to include finite inertia effects.

Findings

Derived the mean-field limit for particles with inertia.

Proved the limit holds for times inversely proportional to the Lipschitz constant.

Improved upon previous perturbative derivations by allowing finite inertia.

Abstract

We consider a microscopic model of spherical particles with inertia in a Stokes flow. As the particle number grows to infinity and their size goes to zero we derive the monokinetic Vlasov-Stokes equations as mean-field limit. We do this under the assumption that the particles have initial velocities given by a Lipschitz velocity profile and prove the mean-field limit for times of the order of the inverse Lipschitz constant. Notably this is not a perturbative result. In particular, we do not require the inertia of the particles to vanish in the limit. Thereby the result improves upon the perturbative derivation in [HS23] in the case of a monokinetic limit density.