A pathwise approach to the enhanced dissipation of passive scalars advected by shear flows

TL;DR

This paper introduces a probabilistic, pathwise framework using stochastic differential equations and Girsanov's theorem to analyze and quantify the enhanced dissipation of passive scalars in shear flows, including singular and critical cases.

Contribution

It develops a novel probabilistic approach for studying enhanced dissipation in shear flows, extending results to singular and critical shear profiles and providing local decay estimates.

Findings

Recovered classical enhanced dissipation timescales for smooth shear flows on torus

Extended results to radially symmetric shear flows with singularities

Provided local decay rate estimates along streamlines

Abstract

We develop a framework for studying the enhanced dissipation of passive scalars advected by shear flows based on analyzing the particle trajectories of the stochastic differential equation associated with the governing drift-diffusion equation. We consider both shear flows on $\mathbb{T}^2$ and radially symmetric shears on $\mathbb{R}^2$ or the unit disk. Using our probabilistic approach, we are able to recover the well-known enhanced dissipation timescale for smooth shear flows on $\mathbb{T}^2$ with finite-order vanishing critical points [1, 5, 34, 36] and a generalized version of the results for radially symmetric shear flows from [42]. We also obtain results for shear flows with singularities and critical points where the derivative vanishes to infinite order. The proofs are all based on using Girsanov's theorem to reduce enhanced dissipation to a quantitative control problem that can be solved by leveraging the shearing across streamlines. Our method also has the feature that it is local in space, which allows us also to obtain estimates on the precise decay rate of solutions along each streamline in terms of the local shear profile.