Mixing and Un-mixing by Incompressible Flows

We consider the questions of efficient mixing and un-mixing by incompressible flows which satisfy periodic, no-flow, or no-slip boundary conditions on a square. Under the uniform-in-time constraint $\|\nabla u(\cdot,t)\|_p\leq 1$ we show that any function can be mixed to scale $\epsilon$ in time $O(|\log\epsilon|^{1+\nu_p})$, with $\nu_p=0$ for $p<\tfrac{3+\sqrt 5}2$ and $\nu_p\leq \tfrac 13$ for $p\geq \tfrac{3+\sqrt 5}2$. Known lower bounds show that this rate is optimal for $p\in(1,\tfrac{3+\sqrt 5}2)$. We also show that any set which is mixed to scale $\epsilon$ but not much more than that can be un-mixed to a rectangle of the same area (up to a small error) in time $O(|\log\epsilon|^{2-1/p})$. Both results hold with scale-independent finite times if the constraint on the flow is changed to $\|u(\cdot,t)\|_{\dot W^{s,p}}\leq 1$ with some $s<1$. The constants in all our results are independent of the mixed functions and sets.