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A universal total anomalous dissipator

Elias Hess-Childs, Keefer Rowan

TL;DR

This work addresses anomalous dissipation for a passive scalar θ^κ advected by a divergence-free velocity field V on the torus, proving the existence of an explicit V with Hölder regularity that enforces asymptotic total dissipation for every mean-zero initial data as the diffusivity $\kappa\to0$. The authors construct a two-cell dissipator from a known perfect-mixing flow and then assemble a universal, self-similar dissipator by space-time tiling and rescaling, achieving unit-time dissipation for any data via a geometric cascade of scales. They establish a precise algebraic dissipation rate $\|\theta^\kappa(1,\cdot)\|_{L^1} \le C\kappa^{(1-\alpha)^2/72}\|\theta_0\|_{TV}$ and show asymptotic total dissipation (limiting energy vanishes) with an explicitly atomic dissipation measure, together with corollaries in $L^p$ and Sobolev norms and adjoint flows. The approach relies on stability estimates via stochastic representations of drift-diffusion equations, a detailed analysis of advection-diffusion versus pure transport, and a maximal spreading result for stochastic trajectories under $V\in L^\infty_t C^\alpha_x$. This provides a rare, explicit, universal mechanism for total dissipation in a passive-scalar setting, offering insight into mixing-driven dissipation in turbulence while highlighting the delicate regularity constraints required for such phenomena.

Abstract

For all $α\in(0,1)$, we construct an explicit divergence-free vector field $V\in L^\infty_tC^α_x \cap C^{\fracα{1-α}}_t L^\infty_x$ so that the solutions to the drift-diffusion equations $$\partial_tθ^κ-κΔθ^κ+V\cdot\nablaθ^κ=0$$ exhibit asymptotic total dissipation for all mean-zero initial data: $\lim_{κ\rightarrow 0}\|θ^κ(1,\cdot)\|_{L^2}=0$. Additionally, we give explicit rates in $κ$ and uniform dependence on initial data.

A universal total anomalous dissipator

TL;DR

This work addresses anomalous dissipation for a passive scalar θ^κ advected by a divergence-free velocity field V on the torus, proving the existence of an explicit V with Hölder regularity that enforces asymptotic total dissipation for every mean-zero initial data as the diffusivity . The authors construct a two-cell dissipator from a known perfect-mixing flow and then assemble a universal, self-similar dissipator by space-time tiling and rescaling, achieving unit-time dissipation for any data via a geometric cascade of scales. They establish a precise algebraic dissipation rate and show asymptotic total dissipation (limiting energy vanishes) with an explicitly atomic dissipation measure, together with corollaries in and Sobolev norms and adjoint flows. The approach relies on stability estimates via stochastic representations of drift-diffusion equations, a detailed analysis of advection-diffusion versus pure transport, and a maximal spreading result for stochastic trajectories under . This provides a rare, explicit, universal mechanism for total dissipation in a passive-scalar setting, offering insight into mixing-driven dissipation in turbulence while highlighting the delicate regularity constraints required for such phenomena.

Abstract

For all , we construct an explicit divergence-free vector field so that the solutions to the drift-diffusion equations exhibit asymptotic total dissipation for all mean-zero initial data: . Additionally, we give explicit rates in and uniform dependence on initial data.

Paper Structure

This paper contains 23 sections, 16 theorems, 153 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Previous work
  4. Anomalous dissipation examples
  5. Connections with other problems
  6. Contributions
  7. Outline of contents
  8. Acknowledgments
  9. Definition of flows and proof outline
  10. Heuristic argument
  11. Constructing the two-cell dissipator
  12. The [ACM19] perfect mixing flow
  13. The two-cell dissipating flow
  14. Sketch of the proof of Theorem \ref{['thm:two-cell-dissipation']}
  15. Constructing the universal total dissipator
  16. ...and 8 more sections

Key Result

Theorem 1.1

For all $\alpha \in (0,1)$, there exists a constant $C(\alpha)>0$ and a divergence-free vector field $V\in L^\infty([0,1], C^\alpha(\mathbb{T}^2)) \cap C^{\frac{\alpha}{1-\alpha}}([0,1], L^\infty(\mathbb{T}^2))$ so that for all mean-zero $\theta_0 \in TV(\mathbb{T}^2)$ and $\kappa>0$, the unique sol

Figures (4)

  • Figure 2.1: The two-cell dissipator.
  • Figure 2.2: The universal total dissipator. The red arrows denote the action of the two-cell dissipator.
  • Figure 2.3: A self-similar mixing flow. The arrows represent the flow of the solution under the transport equation for unit time.
  • Figure 2.4: The left box is $A_0 = B$. The middle box is made up of the collection of sub-boxes $(A_1+x_1)_{x_1\in \Lambda_1}$. The right box is made up of the collection of sub-boxes $(A_2+x_2)_{x_2\in \Lambda_2}$. The $\Gamma$ tracks how the orientation of the box transforms under repeated applications of $\mathcal{R}.$

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 1.2
  • Corollary 1.3
  • Remark 2.1: Dissipation measure
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.6: alberti_exponential_2019
  • Definition 2.7
  • ...and 28 more