  We show that the classical Shinbrot's criteria to guarantee that a Leray-Hopf
solution satisfies the energy equality follows trivially from the $L^4(
(0\,,T)\times\Omega))$ Lions-Prodi particular case. Moreover we extend
Shinbrot's result to space coefficients $ r \in (3,\,4)\,.$ In this last case
our condition coincides with Shinbrot condition for $r=4$, but for $r<4$ it is
more restrictive than the classical one, $ 2/p + 2/r = 1\,.$ It looks
significant that in correspondence to the extreme values $r=3$ and $r=\infty$,
and just for these two values, the conditions become respectively $u \in
L^\infty(L^3)$ and $u \in L^2(L^\infty)$, which imply regularity by appealing
to classical Ladyzhenskaya-Prodi-Serrin (L-P-S) type conditions. However, for
values $r\in (3,\infty)$ the L-P-S condition does not apply, even for the more
demanding case $\,3<r<4\,.$ The proofs are quite trivial, by appealing to
interpolation, with $L^\infty(L^2)$ in the first case and with $L^2(L^6)$ in
the second case. The central position of this old classical problem in
Fluid-Mechanics, together with the simplicity of the proofs (in particular the
novelty of the second result) looks at least curious. This may be considered a
merit of this very short note.
