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(∣logϵ∣1+νp), with νp=0 for p<23+5 and
νp≤31 for p≥23+5. Known lower bounds show
that this rate is optimal for p∈(1,23+5). 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(∣logϵ∣2−1/p). Both results hold with scale-independent finite
times if the constraint on the flow is changed to ∥u(⋅,t)∥W˙s,p≤1 with some s<1. The constants in all our results are
independent of the mixed functions and sets.Comment: 37 pages, 5 figure