  We show that a Leray-Hopf weak solution to the 3D Navier-Stokes Cauchy
problem belonging to the space $L^\infty(0,T; B^{-1}_{\infty,\infty}(\mathbb
R^3))$ is regular in $(0,T]$. As a consequence, it follows that any Leray-Hopf
weak solution to the 3D Navier-Stokes equations is regular while it is
temporally bounded in the largest critical space
$\dot{B}^{-1}_{\infty,\infty}(\mathbb R^3)$ as well as in any critical spaces.
  For the proof we present a new elementary method which is to superpose the
energy norm of high frequency parts in an appropriate way to generate higher
order norms. Thus, starting from the energy estimates of high frequency parts
of a weak solution, one can obtain its estimates of higher order norms.
  By a linear energy superposition we get very simple and short proofs for
known regularity criteria for Leray-Hopf weak solutions in endpoint Besov
spaces $B^{\sigma}_{\infty,\infty}$ for $\sigma\in [-1,0)$, the extension of
Prodi-Serrin conditions. The main result of the paper is proved by applying
technique of a nonlinear energy superposition and linear energy superpositions,
repeatedly. The energy superposition method developed in the paper can also be
applied to other supercritical nonlinear PDEs.
