2024/03/28 by William Cooperman, Cooperman, William, Gautam Iyer +3 · 5 citations
Mathematics · #37A25 (Primary) 60J05 #76R99 (Secondary) #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #History and Theory of Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2403.19858
openalex publication_date 2024/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 κ denotes the strength of the diffusion, then we show that with probability at least 1 - o(κN) enhanced dissipation occurs on time scales of order |ln κ|, 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.