The Boussinesq system arises in Fluid Mechanics when motion is governed by density gradients caused by temperature or concentration differences. In the former case, and when thermodynamical coefficients are regarded as temperature dependent, the system consists of the Navier-Stokes equations and the non linear heat equation coupled through the viscosity, bouyancy and convective terms. According to the balance between specific heat and thermal conductivity the diffusion term in the heat equation may lead to a singular or degenerate parabolic equation. In this paper we prove the existence of solutions of the general problem as well as the uniqueness of solutions when the spatial dimension is two.

Nonlinear parabolic equations (msc 35K55), Generalized solutions (msc 35Dxx), Dependence of solutions on initial and boundary data, parameters (msc 35B30), Free convection (msc 76R10)
Modelling, Analysis and Simulation [MAS]

Díaz, J.I, & Galiano, G. (1997). Existence and uniqueness of solutions to the Boussinesq system with nonlinear thermal diffusion. Modelling, Analysis and Simulation [MAS]. CWI.