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.

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

Flandoli F.;Leocata M.;Ricci C.
2019

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.
2019
Kinetic theory; Mean-Field; Particle-system; Scaling limits; Vlasov-Navier-Stokes
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/82034
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