Abstract:
I will report on some joint work with G. Alberti (University of Pisa,
Italy) and A. L. Mazzucato (Penn State) regarding the study of 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 Lipschitz case, 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 for
the decay in time of both scales. We also obtain several consequences
for the geometry of regular Lagrangian flows associated to Sobolev
velocity fields and for the loss of regularity for continuity
equations with non-Lipschitz velocity field. |