## Mixing and un-mixing by incompressible flows

#### Yao YaoUniversity of Wisconsin-Madison

In this talk, we consider the questions of efficient mixing and un-mixing by incompressible flows, under the constraint that the W^{1,p} Sobolev norm of flow is bounded uniformly in time. We construct some explicit flows to show that any initial data can be mixed to scale epsilon in time O(|log(epsilon)|) for p<(3+sqrt{5})/2 and in time O(|log(epsilon)|^{1/3}) for p>=(3+sqrt{5})/2. Known lower bounds show that this rate is optimal for p between 1 and (3+sqrt{5})/2. For un-mixing, we 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). The constants in all our results are independent of the initial data. This is a joint work with Andrej Zlatos.

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