Coupled problems on stationary flow of electrorheological fluids under the conditions of nonhomogeneous temperature distribution

We set up and study a coupled problem on stationary non-isothermal flow of electrorheological fluids. The problem consist in finding functions of velocity, pressure and temperature which satisfy the motion equations, the condition of incompressibility, the equation of the balance of thermal energy and boundary conditions. We introduce the notions of a $P$-generalized solution and generalized solution of the coupled problem. In case of the $P$-generalized solution the dissipation of energy is defined by the regularized velocity field, which leads to a nonlocal model. Under weak conditions, we prove the existence of the $P$ -generalized solution of the coupled problem. The existence of the generalized solution is proved under the conditions on smoothness of the boundary and on smallness of the data of the problem