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

Abstract

We consider the advection-diffusion equation \[ ϕ_t + Au \cdot \nabla ϕ= Δϕ, \qquad ϕ(0,x)=ϕ_0(x) \] on ${\mathbb{R}}^2$, with $u$ a periodic incompressible flow and $A\gg 1$ its amplitude. We provide a sharp characterization of all $u$ that optimally enhance dissipation in the sense that for any initial datum ${\varphi }_0\in L^p({\mathbb{R}}^2)$, $p<\infty$, and any $\tau >0$, \[ \|ϕ(\cdot,τ)\|_{L^\infty(\bbR^2)} \to 0 \qquad \text{as $A\to \infty$.} \] 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 ${\varphi }_0$ in the unit ball of $L^p({\mathbb{R}}^2)$, $p<\infty$, and $\Vert \cdot {\Vert }_{\infty }$ can be replaced by any $\Vert \cdot {\Vert }_q$, $q>p$. Extensions to higher dimensions and applications to reaction-advection-diffusion equations are also considered.

35 pp