On a Class of Generalised Solutions to the Kinetic Hookean Dumbbell Model for Incompressible Dilute Polymeric Fluids: Existence and Macroscopic Closure

Abstract

We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier–Stokes equations in a bounded Lipschitz domain, coupled to a Fokker–Planck-type parabolic equation with a centre-of-mass diffusion term, for the probability density function, modelling the evolution of the configuration of noninteracting polymer molecules in the solvent. The micro–macro interaction is reflected by the presence of a drag term in the Fokker–Planck equation and the divergence of a polymeric extra-stress tensor in the Navier–Stokes balance of momentum equation. We introduce the concept of generalised dissipative solution—a relaxation of the usual notion of weak solution, allowing for the presence of a, possibly nonzero, defect measure in the momentum equation. This defect measure accounts for the lack of compactness in the polymeric extra-stress tensor. We prove global existence of generalised dissipative solutions satisfying additionally an energy inequality for the macroscopic deformation tensor. Using this inequality, we establish a conditional regularity result: any generalised dissipative solution with a sufficiently regular velocity field is a weak solution to the Hookean dumbbell model. Additionally, in two space dimensions we provide a rigorous derivation of the macroscopic closure of the Hookean model and discuss its relationship with the Oldroyd-B model with stress diffusion. Finally, we improve a result by Barrett and Süli (Nonlinear Anal. Real World Appl. 39:362–395, 2018) by establishing the global existence of weak solutions for a larger class of initial data.