Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations

We study the local equation of energy for weak solutions of three-dimensional incompressible Navier-Stokes and Euler equations. We define a dissipation term D(u) which stems from an eventual lack of smoothness in the solution u. We give in passing a simple proof of Onsager's conjecture on energy conservation for the three-dimensional Euler equation, slightly weakening the assumption of Constantin et al. We suggest calling weak solutions with non-negative D(u) 'dissipative'.