I will start by explaining the Onsager's conjecture on the conservation of the kinetic energy for the incompressible non viscous fluids. I will also show how the same structure of the equation, that allows to prove energy conservation above certain regularity threshold, also gives a peculiar regularity of the energy in the range in which it is not necessarily constant.

If time permits, I will describe a recent result (in collaboration with R. Tione) on the sharpness of such energy regularity.