Instantaneous Total Enhanced Dissipation For Very Rough Shear Flows
Marco Romito, Leonardo Roveri
TL;DR
This work addresses enhanced dissipation for a passive scalar transported by very rough horizontal shear flows on the 2D torus, focusing on flows that live in negative Besov spaces. The authors develop a hypoelliptic framework, deploy Feynman–Kac and Itô techniques, and introduce a generalized Wei irregularity index to quantify roughness, proving that the dissipation rate $r(\nu)$ diverges as $\nu\to0$ for sufficiently irregular velocities. They establish both upper and lower bounds on $r(\nu)$ and show that diffusion enhancement occurs with rate $r(\nu) \gtrsim \nu^{\alpha/(\alpha+2)}$ under an $\alpha$-irregularity condition, with the irregular velocity set shown to be prevalent in $B_{1,\infty}^{\alpha}$. The analysis links Wei’s irregularity to Hölder roughness and stochastic prevalence, concluding that almost every velocity in these Besov spaces produces enhanced dissipation, with implications for inviscid limits and mixing in rough 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.