Publication 24-CNA-004
A Harris Theorem for Enhanced Dissipation, and an Example of Pierrehumbert
William Cooperman
Courant Institute of Mathematical Sciences
New York University
NY 10003
bill@cprmn.org
Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu
Seungjae Son
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA, 15213
seungjas@andrew.cmu.edu
Abstract: In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium - a phenomenon known as
enhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain
Harris conditions. If
k denotes the strength of the diffusion, then we show that with probability at least 1 -
o(kN) enhanced dissipation occurs on time scales of order |ln
k|, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surely
independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert '94.
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