On the dynamics of charged particles in an incompressible flow: From kinetic-fluid to fluid-fluid models

In this paper, we are interested in the dynamics of charged particles interacting with the incompressible viscous flow. More precisely, we consider the Vlasov-Poisson or Vlasov-Poisson-Fokker-Planck equation coupled with the incompressible Navier-Stokes system through the drag force. For the proposed kinetic-fluid model, we study the asymptotic regime corresponding to strong local alignment and diffusion forces. Under suitable assumptions on well-prepared initial data, we rigorously derive a coupled isothermal/pressureless Euler-Poisson system and incompressible Navier-Stokes system (EPNS system). For this hydrodynamic limit, we employ the modulated kinetic, internal, interaction energy estimates. We also construct a global-in-time strong solvability for the isothermal/pressureless EPNS system. In particular, this global-in-time solvability gives the estimates of hydrodynamic limit hold for all times.