Problem in natural convection with nonlinear boundary conditions

Existence theorems are established for generalized solutions of the conjugate problem of steady natural convection of a viscous incompressible fluid filling a vessel with rigid walls with nonlinear heat transfer at the outer surface of the vessel. The problem is formulated in appropriate Sobolev and Hilbert spaces. The generalized solution must satisfy an integral identity. Existence of the generalized solution is proved for a two-dimensional and a three-dimensional domain