Mixing and Un-mixing by Incompressible Flows

Abstract

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 $\Vert \nabla u(\cdot ,t){\Vert }_p\le 1$ we show that any function can be mixed to scale $\epsilon$ in time $O(|\log \epsilon {|}^{1+{\nu }_p})$, with ${\nu }_p=0$ for $p<\frac{3+\sqrt{5}}{2}$ and ${\nu }_p\le \frac{1}{3}$ for $p\ge \frac{3+\sqrt{5}}{2}$. Known lower bounds show that this rate is optimal for $p\in (1,\frac{3+\sqrt{5}}{2})$. 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 $\Vert u(\cdot ,t){\Vert }_{{\dot{W}}^{s,p}}\le 1$ with some $s<1$. The constants in all our results are independent of the mixed functions and sets.

37 pages, 5 figures

References in corpus (1)

• A Lemma and a Conjecture on the Cost of Rearrangements