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

Franco Flandoli; Marta Leocata; Cristiano Ricci

doi:10.3934/dcdsb.2018313

Article Contents

2019, Volume 24, Issue 8: 3741-3753. Doi: 10.3934/dcdsb.2018313

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

1.
Scuola Normale Superiore of Pisa - Italy
2.
University of Pisa - Italy
3.
University of Florence - Italy

Received: March 31, 2018

Revised: June 30, 2018

Early access: October 2018

Published: July 2019

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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.

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Mathematics Subject Classification: Primary: 35J05, 76T20, 60K35.

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