Derivation of models for thin sprays from a multiphase Boltzmann model

We shall review the validation of a class of models for thin sprays wherea Vlasov type equation is coupled to an hydrodynamic equation of Navier-Stokesor Stokes type. We present a formal derivation of these models from a multiphaseBoltzmann system for a binary mixture: under suitable assumptions on the collisionkernels and in appropriate asymptotics (resp. for the two different limit models), weprove the convergence of solutions to the multiphase Boltzmann model to distributionalsolutions to the Vlasov-Navier-Stokes or Vlasov-Stokes system. The proofsare based on the procedure followed in [BardosGolseLevermore1991] and explicitevaluations of the coupling terms due to the interaction between the two componentsof the mixture. The results reviewed in this article are proved in detail in two previous papers, joint works with E. Bernard, L. Desvillettes and F. Golse. [BDGR2016a], [BDGR2016b]