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 H1/2 holds sharply with rate r(t) \sim t1/(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 [Stochastic Process. Appl., 126 (2016), pp. 2323-2366], and provide some new insights compared to the behavior predicted by Colombo, Zelati, and Widmayer [Ars Inveniendi Anal. (2021)].