Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off - Barrett, J; Suli, E (2008)
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 NavierStokes 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 FokkerPlanck-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 potential. A key ingredient of the argument is a special testing procedure in the weak formulation of the FokkerPlanck equation, based on the convex entropy function $s ε ℝ ≥ 0 → F(s) := s(ln s - 1) + 1 ε ℝ ≥ 0. In the case of a corotational drag term, passage to the limit as L → ∞ recovers the NavierStokesFokkerPlanck model with centre-of-mass diffusion, without cut-off. © 2008 World Scientific Publishing Company.