  This article addresses mixing and diffusion properties of passive scalars
advected by rough ($C^\alpha$) shear flows. We show that in general, one cannot
expect a rough shear flow to increase the rate of inviscid mixing to more than
that of a smooth shear without critical points. On the other hand, diffusion
may be enhanced at a much faster rate. This shows that in the setting of low
regularity, the interplay between inviscid mixing properties and enhanced
dissipation is more intricate, and in fact contradicts some of the natural
heuristics that are valid in the smooth setting.
