Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows

TL;DR

This paper investigates how shear flows enhance dissipation and regularization in 2D drift-diffusion equations, revealing new effects in small noise inviscid limits and demonstrating instant Gevrey regularity with partial diffusion.

Contribution

It provides a detailed analysis of enhanced dissipation, hypoelliptic regularization, and small noise limits in shear flows, introducing new spectral gap techniques and extending understanding of inviscid limits.

Findings

Quantifies enhanced dissipation due to shear flows.

Shows instant Gevrey regularity with partial diffusion.

Establishes convergence of viscous invariant measures to inviscid limits.

Abstract

We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic $\mathbb{T}^2$ setting and the case of a bounded channel $\mathbb{T}\times [0,1]$ with no-flux boundary conditions. In the infinite P\'eclet number limit (diffusivity $\nu\to 0$), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Gevrey regular even with only partial diffusion. The proofs rely on localized spectral gap inequalities and ideas from hypocoercivity with an augmented energy functional with weights replaced by pseudo-differential operators (of a rather simple form). As an application, we study small noise inviscid limits of invariant measures of stochastic perturbations of passive scalars, and show that the classical Freidlin scaling between noise and diffusion can be modified. In particular, although statistically stationary solutions blow up in $H^1$ in the limit $\nu\to0$, we show that viscous invariant measures still converge to a unique inviscid measure.