Mixing for generic rough shear flows

TL;DR

This paper investigates the mixing and diffusion of passive scalars in generic rough shear flows, establishing sharp bounds on mixing rates and enhanced dissipation, and introduces new insights into the behavior of such flows.

Contribution

It provides the first sharp bounds for mixing and dissipation rates in generic rough shear flows, utilizing the concept of ta-irregularity.

Findings

Inviscid mixing in H^{1/2} occurs with rate t^{1/(2 lpha)}

Enhanced dissipation occurs with rate mbda^{lpha/(lpha+2)}

Introduces new insights into flow behavior using ta-irregularity

Abstract

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).