Onsager's Conjecture for Euler equations in critical Besov spaces
Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in R3 conserve energy only if they have a smoothness of order higher than 1/3 fractional derivatives and that they dissipate energy if they are rougher. We prove that energy is in fact conserved for velocities in a Besov space of the critical smoothness 1/3 with decaying Littlewood-Paley components. In particular, it holds in B1/33,r, for all 1 < r < ¥. We show that this space is sharp in a natural sense. Specifically, the energy flux may not vanish if r=¥. The energy flux will be discussed in detail. We present new estimates on the trilinear term that show strong exponential locality of scales that participate in transporting the energy from one dyadic shell to another. This locality is shown to hold also for the helicity flux; as a consequence, every weak solution of the Euler equations that belongs to B2/33,r, 1 < r < ¥ conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range, and a weak locality holds in the infrared range. Applications to the energy equality for Leray-Hopf solutions of the 3D Navier-Stokes equations and to turbulence and will be presented.