Global existence of solutions for the coupled Vlasov and Navier-Stokes equations. (English) Zbl 1240.35403

The authors are interested in the problem of existence of weak solutions of the three-dimensional incompressible Vlasov-Navier-Stokes equations. The periodic boundary conditions are considered. The coupling is done through a drag force which is linear with respect to the relative velocity of the fluid and particles. The proof of existence is split into three main steps. In the first one, they build a sequence of approximate solutions, where the convection velocity of the Navier-Stokes equations and the fluid velocity appearing in the Vlasov equation are regularized. In the second step, they prove that the approximate solutions satisfy uniform bounds on a certain time interval. This interval allows us to pass to the limit when the parameter of regularization tends to 0. Finally, they prove the existence of global solution.


35Q35 PDEs in connection with fluid mechanics
76T10 Liquid-gas two-phase flows, bubbly flows
76D05 Navier-Stokes equations for incompressible viscous fluids
35D30 Weak solutions to PDEs