**Onsager's Conjecture for Euler equations in critical Besov spaces**

**Abstract:**

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 B^{1/3}_{3,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 B^{2/3}_{3,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.