Weak continuity in convection problems (Q1090563)

The Darcy-Boussinesq system of equations, which governs the steady natural convection in fluid-saturated porous media, is studied. The existence of the weak solutions is proved by a slight generalization of \textit{J. P. Gossez}'s theorem [Acad. Roy. Belgique, Bull. Cl. Sci., V. Sér. 52, 1073-1077 (1966; Zbl 0148.387)]. A maximum principle is obtained by the technique of inequalities in Sobolev spaces. Further regularity properties and an uniqueness criterion (in terms of the dimensionless Rayleigh number) are presented.