On the α-Navier–Stokes–Vlasov and the α-Navier–Stokes–Vlasov–Fokker–Planck equations

We consider the α-Navier–Stokes equations coupled with a Vlasov type equation to model the flow of an incompressible fluid containing small particles. We prove the existence of global weak solutions to the coupled system subject to periodic boundary conditions. Moreover, we investigate the regularity of weak solutions and the uniqueness of regular solutions. The convergence of its solutions to that of the Navier–Stokes–Vlasov equations when α tends to zero is also established. The results are extended to the model with the diffusion of spray, i.e., to the α-Navier–Stokes–Vlasov–Fokker–Planck equations.