Model for Aqueous Polymer Solutions with Damping Term: Solvability and Vanishing Relaxation Limit

The main aim of this paper is to investigate the solvability of the steady-state flow model for low-concentrated aqueous polymer solutions with a damping term in a bounded domain under the no-slip boundary condition. Mathematically, the model under consideration is a boundary value problem for the system of strongly nonlinear partial differential equations of third order with the zero Dirichlet boundary condition. We propose the concept of a full weak solution (velocity–pressure pair) in the distributions sense. Using the method of introduction of auxiliary viscosity, the acute angle theorem for generalized monotone nonlinear operators, and the Krasnoselskii theorem on the continuity of the superposition operator in Lebesgue spaces, we obtain sufficient conditions for the existence of a full weak solution satisfying some energy inequality. Moreover, it is shown that the obtained solutions of the original problem converge to a solution of the steady-state damped Navier–Stokes system as the relaxation viscosity tends to zero.