# Meeting Details

Title: Optimal mixing and optimal stirring for fixed energy, fixed power or fixed palenstrophy flows Computational and Applied Mathematics Colloquium Evelyn Lunasin, U. Michigan Mathematics We consider passive scalar mixing by a prescribed divergence-free velocity vector field in a periodic box and address the following question: Starting from a given initial inhomogeneous distribution of passive tracers, and given a certain energy budget, power budget or finite palenstrophy budget, what incompressible flow field best mixes the scalar quantity? We focus on the optimal stirring strategy recently proposed by Lin, Thiffeault and Doering (2011) that determines the flow field that instantaneously maximizes the depletion of the  $H^{-1}$ mix-norm. We present an explicit example demonstrating finite-time perfect mixing with a finite energy constraint on the stirring flow. On the other hand, we establish that the $H^{-1}$ mix-norm decays at most exponentially in time if the two-dimensional incompressible flow is constrained to have constant palenstrophy. Finite-time perfect mixing is thus ruled out when too much cost is incurred by small scale structures in the stirring. We conjecture an exponential lower bound on the $H^{-1}$ mix-norm in the case of finite power constraint and discuss some related problems from other areas of analysis that are similarly suggestive of this conjecture.