Well-posedness of slightly compressible Boussinesq’s flow in Darcy–Bénard problem

In the present article, we show the existence, uniqueness, and behavior in time of solutions to the Darcy–Bénard problem for an extended-quasi-thermal-incompressible fluid (extended Darcy–Bénard problem) uniformly heated from below, modeling the phenomenon of thermal convection in porous media. First and foremost, for the sake of a more realistic description of the phenomenon under analysis, we here assume the fluid’s density to depend on pressure and temperature, unlike the classical problem where the density is solely a function of temperature. This new constitutive law for the density, proposed by Guoin and Ruggeri in 2012, arises as a necessary requirement for the thermodynamic consistency of the model and introduces a slight compressibility making the model more closely aligned with reality. Moreover, the dependence of the fluid density on both temperature and pressure fields, from the mathematical viewpoint, adds an interesting challenge, requiring a new approach for its resolution. The well-posedness results establish a foundation for further analyses of the hydrodynamic stability properties of the solutions of the extended Darcy–Bénard problem, with particular regard to the effect of compressibility factor, which has great interest in applications.