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 advection–diffusion equation. The proof relies on the renormalization techniques, the method of moments and a velocity averaging lemma. Here, we extend the works of Goudon, Jabin and Vasseur [Hydrodynamic limit for the Vlasov-Navier-Stokes equations. Part I: Light particles regime, Indiana Univ. Math. J. 53(6) (2004) 1495–1516] and Cui, Wang and Yao [Asymptotic analysis for 1D compressible Navier–Stokes–Vlasov equations, Commun. Pure Appl. Anal. 19(5) (2020) 2737–2750] to inhomogeneous incompressible Navier–Stokes/Vlasov–Fokker–Planck equations.