Dissipation anomaly and anomalous dissipation in incompressible fluid flows

TL;DR

This paper investigates the phenomenon of dissipation anomaly and anomalous dissipation in incompressible fluid flows, demonstrating various scenarios of energy loss in the zero-viscosity limit and establishing connections with turbulence theory.

Contribution

It proves the existence of multiple dissipation scenarios in the zero-viscosity limit, including loss of energy and multiple Euler solutions, expanding understanding of turbulence and energy dissipation.

Findings

Existence of total and partial energy loss due to dissipation anomaly

Identification of anomalous dissipation without dissipation anomaly

Existence of infinitely many Euler solutions in the zero-viscosity limit

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 Bru\`e 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.