In this talk, I consider a model for mixing passive scalars under an imcompressible flow subject to a constraint on the viscous dissipation budget. I present rigorous lower bounds on two type of mixing norms, a Monge-Kantorovich distance and the $H^{-1}$ norm, which show that mixing cannot proceed faster than exponentially in time. The exponential decay rate is uniform in the initial data. I finally discuss some recent examples which indicate that these bounds are optimal.
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