Global existence of weak and classical solutions for the Navier–Stokes–Vlasov–Fokker–Planck equations

We consider a system coupling the incompressible Navier–Stokes equations to the Vlasov–Fokker–Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov–Stokes system.