Mixing at the Batchelor Scale for White-In-Time Flows

2025/12/03 by Robin Chemnitz, Chemnitz, Robin, Dennis Chemnitz +1
Mathematics · Economics, Econometrics and Finance · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and financial applications #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.2512.04297

Abstract

We consider the mixing properties of solutions to the advection-diffusion equation of a white-in-time velocity field on the 2-dimensional torus with four forced modes. As the diffusivity parameter goes to zero, we show that the almost-sure exponential dissipation rate stays bounded from below. Together with the corresponding upper bound established by Gess and Yaroslavtsev, this constitutes an example of a velocity field for which the Batchelor scale conjecture can be verified. In addition, we characterize the exponential mixing rate without diffusion of this system. Our results are not restricted to two dimensions, and we construct a three-dimensional white-in-time velocity field with the same properties.

Citations

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Related

• Exponential mixing by random cellular flows
• Cellular mixing with bounded palenstrophy
• On the relation between enhanced dissipation time-scales and mixing\n rates
• Exponential self-similar mixing by incompressible flows
• Quantitative dependence of the Pierrehumbert flow's mixing rate on the amplitude