Does a fast mixer really exist?
In this paper, we are concerned about the limit behavior of the decay rate of variance of a passive and diffusive scalar in a flow field as the diffusivity of the scalar goes to zero. Motivated by the concept of the fast dynamo in the dynamo theory, we term a flow as fast mixer if the decay rate remains away from zero as the diffusivity goes to zero. We first repeat numerical simulations with flow maps and velocity fields used in the existing literature, including the lattice map, the 1D baker's map, and the sinusoidal shear flow. Our simulations shows that, in all cases, the decay rate tends to zero as the diffusivity goes to zero. For the closed flows in a bounded domain, we then theoretically proved this result under certain plausible conditions on the flows. For the open flows in the whole space, we show that the effective diffusivity matrix tends to zero in the limit without the conditions for the closed flow. In conclusion, although a fast mixer might exist, it could be very difficult to find one.
Liu
Weijiu
W
Department of Mathematics, University of Central Arkansas, 201 Donaghey Avenue, Conway, Arkansas 72034, USA. liuweijiu@hotmail.com