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 microscopicmacroscopic 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 Ω ⊂ Rd, 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 ψ 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 βL(ψ) = min(ψ,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 (FENE) 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 ∈ R≥0 7→ F(s) := s (ln s − 1) + 1 ∈ R≥0. In the case of a corotational drag term, passage to the limit as L → ∞ recovers the Navier–Stokes–Fokker–Planck model with centre-of-mass diffusion, without cut-off.