Affiliations: Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, Institut Universitaire de France, Créteil, France. E-mail: raphael.danchin@u-pec.fr | Department of Mathematical Sciences, Tsinghua University, Beijing, China. E-mail: lbhe@math.tsinghua.edu.cn
Note: [] Corresponding author: Raphaël Danchin, Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, Institut Universitaire de France, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France. E-mail: raphael.danchin@u-pec.fr
Abstract: 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 fluids 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.
