Instantaneous Total Enhanced Dissipation For Very Rough Shear Flows

TL;DR

This paper demonstrates that very rough shear flows cause the dissipation rate of a passive scalar to become infinite as viscosity approaches zero, extending previous results to more irregular flows in negative Besov spaces.

Contribution

It extends enhanced dissipation results to generic, highly irregular shear flows in negative Besov spaces, providing bounds on dissipation rates and implications for triviality of solutions.

Findings

Dissipation rate increases to infinity as viscosity vanishes.

Upper bounds hold for irregular velocities satisfying Wei's irregularity index.

Vanishing viscosity solutions are trivial for truly rough shear flows.

Abstract

This paper investigates enhanced dissipation for a passive scalar advected by "very rough" horizontal shear flows, described by an advection-diffusion equation on the 2D torus. The authors extend results of Galeati and Gubinelli (2023) to generic flows in negative Besov spaces, proving that the dissipation rate increases to infinity as viscosity vanishes. This is obtained by deriving (non-sharp) upper and lower bounds on the dissipation rate. The upper bound holds for truly irregular velocities, namely those verifying a suitable version of the Wei irregularity index (Wei (2021)). As a by-product, it follows that for truly rough shear flows the vanishing viscosity solution to the corresponding inviscid equation is trivial.