The Vlasov-Navier-Stokes equations as a mean field limit (Q2321082)

A model of particles dispersed in an incompressible viscous fluid is investigated. A forcing term, containing the density of particles immersed in the fluid (denoted by \(F\)), appears in the initial 2D Navier-Stokes equations. The paper is a continuation of the work [the first author, ``A fluid-particle system related to Vlasov-Navier-Stokes equations'', to appear in Lecture Notes RIMS Kyoto]. The particle number \(N\) is coupled with the parameter \(\epsilon\) which describe the size of interaction of the particle. The limits \(N \rightarrow \infty\) and \(\epsilon \rightarrow 0 \) are studied. The joint limit introduces great difficulties. The point is to use the interaction between particles and fluid, represented by the forcing term, involving also the Stokes drag force. The considered differential system reminds a mean field model. A specific equation is proposed for the particle system. The new element is the introduction of the additive noise. It is mentioned that ``A weak-strong uniqueness theorem would be sufficient, proving that our weaker solutions coincide with the stronger ones provided by [\textit{C. Yu}, J. Math. Pures Appl. (9) 100, No. 2, 275--293 (2013; Zbl 1284.35119)]. We hope to overcome these technical problems in future research.'' The main mathematical tools are: relative compactness of measures (Prohorov's Theorem), the Aubin-Lions lemma, Portemanteau's theorem. Some very interesting open problems are outlined in the last part.