Almost-sure exponential mixing of passive scalars by the stochastic Navier–Stokes equations

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