Weak Solutions for a Class of Non-Newtonian Fluids with Energy Transfer

In the present paper, we shall consider a nonlinear thermoconvection problem consisting of a coupled system of nonlinear partial differential equations due to temperature dependent coefficients. We prove that weak solutions exist in appropriate Sobolev spaces under mild hypothesis on the regularity of the data. This result is established through a fixed point theorem for multivalued functions, which requires a detailed analysis of the continuous dependence of auxiliary problems, including the associated Lagrange multipliers of the generalized Navier—Stokes system.