Energy equality for the Navier-Stokes equations in weak-in-time Onsager spaces

Onsager's conjecture for the 3D Navier-Stokes equations concerns the validity of energy equality of weak solutions with regards to their smoothness. In this note, we establish the energy equality for weak solutions in a large class of function spaces. These conditions are weak-in-time with optimal space regularity and therefore weaker than previous classical results. Heuristics using intermittency argument and divergence-free counterexamples are given, indicating the possible sharpness of our conditions.