ALMOST-SURE EXPONENTIAL MIXING OF PASSIVE SCALARS BY THE STOCHASTIC NAVIER-STOKES EQUATIONS

• 成果类型: Article
• 署名作者: Bedrossian, Jacob; Blumenthal, Alex; Punshon-Smith, Samuel
• 署名单位: University System of Maryland; University of Maryland College Park; Brown University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1533
• 发表日期: 2022
• 页码: 241-303
• 关键词: coupling approach; large deviations; ergodicity; THEOREM; sample; FLOWS; fluid; decay

摘要

We deduce almost-sure exponentially fast mixing of passive scalars advected by solutions of the stochastically-forced 2D Navier-Stokes equations and 3D hyper-viscous Navier-Stokes equations in T-d subjected to nondenegenerate H-sigma-regular noise for any sigma sufficiently large. That is, for all s > 0 there is a deterministic exponential decay rate such that all mean-zero H-s passive scalars decay in H-s at this same rate with probability one. This is equivalent to what is known as quenched correlation decay for the Lagrangian flow in the dynamical systems literature. This is a follow-up to our previous work, which establishes a positive Lyapunov exponent for the Lagrangian flow-in general, almost-sure exponential mixing is much stronger than this. Our methods also apply to velocity fields evolving according to finite-dimensional models, for example, Galerkin truncations of Navier-Stokes or the Stokes equations with very degenerate forcing. For all 0 <= k <= infinity, this exhibits many examples of (CtCx infinity)-C-k random velocity fields that are almost-sure exponentially fast mixers.