Solvability of one non-Newtonian fluid dynamics model with memory

In the present paper we establish the existence of weak solutions to the initial-boundary value problem for one viscoelastic model of Oldroyd's type fluid with memory along trajectories of the velocity field. Previously such problem has been studied for corresponding regularized models. The reason of the regularization was the lack of results on the solvability of the Cauchy problem with not sufficiently smooth velocity field. However, recent results about regular Lagrangian flows (generalization of classical solutions of a Cauchy problem) allow to establish the existence theorem for the original problem. We use topological approximation method which involves the approximation of the original problem by regularized operator equations with consequent application of topological degree theory for its solvability. This allows to establish the existence of weak solutions of considered problem on the base of a priori estimates and passing to the limit.