We consider a passive scalar $f$ advected by a strictly monotone shear flow
and with a diffusivity parameter $\nu\ll 1$. We prove an estimate on the
homogeneous $\dot{H}^{-1}$ norm of $f$ that combines both the $L^2$ enhanced
diffusion effect at a sharp rate proportional to $\nu^{1/3}$, and the sharp
mixing decay proportional to $t^{-1}$ of the $\dot{H}^{-1}$ norm of $f$ when
$\nu=0$. In particular, the estimate is stable in the infinite P\'eclet number
limit, as $\nu\to 0$. To the best of our knowledge, this is the first result of
this kind since the work of Kelvin in 1887 on the Couette flow.
The two key ingredients in the proof are an adaptation of the hypocoercivity
method and the use of a vector field $J$ that commutes with the transport part
of the equation. The $L^2$ norm of $Jf$ together with the $L^2$ norm of $f$
produces a suitable upper bound for the $\dot{H}^{-1}$ norm of the solution
that gives the extra decay factor of $t^{-1}$.