  In this paper we study the validity of the so-called Oberbeck-Boussinesq
approximation for compressible viscous perfect gases in the whole
three-dimensional space. Both the cases of uids with positive heat conductivity
and zero conductivity are considered. For small perturbations of a constant
equilibrium, we establish the global existence of unique strong solutions in a
critical regularity functional framework. Next, taking advantage of Strichartz
estimates for the associated system of acoustic waves, and of uniform estimates
with respect to the Mach number, we obtain all-time convergence to the
Boussinesq system with a explicit decay rate.
