Abstract
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain, and for $x\in\Omega$ let $\tau(x)$ be the expected exit time from $\Omega$ of a diffusing particle starting at x and advected by an incompressible flow u. We are interested in the question which flows maximize $\|\tau\|_{L^\infty(\Omega)}$, that is, they are most efficient in the creation of hotspots inside $\Omega$. Surprisingly, among all simply connected domains in two dimensions, the discs are the only ones for which the zero flow $u\equiv0$ maximizes $\|\tau\|_{L^\infty(\Omega)}$. We also show that in any dimension, among all domains with a fixed volume and all incompressible flows on them, $\|\tau\|_{L^\infty(\Omega)}$ is maximized by the zero flow on the ball.
Highlights
It is well known that mixing by an incompressible flow enhances diffusion in many contexts
This is demonstrated, for instance, by the fact that the effective diffusivity of a periodic incompressible flow is always larger than diffusion in the absence of a flow [5], or that the principal eigenvalue μu of the problem
Classes of flows which are most effective in enhancing diffusion have been studied both on bounded and unbounded domains, and their characterizations have been provided in [3, 12]. It was observed in [9] that an incompressible flow may slow down diffusion in the following sense
Summary
It is well known that mixing by an incompressible flow enhances diffusion in many contexts. Our first result shows that in any bounded connected domain in R2 which is not a disk, there are (regular) incompressible flows which increase the maximum of the expected exit time of Xt from Ω. C3 and such that the stream function ψ only has a single critical point in Ω (which is connected and ψ > 0 on Ω).
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