The present paper deals with the global existence and uniqueness of strong solutions to small perturbation of a constant equilibrium state for the so-called Oberbeck-Boussinesq approximation of compressible viscous perfect gases in three space dimensions. The strong solutions are studied in critical regularity spaces. The Oberbeck-Boussinesq is typically used in meteorology or oceanography to model stratified fluids. When the model paramaters, i.e. the Mach and Froude numbers, tend to zero the limiting equations describe incompessible fluids with constant density. For the initial data the authors consider the so-called ill-prepared data. In order to get all-time converegence to the Boussinesq system with explicit decay rates the authors apply the Strichatz estimates for the resulting acoustic wave equation system. Furthermore, the a priori estimates that are uniform in the small parameter (Mach/Froude number) are necessary. Both cases with zero and positive heat conductivity are considered.
This paper extends the results obtained by E. Feireisl et al. on global weak solutions to the full compressible Navier-Stokes equations, (see, e.g. E. Feireisl [Oxf. Lect. Ser. Math. Appl. 26, 212 p. (2004; Zbl 1080.76001); E. Feireisl and A. Novotný, J. Math. Fluid Mech. 11, No. 2, 274–302 (2009; Zbl 1214.76008); E. Feireisl and A. n Novotný, Singular limits in thermodynamics of viscous fluids. BBasel: Birkhäuser (2009; Zbl 1176.35126)]).