Mixing for generic rough shear flows

We study mixing and diffusion properties of passive scalars driven by $generic$ rough shear flows. Genericity is here understood in the sense of prevalence and (ir)regularity is measured in the Besov-Nikolskii scale $B^{\alpha}_{1, \infty}$, $\alpha \in (0, 1)$. We provide upper and lower bounds, showing that in general inviscid mixing in $H^{1/2}$ holds sharply with rate $r(t) \sim t^{1/(2 \alpha)}$, while enhanced dissipation holds with rate $r(\nu) \sim \nu^{\alpha / (\alpha+2)}$. Our results in the inviscid mixing case rely on the concept of $\rho$-irregularity, first introduced by Catellier and Gubinelli (Stoc. Proc. Appl. 126, 2016) and provide some new insights compared to the behavior predicted by Colombo, Coti Zelati and Widmayer (arXiv:2009.12268, 2020).