Solvability of the Non-Linearly Viscous Polymer Solutions Motion Model

In this paper we consider the initial–boundary value problem describing the motion of weakly concentrated aqueous polymer solutions. The model involves the regularized Jaumann’s derivative in the rheological relation. Also this model is considered with non-linear viscosity. On the basis of the topological approximation approach to the study of hydrodynamics problems the existence of weak solutions is proved. Also we consider an optimal feedback control problem for this initial–boundary value problem. The existence of an optimal solution minimizing a given performance functional is proved.