text: 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.
  @cite_5: The present paper is dedicated to the study of the global existence for the inviscid two-dimensional Boussinesq system. We focus on finite energy data with bounded vorticity and we find out that, under quite a natural additional assumption on the initial temperature, there exists a global unique solution. No smallness conditions are imposed on the data. The global existence issues for infinite energy initial velocity, and for the Benard system are also discussed.
  @cite_10: Abstract Motivated by the study on the uniqueness problem of the coupled model, in this paper, we revisit 2d incompressible Navier–Stokes equations in bounded domains. In fact, we establish some new smoothing estimates to the Leray solution based on the spectral analysis of Stokes operator. To understand well these estimates, on one hand, we establish some new Brezis–Waigner type inequalities in general domain and in any dimension and disclose the connection between both of them. On the other hand, we show that these new estimates can be applied to prove the existence and uniqueness of the weak solutions for two coupled models: Boussinesq system with partial viscosity (no dissipation for the temperature) and Fluid Particle system, in two dimension and in bounded domains.
  @cite_17: 
summary: There is an important literature dedicated to the limit system, that is the Oberbeck-Boussinesq equations , and , under various hypotheses over the coefficients @math and @math and the potential @math (although the most common assumption is that @math ). Loosely speaking the classical results concerning the existence issue are (see e.g. @cite_17 @cite_5 @cite_10 and the references therein): Dimension @math : Global existence of strong solutions if @math Dimension @math with @math : Global weak solutions and local strong solutions (which become global if the data are small). Dimension @math with @math : only local-in-time strong solutions are available.