Asymptotic decay in chaotic flows

In the presence of advection and diffusion in a bounded domain the concentration field becomes more and more uniform in space. The distribution can be characterized by its variance, that in the case of chaotic advection decays exponentially in time. 

In the simplest case when chaotic advection is generated by a time-independent flow (e.g. in a three dimensional system) the asymp- 

Since the average concentration, or total amount, is conserved by advection and diffusion, a zero eigenvalue always exists. The asymptotic decay of the non-uniform component of the concentration field is determined by the second, i.e. the largest non-zero eigenvalue 

The situation is similar in the case of time-periodic flows (Liu and Haller, 2004), where the eigenmodes are also time-dependent. The same advection and diffusion process is applied to the concentration field during each period T, that is given by the linear propagator 

Assuming that this operator has a discrete spectrum of time-periodic eigenfunctions Ci that satisfy 

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