  We prove new results for general weak solutions of the incompressible
Navier-Stokes equations on the three-dimensional torus, without using the
strong energy inequality or other regularity assumptions. Specifically, we
prove two types of differential energy equalities, in which $(d/dt)\lVert
u\rVert^2$ is replaced by $2\langle \partial_t u,u\rangle$ and
$\lim_{N\rightarrow\infty}(d/dt)\lVert u_N\rVert^2$, where the $u_N$ are
projections of the original weak solution $u$. We also provide a new
characterization of Leray-Hopf solutions, as weak solutions having a version
for which the energy is monotonically decreasing. In order to prove these
results, we first prove that both $\partial_t u$ and $\Delta u$ are in
$L^{4/3}(\epsilon, T; L^{6/5})$, results that were previously known only for
Leray-Hopf weak solutions.
