The Problem of the Flow of One Type of Non-Newtonian Fluid through the Boundary of a Multiply Connected Domain

The existence of a weak solution of the initial boundary value problem for the equations of motion of a viscoelastic non-Newtonian fluid in a multiply connected domain with memory along trajectories of a nonsmooth velocity field and with an inhomogeneous boundary condition is established. The study assumes Galerkin-type approximations of the original problem followed by passage to the limit based on a priori estimates. The theory of regular Lagrangian flows is used to study the behavior of trajectories of a nonsmooth velocity field.