Diffusion in Fluid Flow: Dissipation Enhancement by Flows in 2D

We consider the advection-diffusion equation [image omitted] on 2, with u a periodic incompressible flow and A >> 1 its amplitude. We provide a sharp characterization of all u that optimally enhance dissipation in the sense that for any initial datum phi 0Lp(2), p , and any 0, [image omitted] Our characterization is expressed in terms of simple geometric and spectral conditions on the flow. Moreover, if the above convergence holds, it is uniform for phi 0 in the unit ball in Lp(2), and center dot can be replaced by any center dot q, with qp. Extensions to higher dimensions and applications to reaction-advection-diffusion equations are also considered.