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On anomalous dissipation induced by transport noise

Antonio Agresti

TL;DR

The paper shows that transport noise, when localized at high frequencies and tuned with viscosity or diffusivity, can induce anomalous dissipation in both 2D NSEs (enstrophy) and diffusion equations (energy of passive scalars) even as the dissipation coefficients vanish. The main technical advance is a refined scaling-limit analysis powered by stochastic Meyers' estimates, which yields uniform-in-time convergence in spaces with positive regularity, enabling precise lower bounds on dissipation via an effective eddy-diffusion limit. The results provide a homogenization-style interpretation: unresolved small-scale fluctuations act as an effective diffusion that prevents complete energy retention, thereby producing nontrivial dissipation in high-Reynolds regimes. The findings illuminate a new dissipation mechanism in stochastic fluid dynamics, distinct from classical Kraichnan–Batchelor pictures, and they furnish quantitative control over how noise intensity and spectral localization influence dissipation.

Abstract

In this paper, we show that suitable transport noises produce anomalous dissipation of both enstrophy of solutions to 2D Navier-Stokes equations and of energy of solutions to diffusion equations in all dimensions. The key ingredients are Meyers' type estimates for SPDEs with transport noise, which are combined with recent scaling limits for such SPDEs. The former enables us to establish, for the first time, uniform-in-time convergence in a space of positive smoothness for such scaling limits. Compared to previous work, one of the main novelties is that anomalous dissipation might take place even in the presence of a transport noise of arbitrarily small intensity. Physical interpretations of our results are also discussed.

On anomalous dissipation induced by transport noise

TL;DR

The paper shows that transport noise, when localized at high frequencies and tuned with viscosity or diffusivity, can induce anomalous dissipation in both 2D NSEs (enstrophy) and diffusion equations (energy of passive scalars) even as the dissipation coefficients vanish. The main technical advance is a refined scaling-limit analysis powered by stochastic Meyers' estimates, which yields uniform-in-time convergence in spaces with positive regularity, enabling precise lower bounds on dissipation via an effective eddy-diffusion limit. The results provide a homogenization-style interpretation: unresolved small-scale fluctuations act as an effective diffusion that prevents complete energy retention, thereby producing nontrivial dissipation in high-Reynolds regimes. The findings illuminate a new dissipation mechanism in stochastic fluid dynamics, distinct from classical Kraichnan–Batchelor pictures, and they furnish quantitative control over how noise intensity and spectral localization influence dissipation.

Abstract

In this paper, we show that suitable transport noises produce anomalous dissipation of both enstrophy of solutions to 2D Navier-Stokes equations and of energy of solutions to diffusion equations in all dimensions. The key ingredients are Meyers' type estimates for SPDEs with transport noise, which are combined with recent scaling limits for such SPDEs. The former enables us to establish, for the first time, uniform-in-time convergence in a space of positive smoothness for such scaling limits. Compared to previous work, one of the main novelties is that anomalous dissipation might take place even in the presence of a transport noise of arbitrarily small intensity. Physical interpretations of our results are also discussed.
Paper Structure (22 sections, 17 theorems, 167 equations)

This paper contains 22 sections, 17 theorems, 167 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Anomalous dissipation results
  3. Heuristics for $\nu$-dependent transport noise
  4. Strategy, novelty, physical interpretation and comparison
  5. Strategy
  6. Novelty -- Improved scaling limits via Meyers' estimates
  7. Physical interpretation -- Connection with homogenization
  8. Further comparison with the literature
  9. Preliminaries
  10. Notation
  11. Structure of the noise
  12. Solution concept and well-posedness
  13. Anomalous dissipation of energy -- Passive scalars
  14. Scaling limit at a fixed diffusivity
  15. Proof of Theorem \ref{['t:anomalous_diss_diff']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $N\geq 1$ and $\delta>0$ be fixed. Then, for all $\mu>0$, there exists a family $(\theta^{\nu})_{\nu\in (0,1)}\subseteq \mathcal{S}_{\ell^2}^0$ such that, for all mean-zero $\zeta_0\in H^{\delta}(\mathbb{T}^2)$ satisfying $\|\zeta_0\|_{H^{\delta}(\mathbb{T}^2)}\leq N$, we have where $\zeta^\nu$ is the unique global smooth solution to eq:NS_2d_vorticity.

Theorems & Definitions (37)

  • Theorem 1.1: Anomalous dissipation of enstrophy by transport noise -- 2D NSEs
  • Theorem 1.2: Anomalous dissipation of energy by transport noise -- Passive scalars
  • Remark 1.3
  • Definition 2.1: Solutions -- Passive scalars
  • Definition 2.2: Solutions -- 2D NSEs vorticity formulation
  • Proposition 3.1: Scaling limit -- Passive scalars
  • Lemma 3.2: Uniform in $\theta$-estimates
  • proof
  • Lemma 3.3: Time-regularity estimate
  • proof : Proof of Proposition \ref{['prop:scaling_diff']}
  • ...and 27 more