Energy conservation and Onsager's conjecture for the Euler equations

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in R-3 conserve energy only if they have a certain minimal smoothness (of the 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-3,c(N)(1/3). 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-3,c(N)(2/3) conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.