  We consider a transport equation by a gradient vector field with a small
viscous perturbation --$\epsilon\Delta_g$. We study uniform observability
(resp. controllability) properties in the (singular) vanishing viscosity limit
$\epsilon\rightarrow 0^+$, that is, the possibility of having a uniformly
bounded observation constant (resp. control cost). We prove with a series of
examples that in general, the minimal time for uniform observability may be
much larger than the minimal time needed for the observability of the limit
equation $\epsilon = 0$. We also prove that the two minimal times coincides for
positive solutions. The proofs rely on a semiclassical reformulation of the
problem together with (a) Agmon estimates concerning decay of eigenfunctions in
the classically forbidden region [HS84] (b) fine estimates of the kernel of the
semiclassical heat equation [LY86].
