Stability of negative-regularity mixing and applications to stochastic fluids

We consider mixing for a scalar advected by random vector fields, such as solutions to stochastic fluid equations. Our main abstract result is that contraction in expectation of a negative Sobolev norm of the scalar (negative-regularity mixing) is stable under perturbations of the joint scalar-velocity process. We apply the result in three settings: mixing for a hyperviscous Boussinesq-type system where the scalar actively drives the velocity, mixing for a passive scalar advected by two-dimensional Navier-Stokes driven by smooth-in-time stochastic forcing, and a black-box proof that negative-regularity mixing implies enhanced dissipation.