On the energy-constrained optimal mixing problem for one-dimensional initial configurations

TL;DR

This paper investigates the limits of how quickly a passive scalar can be perfectly mixed in a one-dimensional setting under energy constraints, showing the lower bounds are sharp for certain initial configurations.

Contribution

It establishes a new sharp lower bound for mixing time in a weak solution setting and introduces convex hull inequalities as a novel tool for analyzing optimal mixing.

Findings

Lower bound for mixing time is sharp for certain initial configurations.

Perfect mixing in finite time is not always possible under energy constraints.

Convex hull inequalities are effective in studying optimal mixing problems.

Abstract

We consider the problem of mixing a passive scalar in a periodic box by incompressible vector fields subject to a fixed energy constraint. In that setting a lower bound for the time in which perfect mixing can be achieved has been given by Lin, Thiffeault, Doering \cite{Lin_Thiffeault_Doering_2011}. While examples by Depauw \cite{Depauw} and Lunasin et al. \cite{Lunasin_etal_2012} show that perfect mixing in finite time is indeed possible, the question regarding the sharpness of the lower bound from \cite{Lin_Thiffeault_Doering_2011} remained open. In the present article we give a negative answer for the special class of initial configurations depending only on one spatial coordinate. The new lower bound holds true for distributional solutions satisfying only the uniform energy constraint for the velocity field and a weak compatibility condition for the passive scalar coming from the transport equation. In that weak setting we also provide an example for which the new bound is sharp. As a new ingredient in the investigation of optimal mixing we utilize the convex hull inequalities of the transport equation with constraints when seen as a differential inclusion.