Parabolic limit for the inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations

In this paper, we study the parabolic limit for the inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations in two-dimensional bounded domain when the initial density of the fluid is bounded away from zero. The limit problem consists of the inhomogeneous incompressible Navier-Stokes equation for the density and velocity of the fluid, while the macroscopic density of the particles verifies an advectiondiffusion equation. The proof relies on the renormalization techniques, the method of moments and a velocity averging lemma. Here, we extend the works of Goudon-Jabin-Vasseur ([18], 2004) and Cui-Wang-Yao ([11], 2020) to inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations.