A tribute to conservation of energy for weak solutions

TL;DR

This paper reviews the energy conservation principle for weak solutions in fluid dynamics, discussing Onsager's conjecture and regularity conditions across various fluid systems.

Contribution

It provides a comprehensive survey of recent results on energy conservation criteria for weak solutions in multiple fluid dynamics models.

Findings

Overview of Onsager's conjecture resolution

Optimal regularity conditions for energy conservation

Comparison across different fluid systems

Abstract

In this article we focus our attention on the principle of energy conservation within the context of systems of fluid dynamics. We give an overview of results concerning the resolution of the famous Onsager conjecture - which states regularity requirements for weak solutions of the incompressible Euler system to conserve energy. Further we survey results providing optimal sufficient regularity conditions for energy conservation for other balance laws: compressible Euler, Navier-Stokes, magnetohydrodynamics and general conservation laws.