Stationary weak solutions for a class of non-Newtonian fluids with energy transfer

Abstract

We prove the existence of weak solutions to the coupled system of stationary equations for a class of general non-Newtonian fluids with energy transfer. In particular, we may include Bingham flows that lead to classical free boundary problems of fluid dynamics. Using convex analysis and L 1-theory for elliptic mixed boundary value problems, we consider separately two auxiliary problems, obtained by prescribing the essential non-linearities in the equations for the velocity and for the energy. We use then a general fixed point theorem, due to Glicksberg, for the multivalued mapping in a product of Banach spaces endowed with the weak topologies.