We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and sufficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. In particular,

more »

... find that weakly mixing flows always enhance dissipation in this sense. The proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form Γ + iAL with a negative unbounded self-adjoint operator Γ, a self-adjoint operator L, and parameter A 1. In particular, they employ the RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian (related to a classical theorem of Wiener on Fourier transforms of measures). Applications to quenching in reaction-diffusion equations are also considered. with |M | being the volume of M . We would like to understand how the speed of convergence to the average depends on the properties of the flow and determine which flows are efficient in enhancing the relaxation process. The question of the influence of advection on diffusion is very natural and physically relevant, and the subject has a long history. The passive scalar model is one of the most