  This article deals with the issues of global-in-time existence and asymptotic
analysis of a fluid-particle interaction model in the so-called bubbling
regime. The mixture occupies the physical space $\Omega \subset \mathbb{R}^3$
which may be unbounded. The system under investigation describes the evolution
of particles dispersed in a viscous compressible fluid and is expressed by the
conservation of fluid mass, the balance of momentum and the balance of particle
density often referred as the Smoluchowski equation. The coupling between the
dispersed and dense phases is obtained through the drag forces that the fluid
and the particles exert mutually by action-reaction principle. We show that
solutions exist globally in time under reasonable physical assumptions on the
initial data, the physical domain, and the external potential. Furthermore, we
prove the large-time stabilization of the system towards a unique stationary
state fully determined by the masses of the initial density of particles and
fluid and the external potential.
