Relaxation enhancement by controlling incompressible fluid flows

TL;DR

This paper introduces a PDE-control approach to enhance diffusive mixing in passive scalar fields by using time-dependent vector fields derived from controllable incompressible Euler flows, avoiding preset fields.

Contribution

It develops a novel control framework for the Euler equations to generate relaxation-enhancing flows, enabling improved mixing without predefined velocity fields.

Findings

Established approximate controllability of Euler flows on 

Proved enhanced relaxation of passive scalars via controllable flows

Demonstrated control of Euler system with localized forces

Abstract

We propose a PDE-controllability based approach to the enhancement of diffusive mixing for passive scalar fields. Unlike in the existing literature, our relaxation enhancing fields are not prescribed $\textit{ab initio}$ at every time and at every point of the spatial domain. Instead, we prove that time-dependent relaxation enhancing vector fields can be obtained as $\textit{state trajectories of control systems described by the incompressible Euler equations}$ either driven by finite-dimensional controls or by controls localized in space. The main ingredient of our proof is a new approximate controllability theorem for the incompressible Euler equations on $\mathbb{T}^2$, ensuring the approximate tracking of the full state all over the considered time interval. Combining this with a continuous dependence result yields enhanced relaxation for the passive scalar field. Another essential tool in our analysis is the exact controllability of the incompressible Euler system driven by spatially localized forces.