Solvability of a Boundary Value Problem for the Stationary Heatand Mass Transfer Equations with Variable Coefficients

A new boundary value problem for the stationary heat and mass transfer equations with variable coefficients is considered. It is assumed that the leading viscosity, thermal conductivity, and diffusion coefficients, as well as the buoyancy force, occurring in the equations depend on the temperature and the concentration of the substance dissolved in the base medium. A mathematical technique based on a variational approach is developed to study this boundary value problem. This technique is used to prove the global existence of a weak solution of the boundary value problem and establish sufficient conditions on the problem data ensuring the local uniqueness of a weak solution under the additional condition that the temperature and concentration are smooth.