Approximation à la Oberbeck-Boussinesq for fluids with pressure-induced stratified density

We consider a model for convection in compressible fluids in two dimensions. A constitutive limit is studied in which both the mechanical compressibility and thermal expansion affect the buoyancy force. The motion is no longer isochoric as in the classical Boussinesq approximation but has a uniform expansion rate associated to the upward motion: ∇ · v = β v_z. By using a perturbative approach we study a Boussinesq-like approximation with pressure-dependent buoyancy force. The existence of weak solutions for the approximated system is proved and their stability is investigated