Steady Boussinesq System with Mixed Boundary Conditions Including Friction Conditions

In this paper we are concerned with the steady Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak, one-sided leak, velocity, vorticity, pressure and stress conditions together and the conditions for temperature may include Dirichlet, Neumann and Robin conditions together. For the problem involving the static pressure and stress boundary conditions, it is proved that if the data of the problem are small enough, then there exists a solution and the solution with small norm is unique. For the problem involving the total pressure and total stress boundary conditions, the existence of a solution is proved without smallness of the data.