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 ‖∇u(·, t)‖p ≤ 1 we show that any function can be mixed to scale ε in time O(| log ε|1+νp), with νp = 0 for p < 3+ √ 5 2 and νp ≤ 13 for p ≥ 3+ √ 5 2 . Known lower bounds show that this rate is optimal for p ∈ (1, 3+ √ 5 2 ). We also show that any set which is mixed to scale ε 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 ε|2−1/p). Both results hold with scale-independent finite times if the constraint on the flow is changed to ‖u(·, t)‖Ẇ s,p ≤ 1 with some s < 1. The constants in all our results are independent of the mixed functions and sets.