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Title:
Exponential mixing for Pierrehumbert’s alternating sine-shear flow with random phase shifts
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Abstract:
The Pierrehumbert alternating sine-shear flow is a classical example of an incompressible flow observed to mix passive scalars. Proving mixing for concrete is a substantial mathematical challenge in the dynamical systems and passive scalar mixing communities, and many open questions remain, even for simple toy models for which mixing is substantiated in numerical experiments. Recently, techniques from random dynamical systems theory have been applied to problems of this kind for typical realizations of incompressible flows subjected to noise (c.f. my previous joint work with Bedrossian and Punshon-Smith). In this talk I will review a recent result, joint with Michele Cot-Zelati and Rishabh Gvalani, proving exponentially fast mixing (a.k.a. H^{-1} decay) for all passive scalars, under the alternating shear flow with randomly chosen phase shifts. To our knowledge, this example provides the first-ever example of a universal exponentially-fast mixer on the periodic box which is uniformly smooth ($C^\infty$) in space and time.
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