On thick spray equations

This note is based on a presentation done at the Séminaire Laurent Schwartz in November 2023. The goal is to review our recent result [20], obtained in collaboration with Daniel Han-Kwan. We consider the equations of thick sprays, modeling the evolution of a gas-particles mixture through a Vlasov-Boltzmann equation for the particles and compressible Navier-Stokes equations for the gas. In the so-called thick spray regime, the volume occupied by the cloud of particles is not negligible in front of that of the gas. Unlike other fluid-kinetic models, the mathematical analysis of such dense sprays is still in its infancy and raises several challenges, the Cauchy theory being one of the first. Inspired by recent works on singular Vlasov equations, we show that the thick spray equations are locally well-posed in Sobolev regularity, provided that the initial data satisfies a suitable Penrose stability condition.