  The global regularity problem for the periodic Navier-Stokes system asks
whether to every smooth divergence-free initial datum $u_0: (\R/\Z)^3 \to \R^3$
there exists a global smooth solution u. In this note we observe (using a
simple compactness argument) that this qualitative question is equivalent to
the more quantitative assertion that there exists a non-decreasing function $F:
\R^+ \to \R^+$ for which one has a local-in-time \emph{a priori} bound $$ \|
u(T) \|_{H^1_x((\R/\Z)^3)} \leq F(\|u_0\|_{H^1_x((\R/\Z)^3)})$$ for all $0 < T
\leq 1$ and all smooth solutions $u: [0,T] \times (\R/\Z)^3 \to \R^3$ to the
Navier-Stokes system. We also show that this local-in-time bound is equivalent
to the corresponding global-in-time bound.
