Analysis & PDE: "Almost-sure exponential mixing of scalars by stochastic Navier-Stokes and other stochastic fluid models with applications to passive scalar turbulence"
In 1959, Batchelor predicted that passive scalars advected in fluids at finite Reynolds number with small diffusivity κ should display a |k|−1 power spectrum over a small-scale inertial range in a statistically stationary experiment. This prediction has been experimentally and numerically tested extensively in the physics and engineering literature and is a core prediction of passive scalar turbulence. Together with Alex Blumenthal and Sam Punshon-Smith, we have provided the first mathematically rigorous proof of this prediction for a scalar field evolving by advection-diffusion in a fluid governed by the 2D Navier-Stokes equations and 3D hyperviscous Navier-Stokes equations in a periodic box subjected to stochastic forcing at arbitrary Reynolds number. The main mathematical step in this proof is a precise understanding of the mixing of passive scalars by the Lagrangian flow map. In particular, we show that almost-surely and uniformly in diffusivity, the advection diffusion equation transfers information from low to high frequency exponentially fast -- known as (almost-sure) exponential mixing. These results are proved by studying the Lagrangian flow map using infinite dimensional extensions of ideas from random dynamical systems.