Solvability of the initial-boundary value problem for the high-order Oldroyd model

The paper considers the solvability in the weak sense of the initial-boundary value problem for the high-order Oldroyd model. For the considered model on the base of the Laplace transform the stress tensor is expressed from the rheological relation. After substituting it into the motion equations, an initial-boundary value problem is obtained for an integro-differential equation with memory along the trajectories of the velocity field. After that, based on the approximation-topological approach to the study of hydrodynamic problems, the existence of a weak solution is proved. In the proof of the assertions, properties of regular Lagrangian flows are essentially used.