  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 an earlier result by Barrett and S\"{u}li by
establishing the global existence of weak solutions for a larger class of
initial data.
