  In this paper we establish a number of implications between various
qualitative and quantitative versions of the global regularity problem for the
Navier-Stokes equations, in the periodic, smooth finite energy, smooth $H^1$,
Schwartz, or mild $H^1$ categories, and with or without a forcing term. In
particular, we show that if one has global well-posedness in $H^1$ for the
periodic Navier-Stokes problem with a forcing term, then one can obtain global
regularity both for periodic and for Schwartz initial data (thus yielding a
positive answer to both official formulations of the problem for the Clay
Millennium Prize), and can also obtain global smooth solutions from smooth
$H^1$ data, and global almost smooth solutions from smooth finite energy data.
Our main new tools are localised energy and enstrophy estimates to the
Navier-Stokes equation that are applicable for large data or long times, and
which may be of independent interest.
