Asymptotic stability of a fluid-particles interaction model with external forces

We consider the Cauchy problem of the Vlasov-Fokker-Planck equation for the dispersed phase coupled to the incompressible Euler equations with external forces deriving from a potential. Global existence and uniqueness of solution are established using classical energy estimate combined with macro-micro decomposition. With external forces, the asymptotic profile is inhomogeneous, and we estimate the decay rate of the solution to the background profile. The decay is optimal in the sense that it equals the decay rate for the corresponding system without forces {See: Carrillo, J. A., Duan, R. J., and Moussa, A. ["Global classical solutions close to equilibrium to the Vlasov-Euler-Fokker-Planck system," Kinet. Relat. Models 4, 227-258 (2011)]}.