We prove that the energy equality holds for weak solutions of the 3D Navier-Stokes equations in the functional class $L^3([0,T);V^{5/6})$, where $V^{5/6}$ is the domain of the fractional power of the Stokes operator $A^{5/12}$.
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.
