Hydrodynamic limit for the inhomogeneous incompressible Navier-Stokes-Vlasov equations

Abstract

We consider the hydrodynamic limit of weak solutions to the inhomogeneous incompressible Navier-Stokes-Vlasov equations in three-dimensional torus domain. It is shown that the global weak solutions of the inhomogeneous incompressible Navier-Stokes-Vlasov equations converge to the smooth solutions of the two-fluid model in the sense that the distribution function fϵ converges to a Dirac distribution in velocity, the fluid density ρϵ and the velocity uϵ converge to ρ and u in some sobolev space, respectively. The proofs mainly involve the relative entropy method, bootstrap argument and delicate energy estimates.