On Some Properties of Trajectories of Non-Smooth Vector Fields

In this paper, we study the properties of trajectories of systems of ordinary differential equations generated by the velocity field of a moving incompressible viscoelastic fluid with memory along the trajectories in a domain with multiple boundary components. The case of a velocity field from a Sobolev space with inhomogeneous boundary conditions is considered. The properties of the maximal intervals of existence of solutions to the Cauchy problem corresponding to a given velocity field are investigated. The study assumes the approximation of a velocity field by a sequence of smooth fields followed by a passage to the limit. The theory of regular Lagrangian flows is used.