# Energy conservation and Onsager's conjecture for the Euler equations

## TL;DR

This paper proves that energy conservation for weak solutions of the 3D Euler equations occurs at a specific regularity threshold, confirming Onsager's conjecture and analyzing flux locality in turbulence.

## Contribution

It establishes the sharp regularity space for energy conservation and explores flux locality for energy and helicity in the Euler equations.

## Key findings

- Energy is conserved for velocities in $B^{1/3}_{3,c(
n)}$ space.
- Energy flux is controlled by local interactions, confirming flux locality.
- Weak solutions in $B^{2/3}_{3,c(
n)}$ conserve helicity.

## Abstract

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in 3D conserve energy only if they have a certain minimal smoothness, (of order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space $B^{1/3}_{3,c(\NN)}$. We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood-Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to $B^{2/3}_{3,c(\NN)}$ conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.