Stable mixing estimates in the infinite P\'eclet number limit

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 $t^{-1}$.