We study the nonlinear almost compressible 2D Oberbeck-Boussinesq system, characterized by an extra buoyancy term where the density depends on the pressure, and a corresponding dimensionless parameter β, proportional to the (positive) compressibility factor β₀. The local in time existence of the perturbation to the conductive solution is proved for any “size” of the initial data. However, unlike the classical problem where β₀ = 0, a smallness condition on the initial data is needed for global in time existence, along with smallness of the Rayleigh number. Removing this condition appears quite challenging, and we leave it as an open question.

We show well-posedness for the equations describing a new model of slightly compressible fluids. This model was recently rigorously derived in Grandi and Passerini (Geophys Astrophys Fluid Dyn, 2020) from the full set of balance laws and falls in the category of anelastic Navier–Stokes fluids. In particular, we prove existence and uniqueness of global regular solutions in the two-dimensional case for initial data of arbitrary “size”, and for “small” data in three dimensions. We also show global stability of the rest state in the class of weak solutions.