Existence of Global Weak Solutions to Some Regularized Kinetic Models for Dilute Polymers

We study the existence of global-in-time weak solutions to a coupled microscopicmacroscopic bead-spring model 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 which satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. The anisotropic Friedrichs mollifiers, which naturally arise in the course of the derivation of the model in the Kramers expression for the extra-stress tensor and in the drag term in the Fokker–Planck equation, are replaced by isotropic Friedrichs mollifiers. 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. We justify also, through a rigorous limiting process, certain classical reductions of this model appearing in the literature which exclude the center-of-mass diffusion term from the Fokker–Planck equation on the grounds that the diffusion coefficient is small relative to other coefficients featuring in the equation. In the case of a corotational drag term we perform a rigorous passage to the limit as the Friedrichs mollifiers in the Kramers expression and the drag term converge to identity operators.