Sharp uniform-in-diffusivity mixing rates for passive scalars in parallel shear flows

We consider the advection-diffusion equation describing the evolution of a passive scalar in a background shear flow. We prove the optimal uniform-in-diffusivity mixing rate $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{-1}}\lesssim ⟨t{⟩}^{-1\mathrm{/}(N+1)}$, $t\ge 0$, where $N$ is the maximal order of vanishing of the derivative $b^{\mathrm{\prime }}(y)$ of the shear profile, e.g., $N=1$ for plane Pouseille flow. Our proof is based on the description of the solution in terms of resolvents and involves pointwise estimates on the resolvent kernel. In the non-degenerate case, we further give a rigorous asymptotic description of generic solutions in terms of shear layers localized around the critical points. This verifies formal asymptotics in [McLaughlin-Camassa-Viotti, $\mathit{\text{Physics}}\mathit{\text{of}}\mathit{\text{Fluids}}$, 22(11), 2010].