Abstract
In this chapter we are concerned with the steady Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, vorticity, stress together and the conditions for temperature may include Dirichlet, Neumann and Robin conditions together. We will get variational formulations consisting of a variational inequality for velocity and a variational equation for temperature, which are equivalent to the original PDE problems for smooth solutions.
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