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In this paper, we prove the local in time well-posedness of thick spray equations in Sobolev spaces, for initial data satisfying a Penrose-type stability condition. This system is a coupling between particles described by a kinetic equation and a surrounding fluid governed by compressible Navier-Stokes equations. In the thick spray regime, the volume fraction of the dispersed phase is not negligible compared to that of the fluid. We identify a suitable stability condition bearing on the initial conditions that provides estimates without loss, ensuring that the system is well-posed. It coincides with a Penrose condition appearing in earlier works on singular Vlasov equations. We also rely on crucial new estimates for averaging operators. Our approach allows to treat many variants of the model, such as collisions in the kinetic equation, non-barotropic fluid or density-dependent drag force.

We study a hydrodynamic limit of the Vlasov-Navier-Stokes system with external gravity force, following the framework introduced by Han-Kwan and Michel in [Han-Kwan - Michel, arXiv:2103.06668]. We answer a question raised in the latter concerning the derivation of a Boussinesq-Navier-Stokes type system for arbitrarily large times, starting from the previous fluid-kinetic system. 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.

This paper is devoted to the large time behavior of weak solutions to the three-dimensional Vlasov-Navier-Stokes system set on the half-space, with an external gravity force. This fluid-kinetic coupling arises in the modeling of sedimentation phenomena. We prove that the local density of the particles and the fluid velocity enjoy a convergence to $0$ in large time and at a polynomial rate. In order to overcome the effect of the gravity, we rely on a fine analysis of the absorption phenomenon at the boundary. We obtain a family of decay estimates for the moments of the kinetic distribution, provided that the initial distribution function has a sufficient decay in the phase space., Comment: 53 pages