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DTSTART:20070311T020000
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DTSTAMP:20260406T120838
DTSTART;TZID=America/Detroit:20260410T150000
DTEND;TZID=America/Detroit:20260410T160000
SUMMARY:Lecture / Discussion:AIM Seminar:  Long-time behavior of optimal mixing in an advection--diffusion shell model*
DESCRIPTION:Abstract:  What is the long-time optimal mixing rate achievable under enstrophy-constrained flows when both advection and diffusion are active? To address this question\, we investigate the long-time behavior of optimal mixing in an advection--diffusion equation using a shell model framework\, a reduced model that captures the essential kinematics of advection and diffusion. Our focus is on quantifying the decay of scalar variance\, measured by the shell-model analogue of the $H^{-1}$ mix-norm\, under enstrophy-constrained stirring. We perform long-time computations using both local-in-time (maximizing the instantaneous mixing rate) and global-in-time (maximizing mixedness at a prescribed final time) optimization strategies. For mixing with diffusion ($\kappa>0$)\, the numerical results for both strategies show that the scalar length scale is eventually limited by a generalized Batchelor scale\, in close agreement with theoretical predictions. In this regime\, the shell-model analogue of the $H^{-1}$ mix-norm decays exponentially in time at a rate independent of the diffusivity $\kappa$. Compared with the purely advective case ($\kappa=0$)\, diffusion significantly enhances the long-time mixing rate\; moreover\, increasing diffusivity further improves mixing efficiency by reducing the prefactor. Guided by these numerical observations\, we derive new conditional lower bounds on the decay rate of the $H^{-1}$ norm that are strictly independent of the diffusivity parameter $\kappa$ when $\kappa>0$. The computed decay rates are consistent with our theoretical bounds.\n\nThis is joint work with Baole Wen\, Christian Seis\, and Charles R. Doering.\n\nContact:  Silas Alben
UID:141683-21889181@events.umich.edu
URL:https://events.umich.edu/event/141683
CLASS:PUBLIC
STATUS:CONFIRMED
CATEGORIES:Mathematics
LOCATION:East Hall - 1084
CONTACT:
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