STABILITY AND ASYMPTOTIC ANALYSIS OF A FLUID-PARTICLE INTERACTION MODEL JOS ´ E A. CARRILLO, THIERRY GOUDON Abstract. We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a Vlasov-Fokker-Planck equation to describe the microscopic motion of the particles coupled to the Euler equations for a compressible fluid. We investigate dissipative quantities, equilibria and their stability properties and the role of external forces. We also study some asymptotic problems, their equilibria and stability and the derivation of macroscopic two-phase models. 1. Introduction Fluid-particle interaction is of primarily importance in sedimentation analysis of disperse suspensions of particles in fluids, one of the issues being the separation of the solid grains from the fluid by external forces: gravity settling processes or centrifugal forces. These procedures find their applica- tions in biotechnology, medicine, waste-water recycling and mineral process- ing [11]. On the other hand, aerosols and sprays can be also modelled by fluid-particle type interactions in which bubbles of suspended substances are seen as solid particles [5, 6]. Eventually, such problems also arises in com- bustion theory, when modelling Diesel engines or rocket propulsors [45, 46]. In what follows, we describe a single specie of disperse particles by a density function f (t, x, ξ ): f (t, x, ξ )dξ dx gives the number of particles en- closed at time t ≥ 0 in the infinitesimal domain of the phase space centered on (x, ξ ) ∈ R 3 × R 3 , with volume dξ dx. A macroscopic description of the dispersed phase is obtained by looking at averages with respect to the ξ variable like the macroscopic density  R 3 f dξ , the macroscopic momen- tum  R 3 ξf dξ and so on. The surrounding fluid is described by its density n(t, x) ≥ 0 and its velocity field u(t, x) ∈ R 3 . In this work, we will consider the fluid as compressible and we will describe it by the compressible Euler 2000 Mathematics Subject Classification. 35Q99 35B25. Key words and phrases. Fluid-Particles Interaction. Vlasov-Euler system. Stability. Hydrodynamic Limit. 1