Multiphase formulation of the Vlasov-Navier-Stokes equations

In this paper, we study a particular family of solutions of the Vlasov-Navier-Stokes system posed on $\mathbb{R}^d$ (with $d\geq 2$), and show their convergence to the unique solution of the pressureless Euler-Navier-Stokes system. A global existence result for the latter system, in the small data regime, was established in \cite{MonENS}. Here we place ourselves in a multiphase framework, introduced and studied by Zakharov in \cite{Zakharov1,Zakharov2}, in order to carry out an analogous analysis for a system that we will call multiphase pressureless Euler-Navier-Stokes. We then study the single-phase limit and obtain a rigorous link between the Vlasov-Navier-Stokes system and the pressureless Euler-Navier-Stokes system.