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

Björn Gebhard

TL;DR

This paper analyzes energy-constrained optimal mixing of a passive scalar on the torus, showing that the classical lower bound $T_{mix}\ge E^{-1/2}\|\rho_0\|_{H^{-1}}$ is not sharp for one-dimensional initial data. By passing to vertical averages and formulating a differential inclusion, the authors derive a one-dimensional variational problem whose maximizers yield a closed-form decay bound and a sharper lower bound $T_{mix}^{1sub}\ge h_{max}(L^{d-1}/E)^{1/2}\sqrt{2}\,S(\alpha_0)$ with $h_{max}=L^{3/2}/\sqrt{48}$ and $\alpha_0=(1-\sqrt{1-q_0^2})^{1/2}$. They prove sharpness for a maximally unmixed initial datum $\hat{\rho}_0$ on an averaged level and then realize nearly optimal mixing times in the full weak setting via convex integration, establishing that the averaged bound can be achieved asymptotically by distributional solutions. The work combines convex-hull analysis of the transport constraint, symmetric and odd rearrangements, and a nonlocal 1D conservation-law framework to illuminate the limits of energy-based lower bounds in optimal mixing. Overall, the paper provides a precise mechanism for improving mixing-time bounds in 1D settings and demonstrates sharpness in a broad, non-smooth regime with potential extensions to more general initial data and constraints.

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.

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

TL;DR

This paper analyzes energy-constrained optimal mixing of a passive scalar on the torus, showing that the classical lower bound is not sharp for one-dimensional initial data. By passing to vertical averages and formulating a differential inclusion, the authors derive a one-dimensional variational problem whose maximizers yield a closed-form decay bound and a sharper lower bound with and . They prove sharpness for a maximally unmixed initial datum on an averaged level and then realize nearly optimal mixing times in the full weak setting via convex integration, establishing that the averaged bound can be achieved asymptotically by distributional solutions. The work combines convex-hull analysis of the transport constraint, symmetric and odd rearrangements, and a nonlocal 1D conservation-law framework to illuminate the limits of energy-based lower bounds in optimal mixing. Overall, the paper provides a precise mechanism for improving mixing-time bounds in 1D settings and demonstrates sharpness in a broad, non-smooth regime with potential extensions to more general initial data and constraints.

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.
Paper Structure (23 sections, 25 theorems, 162 equations)

This paper contains 23 sections, 25 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Lower bounds
  3. Weak setting
  4. Sharpness
  5. Outline of the strategy
  6. Background and related results
  7. Further questions
  8. Improved lower bound for one-dimensional initial data
  9. Vertical averages
  10. General one-dimensional subsolutions
  11. Estimating the decay rate: Steepest descent
  12. Estimating the decay rate: Variational problems
  13. Symmetric decreasing rearrangement
  14. Odd rearrangement
  15. Problem 1
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $\rho_0\in L^\infty(\mathbb{T}^d)$ satisfy eq:rho_0_conditions and assume that $\rho_0$ depends only on one space-variable. Then $\left\Vert\rho_0\right\Vert_{H^{-1}(\mathbb{T}^d)}\in [0, H_{max}]$ with $H_{max}:=L^{\frac{d+2}{2}}/\sqrt{48}$ and there holds where $r_0:=\left\Vert\rho_0\right\Vert_{H^{-1}(\mathbb{T}^d)}/H_{max}\in[0,1]$ and

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 45 more