Global existence of weak solutions to the FENE dumbbell model of polymeric flows

Systems coupling fluids and polymers are of great interest in many branches of sciences. One of the models to describe them is the FENE (Finite Extensible Nonlinear Elastic) dumbbell model. We prove global existence of weak solutions to the FENE dumbbell model of polymeric flows for a very general class of potentials. The main problem is the passage to the limit in a nonlinear term that has no obvious compactness properties. The proof uses many weak convergence techniques. In particular it is based on the control of the propagation of strong convergence of some well chosen quantity by studying a transport equation for its defect measure.