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

TL;DR

This paper investigates the behavior of charged particles in incompressible flows, deriving a coupled Euler-Poisson and Navier-Stokes system as a hydrodynamic limit, and establishes global solvability for the resulting models.

Contribution

It rigorously derives a coupled hydrodynamic model from kinetic equations and proves global-in-time strong solvability for this system.

Findings

Derivation of the EPNS system as a hydrodynamic limit.

Global-in-time strong solvability of the EPNS system.

Establishment of estimates for the hydrodynamic limit over all time.

Abstract

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 the initial data, we rigorously derive a coupled isothermal/pressureless Euler-Poisson system and incompressible Navier-Stokes system(in short, EPNS system). For this hydrodynamic limit, we employ the modulated kinetic energy estimate together with the relative entropy method and the bounded Lipschitz distance. 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 time.