Mixing and un-mixing by incompressible flows

摘要

We consider the questions of efficient mixing and un-mixing by incompressible flows that satisfy periodic, no-flow, or no-slip boundary conditions on a square. Under the uniform-in-time constraint ||del u(., t)||(p) <= 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 < (3 + root 5)/2 and nu(p) <= 1/3 for p >= (3 + root 5)/2. Known lower bounds show that this rate is optimal for p is an element of (1, (3 + root 5)/2). We also show that any set that is mixed to scale epsilon but not much more 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(., t)||((W) over dot) (s,p) <= 1 with some s < 1. The constants in all our results are independent of the mixed functions and sets.