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In these notes we want, in addition to presenting some new results, to bothclean up and refine some reflections on a couple of articles published on papera few years ago, 2019-20. These papers concerned integral sufficient conditionson $u\,,$ $\n u\,,$ and mixed, to guarantee the equality of the energy, EE inthe sequel, for solutions of the Navier-Stokes equations under the classicalnon-slip boundary condition. Concerning the $\n u$ case a crucial role wasenjoyed by a previous well known Berselli and Chiodaroli's pioneering 2019 workon the subject. The above three papers are the main sources of these notes.References will be mostly concentrated on their direct relation to the abovepapers at the time of pubblication. More recent results will be not statedthroughout the article. However, in the last section, the reader will besuitably sent to the more recent bibliography. Below, we also turn back to theinnovative interpretation of some main parameters which allowed to overcometheir apparent incongruence. Non-Newtonian fluids were also considered in our2019 paper, maybe for the first time in the above Berselli-Chiodaroli'sparticular $\n u\,$ context. However we will stick mostly to the Newtonian casesince in the end we come to the conclusion that there are not particularadditional obstacles to extend the present results from Newtonian tonon-Newtonian fluids. Hence we avoid to go further in this direction.

On a set of some recent contributions to energy equality for the Navier-Stokes equations