Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion

Abstract

The Boussinesq system of hydrodynamics equations [3], [26] arises from a zero order approximation to the coupling between the Navier–Stokes equations and the thermodynamic equation [25]. Presence of density gradients in a fluid allows the conversion of gravitational potential energy into motion through the action of buoyant forces. Density gradients are induced, for instance, by temperature differences arising from non-uniform heating of the fluid. In the Boussinesq approximation of a large class of flow problems, thermodynamical coefficients such as viscosity, specific heat and thermal conductivity may be assumed to be constants, leading to a coupled system of parabolic equations with linear second order operators, see, e.g. [11], [12], [17], [31]. However, there are some fluids such as lubrificants or some plasma flow for which this is not an accurate assumption [16], [29] and a quasilinear parabolic system has to be considered. In this paper we present some results on existence and uniqueness of weak solutions for this kind of models. Results on some qualitative properties related with spatial and time localization of the support of solutions will be published elsewhere, see