Enhanced Dissipation via time-modulated velocity fields

TL;DR

This paper investigates how time-modulated velocity fields can significantly enhance dissipation in advection-diffusion processes, providing quantitative decay estimates and demonstrating super-enhanced dissipation under certain conditions.

Contribution

It introduces a framework for deriving decay rates in non-autonomous flows with time-dependent velocity modulation, revealing conditions for super-enhanced dissipation and analyzing switching flow scenarios.

Findings

Super-enhanced dissipation occurs for certain bounded, increasing modulation functions.

Dissipation rates in switching flows are comparable to autonomous cases.

Application of hypocoercivity with time-dependent weights ensures robust decay estimates.

Abstract

Motivated by mixing processes in analytical laboratories, this work investigates enhanced dissipation in non-autonomous flows. We study the evolution of concentrations governed by the advection-diffusion equation, where the velocity field is modelled as the product of a shear flow and a time-dependent modulation function $\xi(t)$. The main objective of this paper is to derive quantitative estimates for the energy decay rates, which are shown to depend sensitively on the properties of $\xi$. We identify a class of time-dependent functions that are bounded by increasing functions, for which we demonstrate super-enhanced dissipation, characterized by energy decay rates faster than those observed in autonomous cases. Additionally, we explore the case of velocity fields that may be switched on and off over time. Here, the dissipation rates are comparable to those of autonomous flows. To illustrate our results, we analyse two prototypical flows of this class: one exhibiting a gradual turn-on and turn-off phase, and another that undergoes a significant acceleration following a slow initial activation phase. Both results are achieved through the application of the hypocoercivity framework, adapted to an augmented functional with time-dependent weights. These weights are designed to dynamically counteract the potential growth of $\xi$, ensuring robust decay estimates.