In this memoir, the authors 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.

The authors identify a suitable stability condition bearing on the initial data that provides estimates without loss, ensuring that the system is well posed. This condition coincides with a Penrose condition appearing in earlier works on singular Vlasov equations. The authors also rely on crucial new estimates for averaging operators. Their approach allows them to treat many variants of the model, such as collisions in the kinetic equation, non-barotropic fluid or density-dependent drag force.