Geodesic
Mixing and un-mixing by incompressible flows
Journal of the European Mathematical Society, Tome 19 (2017) no. 7, pp. 1911-1948 Cet article a éte moissonné depuis la source EMS Press

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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 ∥∇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 p23+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 s1. The constants in all our results are independent of the mixed functions and sets.
DOI : 10.4171/jems/709
Classification : 35-XX, 76-XX
Keywords: Incompressible flow, mixing, un-mixing 
@article{JEMS_2017_19_7_a0,
     author = {Yao Yao and Andrej Zlato\v{s}},
     title = {Mixing and un-mixing by incompressible flows},
     journal = {Journal of the European Mathematical Society},
     pages = {1911--1948},
     year = {2017},
     volume = {19},
     number = {7},
     doi = {10.4171/jems/709},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/709/}
}
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Yao Yao; Andrej Zlatoš. Mixing and un-mixing by incompressible flows. Journal of the European Mathematical Society, Tome 19 (2017) no. 7, pp. 1911-1948. doi: 10.4171/jems/709

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