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Norm Growth, Non-uniqueness, and Anomalous Dissipation in Passive Scalars

Tarek M. Elgindi, Kyle Liss

TL;DR

The paper constructs a divergence-free velocity field on the torus with near-Lipschitz spatial regularity that induces anomalous dissipation for the advection-diffusion equation for every smooth initial data. The approach reduces anomalous dissipation to an $H^1$-growth mechanism for the transport equation, implemented via a carefully designed sequence of alternating shear maps and a forward-backward growth argument, complemented by a balanced-growth upper bound. It also shows that, with a modification of the velocity field, one can obtain scalar regularity bounds approaching the Obukhov-Corrsin threshold, linking dissipation anomalies to turbulence-style scaling. The study situates these constructions among prior works on anomalous dissipation and non-uniqueness, discusses potential extensions to autonomous flows and higher dimensions, and poses open questions about regularity thresholds and forward-backward dynamics in this context.

Abstract

We construct a divergence-free velocity field $u:[0,T] \times \mathbb{T}^2 \to \mathbb{R}^2$ satisfying $$u \in C^\infty([0,T];C^α(\mathbb{T}^2)) \quad \forall α\in [0,1)$$ such that the corresponding drift-diffusion equation exhibits anomalous dissipation for every smooth initial data. We also show that, given any $α_0 < 1$, the flow can be modified such that it is uniformly bounded only in $C^{α_0}(\mathbb{T}^2)$ and the regularity of solutions satisfy sharp (time-integrated) bounds predicted by the Obukhov-Corrsin theory. The proof is based on a general principle implying $H^1$ growth for all solutions to the transport equation, which may be of independent interest.

Norm Growth, Non-uniqueness, and Anomalous Dissipation in Passive Scalars

TL;DR

The paper constructs a divergence-free velocity field on the torus with near-Lipschitz spatial regularity that induces anomalous dissipation for the advection-diffusion equation for every smooth initial data. The approach reduces anomalous dissipation to an -growth mechanism for the transport equation, implemented via a carefully designed sequence of alternating shear maps and a forward-backward growth argument, complemented by a balanced-growth upper bound. It also shows that, with a modification of the velocity field, one can obtain scalar regularity bounds approaching the Obukhov-Corrsin threshold, linking dissipation anomalies to turbulence-style scaling. The study situates these constructions among prior works on anomalous dissipation and non-uniqueness, discusses potential extensions to autonomous flows and higher dimensions, and poses open questions about regularity thresholds and forward-backward dynamics in this context.

Abstract

We construct a divergence-free velocity field satisfying such that the corresponding drift-diffusion equation exhibits anomalous dissipation for every smooth initial data. We also show that, given any , the flow can be modified such that it is uniformly bounded only in and the regularity of solutions satisfy sharp (time-integrated) bounds predicted by the Obukhov-Corrsin theory. The proof is based on a general principle implying growth for all solutions to the transport equation, which may be of independent interest.
Paper Structure (24 sections, 12 theorems, 117 equations, 1 figure)

This paper contains 24 sections, 12 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Previous works and discussion
  4. Previous work
  5. Discussion
  6. Main ideas of the proof
  7. Sobolev space and Fourier analysis conventions
  8. Proof of main theorems
  9. Criteria for anomalous dissipation
  10. Construction and regularity of the velocity field
  11. Definition of $u:[0,T_*]\times \mathbb{T}^2 \to \mathbb{R}^2$
  12. Choice of parameters and regularity
  13. Norm growth
  14. Proof of discrete-time $H^1$ growth
  15. Proof of continuous time $H^1$ growth
  16. ...and 9 more sections

Key Result

Theorem 1

Fix $T > 0$. There exists a divergence-free velocity field $u:[0,T] \times \mathbb{T}^2 \to \mathbb{R}^2$ with such that, for every mean-zero and smooth initial data $f_0,$ the solutions $f^\kappa$ exhibit anomalous dissipation on $[0,T]$. Solutions to the corresponding transport equation are non-unique while $u$ satisfies for all $t\in [0,T]$ and $z,z'\in\mathbb{T}^2$, with $\omega(s)=s(1+|\log

Figures (1)

  • Figure 1: Numerical simulation of the transport equation \ref{['eq:AE']} with $u$ a regularized version of the alternating shear flow used in the proof of Theorem \ref{['thrm:main']} and initial condition $f_0(x,y) = \sin(x)$, displayed in the upper left image. Moving from left to right, the scalar is shown after each successive shearing direction is applied.

Theorems & Definitions (26)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2
  • Lemma 1.1
  • Proposition 2.1
  • Lemma 2.2: $H^1$ growth
  • Lemma 2.3
  • ...and 16 more