Weak solvability of irregularized model of viscoelastisity with memory

In the present paper we establish the existence of weak solutions of the initial-boundary value problem for equations of a viscoelastic model of fluid with memory along the trajectories of the velocity field. We use approximation-topological method which involves replacement of the given problem by operator equation, an approximation of equation in a weak sense, and the application of topological degree theory which allows to establish the existence of solutions on the base of a priori estimates and passing to the limit statements.