The Vlasov-Navier-Stokes equations as a mean field limit

TL;DR

This paper investigates the convergence of particle systems to the Vlasov-Navier-Stokes equations, proving partial results under modified interactions and establishing support on weak solutions, advancing understanding of mean field limits in fluid-particle systems.

Contribution

It introduces a modified particle-fluid interaction model, proves tightness of the system's laws, and shows limits supported on weak solutions of the Vlasov-Navier-Stokes system.

Findings

Proved tightness of the particle system laws.

Established support on weak solutions of the Vlasov-Navier-Stokes system.

Outlined open problems like weak-strong uniqueness.

Abstract

Convergence of particle systems to the Vlasov-Navier-Stokes equations is a difficult topic with only fragmentary results. Under a suitable modification of the classical Stokes drag force interaction, here a partial result in this direction is proven. A particle system is introduced, its interaction with the fluid is modelled and tightness is proved, in a suitable topology, for the family of laws of the pair composed by solution of Navier-Stokes equations and empirical measure of the particles. Moreover, it is proved that every limit law is supported on weak solutions of the Vlasov-Navier- Stokes system. Open problems, like weak-strong uniqueness for this system and its relevance for the convergence of the particle system, are outlined.