Initial–Boundary Value Problem for Flows of a Fluid with Memory in a 3D Network-Like Domain

We consider an initial–boundary value problem for an integro-differential system that describes 3D flows of a non-Newtonian fluid with memory in a network-like domain. The problem statement uses the Dirichlet boundary conditions for the velocity and pressure fields as well as Kirchhoff-type transmission conditions at the internal nodes of the network. A theorem on the existence and uniqueness of a time-continuous weak solution is proved. In addition, an energy equality for this solution is derived.