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We consider the Hookean dumbbell model, a system of nonlinear PDEs arising inthe kinetic theory of homogeneous dilute polymeric fluids. It consists of theunsteady incompressible Navier-Stokes equations in a bounded Lipschitz domain,coupled to a Fokker-Planck-type parabolic equation with a centre-of-massdiffusion term, for the probability density function, modelling the evolutionof the configuration of noninteracting polymer molecules in the solvent. Themicro-macro interaction is reflected by the presence of a drag term in theFokker-Planck equation and the divergence of a polymeric extra-stress tensor inthe Navier-Stokes balance of momentum equation.  We introduce the concept of generalised dissipative solution - a relaxationof the usual notion of weak solution, allowing for the presence of a, possiblynonzero, defect measure in the momentum equation. This defect measure accountsfor the lack of compactness in the polymeric extra-stress tensor. We proveglobal existence of generalised dissipative solutions satisfying additionallyan energy inequality for the macroscopic deformation tensor. Using thisinequality, we establish a conditional regularity result: any generaliseddissipative solution with a sufficiently regular velocity field is a weaksolution to the Hookean dumbbell model. Additionally, in two space dimensionswe provide a rigorous derivation of the macroscopic closure of the Hookeanmodel and discuss its relationship with the Oldroyd-B model with stressdiffusion. Finally, we improve an earlier result by Barrett and S\"{u}li byestablishing the global existence of weak solutions for a larger class ofinitial data.

On a class of generalised solutions to the kinetic Hookean dumbbell  model for incompressible dilute polymeric fluids: existence and macroscopic  closure