Exponential scalar mixing for the 2D Navier-Stokes equations with degenerate stochastic forcing

TL;DR

This paper proves exponential mixing of passive scalars in 2D stochastic Navier-Stokes equations with limited forcing modes, using advanced probabilistic and PDE techniques.

Contribution

It introduces a novel combination of the asymptotic strong Feller property and mixing theory to establish exponential mixing under degenerate stochastic forcing.

Findings

Exponential mixing is achieved for passive scalars in 2D stochastic Navier-Stokes.

The method applies to systems with finitely many forced modes satisfying hypoellipticity.

The proof combines techniques from Hairer and Mattingly with Bedrossian et al.'s mixing theory.

Abstract

We show exponential mixing of passive scalars advected by a solution to the stochastic Navier-Stokes equations with finitely many (e.g. four) forced modes satisfying a hypoellipticity condition. Our proof combines the asymptotic strong Feller framework of Hairer and Mattingly with the mixing theory of Bedrossian, Blumenthal, and Punshon-Smith.