Derivation of models for thin sprays from a multiphase Boltzmann model

We shall review the validation of a class of models for thin sprays where a Vlasov type equation is coupled to an hydrodynamic equation of Navier-Stokes or Stokes type. We present a formal derivation of these models from a multiphase Boltzmann system for a binary mixture: under suitable assumptions on the collision kernels and in appropriate asymptotics (resp. for the two different limit models), we prove the convergence of solutions to the multiphase Boltzmann model to distributional solutions to the Vlasov-Navier-Stokes or Vlasov-Stokes system. The proofs are based on the procedure followed in [BardosGolseLevermore1991] and explicit evaluations of the coupling terms due to the interaction between the two components of 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