On uniform observability of gradient flows in the vanishing viscosity limit

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 $\epsilon\to 0^+$ , 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].