Existence of global weak solutions for some polymeric flow models

We study the existence of global-in-time weak solutions to a coupled microscopic-macroscopic bead-spring model which arises from the kinetic theory of diluted solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Omega subset of R-d, d = 2,3, for the velocity and the pressure of the fluid, with an extra-stress tensor as right-hand side in the momentum equation. The extra-stress tenser stems from the random movement of the polymer chains and is defined through the associated probability density function which satisfies a Fokker-Planck type degenerate parabolic equation. Upon appropriate smoothing of the convective velocity field in the Fokker-Planck equation, and in some circumstances, of the extra-stress tensor, we establish the existence of global-in-time weak solutions to this regularised bead-spring model for a general class of spring-force-potentials including in particular the widely used FENE (Finitely Extensible Nonlinear Elastic) model.