On uniform observability of gradient flows in the vanishing viscosity limit

We consider a transport equation by gradient vector field with small viscous perturbation -εΔ g . study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit ε→0 + , that is, possibility of having uniformly bounded observation constant control cost). prove series examples general, minimal time for may be much larger than needed ε=0. also two times coincide positive solutions. The proofs rely on semiclassical reformulation problem together (a) Agmon estimates concerning decay eigenfunctions classically forbidden region [HS84] (b) fine kernel heat [LY84].