Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off

We study the existence of global-in-time weak solutions to a coupled microscopic macroscopic bead-spring model with microscopic cut-off which arises from the kinetic theory of dilute 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 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function psi that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term and a cut-off function beta(L)(psi) = min(psi, L) in the drag term, where L >> 1. We establish the existence of global-in-time weak solutions to the model for a general class of spring-force potentials including, in particular, the widely used finitely extensible nonlinear elastic potential. A key ingredient of the argument is a special testing procedure in the weak formulation of the Fokker-Planck equation, based on the convex entropy function s epsilon R (>=) 0 bar right arrow F(s) := s (ln s - 1)+ 1 epsilon R (>=) 0. In the case of a corotational drag term, passage to the limit as L -> infinity recovers the Navier-Stokes-Fokker-Planck model with centre-of-mass diffusion, without cut-off.