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 \|\nabla u(\cdot,t)\|_p\le 1 we show that any function can be mixed to scale \epsilon in time O(|\mathrm {log}\:\epsilon|^{1+\nu_p}) , with \nu_p=0 for p<\tfrac{3+\sqrt 5}2 and \nu_p\le \tfrac 13 for p\ge \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(|\mathrm {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}}\le 1 with some s<1 . The constants in all our results are independent of the mixed functions and sets.