Global existence of weak solutions to the incompressible Vlasov-Navier-Stokes system coupled to convection-diffusion equations

We study the existence of global weak solutions in a three-dimensional time-dependent bounded domain for the incompressible Vlasov-Navier-Stokes system which is coupled with two convection-diffusion equations describing the air temperature and its water vapor mass fraction. This newly introduced model describes respiratory aerosols in the human aiways when one takes into account the hygroscopic effects, also inducing the presence of extra variables in the aerosol distribution function, temperature and size. The mathematical description of these phenomena leads us to make the assumption that the initial distribution of particles does not contain arbitrarily small particles. The proof is based on a regularization and approximation strategy that we solve by deriving several energy estimates, including ones with temperature and size.