On the well-posedness of a stochastic Navier-Stokes-Vlasov-Fokker-Planck system

The work is concerned with the motion of a system of particles evolving in a fluid occupying a three-dimensional bounded domain. The motion of particles is described by a Vlasov-Fokker-Planck equation while the fluid is governed by the incompressible stochastic Navier-Stokes equations. The interaction between particles and fluid is described by Stokes's drag force. We prove the existence of a unique weak probabilistic solution. To achieve our goal, we use the Galerkin approximation combined with some compactness results of probabilistic type, as well as the Yamada-Watanabe theorem.