Solution in weak sense of a boundary value problem describing thermal convection

Boundary value problem for the stationary model thermal convection of a high-viscosity inhomogeneous incompressible fluid in the Boussinesq approximation with irregular boundary data for temperature is investigated. Conditions of uniqueness solvability of the boundary value problem are specified. The smoothness of a weak solution subject to the smoothness of initial data and the smoothness of a boundary of a domain, where solution is sought, is investigated.