Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Anomalous Dissipation at Onsager-Critical Regularity

Alexey Cheskidov, Qirui Peng

TL;DR

The paper advances the understanding of viscous dissipation anomaly in the vanishing-viscosity limit of the 3D Navier–Stokes equations by constructing sequences that realize anomalous dissipation in finite time and by aligning these phenomena with Onsager’s critical regularity. It extends the Bruè–De Lellis 2.5D framework and introduces an exponential time-scale mixing scheme, yielding sharp Besov-scale control and allowing a fully 3D dissipative Euler example under rough forcing. The results establish a sharp flux criterion for energy conservation in 2.5D flows, demonstrate a vanishing-viscosity limit with total viscous dissipation while remaining at Onsager-critical regularity, and provide a separate fully 3D construction with zero anomalous work that sits in the Onsager borderline. Collectively, these constructions illuminate how intermittency and mixing dynamics at critical regularity can drive energy dissipation in the inviscid limit, with potential implications for turbulence models and energy cascade descriptions.

Abstract

We construct solutions to the three-dimensional Euler equations exhibiting anomalous dissipation in finite time through a vanishing viscosity limit. Inspired by \cite{BDL23} and \cite{cheskidov2023dissipation}, we extend the \(2\frac{1}{2}\)-dimensional constructions and establish an Onsager-critical energy criterion adapted to such flows, showing its sharpness. Moreover, we provide a fully three-dimensional dissipative Euler example, sharp in Onsager's sense, driven by a slightly rough external force, following the framework of \cite{CL21}.

Anomalous Dissipation at Onsager-Critical Regularity

TL;DR

The paper advances the understanding of viscous dissipation anomaly in the vanishing-viscosity limit of the 3D Navier–Stokes equations by constructing sequences that realize anomalous dissipation in finite time and by aligning these phenomena with Onsager’s critical regularity. It extends the Bruè–De Lellis 2.5D framework and introduces an exponential time-scale mixing scheme, yielding sharp Besov-scale control and allowing a fully 3D dissipative Euler example under rough forcing. The results establish a sharp flux criterion for energy conservation in 2.5D flows, demonstrate a vanishing-viscosity limit with total viscous dissipation while remaining at Onsager-critical regularity, and provide a separate fully 3D construction with zero anomalous work that sits in the Onsager borderline. Collectively, these constructions illuminate how intermittency and mixing dynamics at critical regularity can drive energy dissipation in the inviscid limit, with potential implications for turbulence models and energy cascade descriptions.

Abstract

We construct solutions to the three-dimensional Euler equations exhibiting anomalous dissipation in finite time through a vanishing viscosity limit. Inspired by \cite{BDL23} and \cite{cheskidov2023dissipation}, we extend the -dimensional constructions and establish an Onsager-critical energy criterion adapted to such flows, showing its sharpness. Moreover, we provide a fully three-dimensional dissipative Euler example, sharp in Onsager's sense, driven by a slightly rough external force, following the framework of \cite{CL21}.
Paper Structure (14 sections, 15 theorems, 212 equations)

This paper contains 14 sections, 15 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Cartesian $2\frac{1}{2}$-dimensional solutions
  5. Littlewood--Paley decomposition
  6. Besov spaces and Onsager's critical energy classes
  7. main Theorem
  8. proof of Theorem \ref{['main_Theorem_1']}
  9. 2${\frac{1}{2}}-$dimensional solutions and mixing construction
  10. The construction through perfect mixing
  11. $2{\frac{1}{2}}$-dimensional solutions to Euler and Navier--Stokes equations
  12. Proof of Theorem \ref{['main_Theorem_2']}
  13. Second construction
  14. Proof of Theorem \ref{['main_Theorem_3']}

Key Result

Lemma 2.1

(Bernstein's inequality) Let $d$ be the spacial dimension, $r\geq s\geq 1$ and $k\geq 0$. Then for all tempered distributions $u$,

Theorems & Definitions (29)

  • Definition 1.1
  • Definition 1.3
  • Definition 1.4
  • Lemma 2.1
  • Theorem 2.2: CCFS08
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • ...and 19 more