Ann. Probab. 50 (1), 241-303, (January 2022) DOI: 10.1214/21-AOP1533
Jacob Bedrossian, Alex Blumenthal, Samuel Punshon-Smith
KEYWORDS: Exponential mixing, quenched correlation decay, stochastic Navier–Stokes equations, passive scalars, spectral theory of Markov semigroups, 37A25, 37A30, 37N10, 76F25, 76D06, 37H15, 37A60, 60H15
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 subjected to nondenegenerate -regular noise for any σ sufficiently large. That is, for all there is a deterministic exponential decay rate such that all mean-zero passive scalars decay in 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 , this exhibits many examples of random velocity fields that are almost-sure exponentially fast mixers.