Existence of global weak solutions for the   Navier-Stokes-Vlasov-Boltzmann equations
  A moderately thick spray can be described by a coupled system of equations consisting of the incompressible Navier-Stokes equations and the Vlasov-Boltzmann equation. We investigate this kind of mathematical model in this paper. In particular, we study the initial value problem for the Navier-Stokes-Vlasov-Boltzmann equations. The existence of global weak solutions is established by a weak convergence method.   The interesting point of our main result is to handle the model with some breakup effects while the velocity of particles is in the whole space. 
Lei Yao, Cheng Yu
math.AP