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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 we show that any function can be mixed to scale in time , with for and for . Known lower bounds show that this rate is optimal for . We also show that any set which is mixed to scale but not much more than that can be un-mixed to a rectangle of the same area (up to a small error) in time . Both results hold with scale-independent finite times if the constraint on the flow is changed to with some . The constants in all our results are independent of the mixed functions and sets.
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Andrej Zlatoš, Yao Yao, Mixing and un-mixing by incompressible flows. J. Eur. Math. Soc. 19 (2017), no. 7, pp. 1911–1948DOI 10.4171/JEMS/709