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

We consider the advection-diffusion equation φt +Au · ∇φ = ∆φ, φ(0, x) = φ0(x) on R, 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 φ0 ∈ L(R), p <∞, and any τ > 0, ‖φ(·, τ)‖∞ → 0 as A→∞. 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 φ0 in the unit ball in L(R), and ‖ · ‖∞ can be replaced by any ‖ · ‖q, q > p. Extensions to higher dimensions and applications to reaction-advection-diffusion equations are also considered.