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I will address the problem of the optimal mixing of a passive scalarÌýunder the action of an incompressible flow in two space dimensions.ÌýThe scalar solves the continuity equation with a divergence-freeÌývelocity field which satisfies a bound in the Sobolev space $W^{s,p}$,Ìýwhere $s \geq 0$ and $1\leq p\leq \infty$. The mixing properties areÌýgiven in terms of a characteristic length scale, called the mixingÌýscale. We consider two notions of mixing scale, one functional,Ìýexpressed in terms of the homogeneous Sobolev norm $\dot H^{-1}$, theÌýother geometric, related to rearrangements of sets. We study rates ofÌýdecay in time of both scales under self-similar mixing. For the caseÌý$s=1$ and $1 \leq p \leq \infty$ (including the Lipschitz case, andÌýthe case of physical interest of enstrophy-constrained flows), weÌýpresent examples of velocity fields and initial configurations for theÌýscalar that saturate the exponential lower bound established inÌýprevious works for the decay in time of both scales. We also obtainÌýseveral consequences for the geometry of regular Lagrangian flowsÌýassociated to Sobolev velocity fields and for the loss of regularityÌýfor continuity equations with non-Lipschitz velocity field. The talkÌýwill be based on joint works with G. Alberti (University of Pisa,ÌýItaly) and A. L. Mazzucato (Penn State).