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

We consider the advection-diffusion equation $$\phi_t + Au \cdot \nabla \phi = \Delta \phi, \qquad \phi(0,x)=\phi_0(x)$$ on $\bbR^2$, 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 $\phi_0\in L^p(\bbR^2)$, $p<\infty$, and any $\tau>0$, $$\|\phi(\cdot,\tau)\|_{L^\infty(\bbR^2)} \to 0 \qquad \text{as $A\to\infty$.}$$ 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 $\phi_0$ in the unit ball of $L^p(\mathbb{R}^2)$, $p<\infty$, and $\|\cdot\|_\infty$ can be replaced by any $\|\cdot\|_q$, $q>p$. Extensions to higher dimensions and applications to reaction-advection-diffusion equations are also considered.