Global derivation of a Boussinesq–Navier–Stokes type system from fluid-kinetic equations

We study a hydrodynamic limit of the Vlasov–Navier–Stokes system with external gravity force. We answer a question raised by Han-Kwan and Michel in [48] concerning the limit towards a Boussinesq–Navier–Stokes type system. Our work provides a rigorous derivation of such hydrodynamic equations for arbitrarily large times, starting from the previous fluid-kinetic coupling. To do so, we consider a particular spatial geometric setting corresponding to the half-space case. Our proof is based on an absorption effect at the boundary which leads to crucial decay in time estimates.