Energy conservation and Onsager's conjecture for the Euler equations
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 . 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 conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.
Cite
@article{arxiv.0704.0759,
title = {Energy conservation and Onsager's conjecture for the Euler equations},
author = {A. Cheskidov and P. Constantin and S. Friedlander and R. Shvydkoy},
journal= {arXiv preprint arXiv:0704.0759},
year = {2007}
}