  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.
