Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dissipation anomaly and anomalous dissipation in incompressible fluid flows

Alexey Cheskidov

Abstract

Dissipation anomaly, a phenomenon predicted by Kolmogorov's theory of turbulence, is the persistence of a non-vanishing energy dissipation for solutions of the Navier-Stokes equations as the viscosity goes to zero. Anomalous dissipation, predicted by Onsager, is a failure of solutions of the limiting Euler equations to preserve the energy balance. Motivated by a recent dissipation anomaly construction for the 3D Navier-Stokes equations by Bruè and De Lellis (2023), we prove the existence of various scenarios in the limit of vanishing viscosity: the total and partial loss of the energy due to dissipation anomaly, absolutely continuous dissipation anomaly, anomalous dissipation without dissipation anomaly, and the existence of infinitely many limiting solutions of the Euler equations in the limit of vanishing viscosity. We also discover a relation between dissipation anomaly and the discontinuity of the energy of the limit solution. Finally, expanding on the obtained total dissipation anomaly construction, we show the existence of dissipation anomaly for long time averages, relevant for turbulent flows, proving that the Doering-Foias (2002) upper bound is sharp.

Dissipation anomaly and anomalous dissipation in incompressible fluid flows

Abstract

Dissipation anomaly, a phenomenon predicted by Kolmogorov's theory of turbulence, is the persistence of a non-vanishing energy dissipation for solutions of the Navier-Stokes equations as the viscosity goes to zero. Anomalous dissipation, predicted by Onsager, is a failure of solutions of the limiting Euler equations to preserve the energy balance. Motivated by a recent dissipation anomaly construction for the 3D Navier-Stokes equations by Bruè and De Lellis (2023), we prove the existence of various scenarios in the limit of vanishing viscosity: the total and partial loss of the energy due to dissipation anomaly, absolutely continuous dissipation anomaly, anomalous dissipation without dissipation anomaly, and the existence of infinitely many limiting solutions of the Euler equations in the limit of vanishing viscosity. We also discover a relation between dissipation anomaly and the discontinuity of the energy of the limit solution. Finally, expanding on the obtained total dissipation anomaly construction, we show the existence of dissipation anomaly for long time averages, relevant for turbulent flows, proving that the Doering-Foias (2002) upper bound is sharp.
Paper Structure (14 sections, 11 theorems, 243 equations, 3 figures)

This paper contains 14 sections, 11 theorems, 243 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Dissipation anomaly for finite time averages
  3. Dissipation anomaly for long time averages
  4. Previous results
  5. Onsager's conjecture and anomalous dissipation
  6. Effect of intermittency on the energy flux
  7. Dissipation Anomaly
  8. Main results
  9. Perfect mixing
  10. Proof of Theorem \ref{['thm:main']}
  11. Proof of Theorem \ref{['thm:main-2']}
  12. Dissipation Anomaly for long time averages: Proof of Theorem \ref{['thm:DA_in_FT']}
  13. Discontinuity of the limit: proof of Theorem \ref{['thm:discont-intro']}
  14. Figures

Key Result

Theorem 1.2

For any nontrivial divergence-free $\phi \in L^\infty(\mathbb{R}; L^2(\mathbb{T}^3))$ there exist constants $c_1$ and $c_2$, such that for every Leray-Hopf weak solution $u^\nu$ of eq:NSE with arbitrary divergence-free finite energy initial data and force $f(x,t)=F \phi^j(\ell^{-1} x,t)$ with arbitr where $\epsilon$, $U$, and $Re$ are defined as in def:eps,U,Re.

Figures (3)

  • Figure 1: Theorem \ref{['thm:main']}: Convergence of the solutions to the NSE (red) to a solution of the Euler equations (blue). The wavenumber $\Lambda\to \infty$ as $\nu \to 0$ schematically shows the extend of the energy cascade for the NSE solutions.
  • Figure 2: Theorem \ref{['thm:main']}: Energy profiles $E(t)$ for various subsequences of solutions of the NSE. The wavenumber $\Lambda$ is heuristically compared to Kolmogorov's dissipation number $\kappa_d$.
  • Figure 3: Theorem \ref{['thm:main-2']}: Convergence of the solutions of the NSE (red) to a solution of the Euler equation (blue).

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • ...and 19 more