On Existence and Uniqueness of the Weak Solution of a Generalized Boussinesq Equation with Press and

In this paper, a generalized Boussinesq equation that couples the mass and heat flows in a viscous incompressible uid is considered. The kinematic viscosity and the heat conductivity are assumed to be dependent on the temperature. The boundary condition on the velocity of fluid is non-standard where the dynamical pressure is given on some part of the boundary, and the temperature of fluid is represented in a mixed boundary condition. The existence of the weak solution is proved by the Galerkin approximation scheme, and the uniqueness is also obtained under the condition on the weak solution that is somehow like the restriction on the Reynold number and the Raleigh number in hydrodynamics.