  The paper is concerned with a class of mathematical models for polymeric
fluids, which involves the coupling of the Navier-Stokes equations for a
viscous, incompressible, constant-density fluid with a parabolic-hyperbolic
integro-differential equation describing the evolution of the polymer
distribution function in the solvent, and a parabolic integro-differential
equation for the evolution of the monomer density function in the solvent. The
viscosity coefficient appearing in the balance of linear momentum equation in
the Navier-Stokes system includes dependence on the shear-rate as well as on
the weight-averaged polymer chain length. The system of partial differential
equations under consideration captures the impact of polymerization and
depolymerization effects on the viscosity of the fluid. We prove the existence
of global-in-time, large-data weak solutions under fairly general hypotheses.
