Almost-sure enhanced dissipation and uniform-in-diffusivity exponential mixing for advection-diffusion by stochastic Navier-Stokes

• 成果类型: Article
• 署名作者: Bedrossian, Jacob; Blumenthal, Alex; Punshon-Smith, Sam
• 署名单位: University System of Maryland; University of Maryland College Park; University System of Georgia; Georgia Institute of Technology; Brown University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-01010-8
• 发表日期: 2021
• 页码: 777-834
• 关键词: passive scalars; lagrangian chaos; power spectra; ergodicity; fluid; hypoellipticity; CONVERGENCE; EQUATIONS; transport; sample

摘要

We study the mixing and dissipation properties of the advection-diffusion equation with diffusivity 0 < kappa << 1 and advection by a class of random velocity fields on T-d, d = {2, 3}, including solutions of the 2D Navier-Stokes equations forced by sufficiently regular-in-space, non-degenerate white-in-time noise. We prove that the solution almost surelymixes exponentially fast uniformly in the diffusivity kappa. Namely, that there is a deterministic, exponential rate (independent of kappa) such that all meanzero H-1 initial data decays exponentially fast in H-1 at this rate with probability one. This implies almost-sure enhanced dissipation in L-2. Specifically that there is a deterministic, uniform-in-kappa, exponential decay in L-2 after time t greater than or similar to vertical bar log kappa vertical bar|. Both the O(vertical bar log kappa vertical bar) time-scale and the uniform-in-kappa exponential mixing are optimal for Lipschitz velocity fields. This work is also a major step in our program on scalar mixing and Lagrangian chaos necessary for a rigorous proof of the Batchelor power spectrum of passive scalar turbulence.