Existence and nonlinear stability of convective solutions for almost compressible fluids in Bénard problem

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.