Analysis & Applied Math

With an increasing level of generality some proofs of local energy conservation for weak solutions to the incompressible Euler equations that are allowed to jump on space-time hypersurfaces are presented. In none of them the classical Constantin, E and Titi commutator argument applies. The result is indeed false in the compressive setting. The proofs make the role of the incompressibility very apparent, suggesting that fine features (mostly geometrical) of hydrodynamic turbulence stand apart from purely dynamical and kinematic arguments. Some emphasis will be put on solutions living at the "Onsager critical" regularity, a setting far from being clear to date.