  The set of nonzero external forces for which the zero function is in the
global attractor of the 2D Navier-Stokes equations is shown to be meagre in a
Fr\'echet topology. A criterion in terms of a Taylor expansion in complex time
is used to characterize the forces in this set. This leads to several relations
between certain Gevrey subclasses of $C^{\infty}$ and a new upper bound for a
Gevrey norm of solutions in the attractor, valid in the strip of analyticity in
time.
