Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows

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 et al. ["Optimal stirring strategies for passive scalar mixing," J. Fluid Mech.675, 465 (2011)]10.1017/S0022112011000292 that determines the flow field that instantaneously maximizes the depletion of the H-1 mix-norm. In this work, we bridge some of the gap between the best available a priori analysis and simulation results. After recalling some previous analysis, we present an explicit example demonstrating finite-time perfect mixing with a finite energy constraint on the stirring flow. On the other hand, using a recent result by Wirosoetisno et al. ["Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations," SIAM J. Numer. Anal.50(1), 126-150 (2012)]10.1137/110834901 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. Direct numerical simulations in two dimensions suggest the impossibility of finite-time perfect mixing for flows with fixed power constraint and we conjecture an exponential lower bound on the H^-1 mix-norm in this case. We also discuss some related problems from other areas of analysis that are similarly suggestive of an exponential lower bound for the H-1 mix-norm.