  In these notes we want, in addition to presenting some new results, to both
clean up and refine some reflections on a couple of articles published on paper
a few years ago, 2019-20. These papers concerned integral sufficient conditions
on $u\,,$ $\n u\,,$ and mixed, to guarantee the equality of the energy, EE in
the sequel, for solutions of the Navier-Stokes equations under the classical
non-slip boundary condition. Concerning the $\n u$ case a crucial role was
enjoyed by a previous well known Berselli and Chiodaroli's pioneering 2019 work
on the subject. The above three papers are the main sources of these notes.
References will be mostly concentrated on their direct relation to the above
papers at the time of pubblication. More recent results will be not stated
throughout the article. However, in the last section, the reader will be
suitably sent to the more recent bibliography. Below, we also turn back to the
innovative interpretation of some main parameters which allowed to overcome
their apparent incongruence. Non-Newtonian fluids were also considered in our
2019 paper, maybe for the first time in the above Berselli-Chiodaroli's
particular $\n u\,$ context. However we will stick mostly to the Newtonian case
since in the end we come to the conclusion that there are not particular
additional obstacles to extend the present results from Newtonian to
non-Newtonian fluids. Hence we avoid to go further in this direction.
