Quantitative mixing and dissipation enhancement property of Ornstein-Uhlenbeck flow

Abstract

This work deals with mixing and dissipation ehancement for the solution of advection-diffusion equation driven by a Ornstein-Uhlenbeck velocity field. We are able to prove a quantitative mixing result, uniform in the diffusion parameter, and enhancement of dissipation over a finite time horizon.

From additive to transport noise in 2D fluid dynamics

“…Let us explain what is added to these works by the present paper. Concerning the passive dynamics, several Wong-Zakai type results of convergence to the white noise transport in Stratonovich form have been proved before (see also [38] for a dissipation enhancement result due to the presence of a Stratonovich-to-Itō corrector), but this seems to be the first work where the velocity field approximating the white noise one is the solution of a nonlinear fluid mechanics equation. Concerning the active dynamics, the results contained in this paper extend and make more precise our previous work [21]: (i) some details in the proof of [21,Proposition 4.1], which after publication appeared not precise, are fixed here in Theorem 2.12; (ii) more importantly, the term u t • ∇ξ t was absent in [21], which therefore should be interpreted more along the research lines of model reduction, inspired by Majda et al [34], instead of multiscale analysis of the full problem.…”

From additive to transport noise in 2D fluid dynamics

“…Let us explain what is added to these works by the present paper. Concerning the passive dynamics, several Wong-Zakai type results of convergence to the white noise transport in Stratonovich form have been proved before (see also [41] for a dissipation enhancement result due to the presence of a Stratonovich-to-Itō corrector), but this seems to be the first work where the velocity field approximating the white noise one is the solution of a nonlinear fluid mechanics equation. Concerning the active dynamics, the results contained in this paper extend and make more precise our previous work [24]: i) some details in the proof of [24,Proposition 4.1], which after publication appeared not precise, are fixed here in Theorem 2.11; ii) more importantly, the term u ǫ t • ∇ξ ǫ t was absent in [24], which therefore should be interpreted more along the research lines of model reduction, inspired by [37], instead of multiscale analysis of the full problem.…”