  In this paper we prove that an operator which projects weak solutions of the
two- or three-dimensional Navier-Stokes equations onto a finite-dimensional
space is determining if it annihilates the difference of two "nearby" weak
solutions asymptotically, and if it satisfies a single appoximation inequality.
We then apply this result to show that the long-time behavior of weak solutions
to the Navier-Stokes equations, in both two- and three-dimensions, is
determined by the long-time behavior of a finite set of bounded linear
functionals. These functionals are constructed by local surface averages of
solutions over certain simplex volume elements, and are therefore well-defined
for weak solutions. Moreover, these functionals define a projection operator
which satisfies the necessary approximation inequality for our theory. We use
the general theory to establish lower bounds on the simplex diameters in both
two- and three-dimensions. Furthermore, in the three dimensional case we make a
connection between their diameters and the Kolmogoroff dissipation small scale
in turbulent flows.
