This talk is a report on ongoing work with Edriss Titi and other coauthors.
The notion of weak convergence for Navier-Stokes and Euler equations presents many similarities
with the notion of average in the statistical theory of turbulence. In particular the counterpart of
Kolmogorov 1/3 law was the Onsager “Holder 1/3" conjecture.

Around 1994 it was proven by Constantin, E and Titi that the 1/3 Hölder regularity implies the
conservation of energy. The interest of such issue was recently emphasized by the construction, of
wild solutions of Hölder regularity less than 1/3 that do not conserve the energy (Issett , Buckmaster,
De Lellis, Szekelyhidi Jr., and Vicol). However the two points of view are not complementary.
One can observe very easily the existence of some other type of solutions not regular at all that do
conserve the energy. However in the zero viscosity limit with no slip boundary condition a theorem
of Kato implies the conservation of 1/3 regularity Hölder, or Besov.., is equivalent to the absence
of anomalous energy dissipation.

This will be the object of my talk which requires the consideration of the Onsager condition
in the presence of boundary and several extensions of the theorem of Kato now dubbed the Kato
Criteria. Such criteria underlines the contribution of no slip boundary condition in the creation
of a turbulent boundary ( also recently analyzed from this point of view by Drivas and Nguyen )
which may contributes to the resolution of the d’Alembert paradox.