  We study the problem of the optimal mixing of a passive scalar under the
action of an incompressible flow in two space dimensions. The scalar solves the
continuity equation with a divergence-free velocity field, which satisfies a
bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq
\infty$. The mixing properties are given in terms of a characteristic length
scale, called the mixing scale. We consider two notions of mixing scale, one
functional, expressed in terms of the homogeneous Sobolev norm $\dot H^{-1}$,
the other geometric, related to rearrangements of sets. We study rates of decay
in time of both scales under self-similar mixing. For the case $s=1$ and $1
\leq p \leq \infty$ (including the case of Lipschitz continuous velocities, and
the case of physical interest of enstrophy-constrained flows), we present
examples of velocity fields and initial configurations for the scalar that
saturate the exponential lower bound, established in previous works, on the
time decay of both scales. We also present several consequences for the
geometry of regular Lagrangian flows associated to Sobolev velocity fields.
