Determining Projections and Functionals for Weak Solutions of the Navier-Stokes Equations

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.