  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 $u\equiv 0$ maximises
$\|\tau\|_{L^\infty(\Omega)}$. 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.
