Note on Onsager's conjecture

TL;DR

This paper critically examines Onsager's conjecture, discussing the conditions under which solutions to the Euler equations conserve energy and highlighting shortcomings in the original ideas of energy dissipation without viscosity.

Contribution

It provides a critical analysis of Onsager's conjecture, clarifying misconceptions and unveiling shortcomings in the concept of anomalous energy dissipation.

Findings

Identifies limitations in Onsager's original conjecture

Highlights issues with the concept of anomalous dissipation

Clarifies the relationship between solution roughness and energy conservation

Abstract

Onsager conjectured that solutions of the incompressible Euler equations possessing a certain degree of roughness do not conserve the kinetic energy. Since, within the physical frame of Onsager's conjecture, the kinetic energy is the only occurring energy, and thus identical with the total energy, the implication would be that the conservation of energy is not absolute, but subject to the properties of mathematical solutions. Further, Onsager introduced the concept of anomalous dissipation of kinetic energy without viscosity. Both these aspects are critically discussed and their shortcomings unveiled.