JournalsjemsVol. 19, No. 7pp. 1911–1948

Mixing and un-mixing by incompressible flows

  • Andrej Zlatoš

    Georgia Institute of Technology, Atlanta, USA
  • Yao Yao

    University of Wisconsin, Madison, USA
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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)p1\|\nabla u(\cdot,t)\|_p\le 1 we show that any function can be mixed to scale ϵ\epsilon in time O(logϵ1+νp)O(|\mathrm {log}\:\epsilon|^{1+\nu_p}), with νp=0\nu_p=0 for p<3+52p<\tfrac{3+\sqrt 5}2 and νp13\nu_p\le \tfrac 13 for p3+52p\ge \tfrac{3+\sqrt 5}2. Known lower bounds show that this rate is optimal for p(1,3+52)p\in(1,\tfrac{3+\sqrt 5}2). We also show that any set which is mixed to scale ϵ\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(logϵ21/p)O(|\mathrm {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˙s,p1\|u(\cdot,t)\|_{\dot W^{s,p}}\le 1 with some s<1s<1. The constants in all our results are independent of the mixed functions and sets.

Cite this article

Andrej Zlatoš, Yao Yao, Mixing and un-mixing by incompressible flows. J. Eur. Math. Soc. 19 (2017), no. 7, pp. 1911–1948

DOI 10.4171/JEMS/709