  We consider the FENE dumbbell polymer model which is the coupling of the
incompressible Navier-Stokes equations with the corresponding
Fokker-Planck-Smoluchowski di ffusion equation. We show global well-posedness
in the case of a 2D bounded domain. We assume in the general case that the
initial velocity is sufficiently small and the initial probability density is
sufficiently close to the equilibrium solution; moreover an additional
condition on the coeffcients is imposed. In the corotational case, we only
assume that the initial probability density is sufficiently close to the
equilibrium solution.
