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

In this paper we prove that an operator which projects weak solutions of the two-or three-dimensional Navier-Stokes equations onto a nite-dimensional space is determining if it annihilates the diierence of two \nearby" weak solutions asymptotically, and if it satisses 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 nite set of bounded linear function-als. These functionals are constructed by local surface averages of solutions over certain simplex volume elements, and are therefore well-deened for weak solutions. Moreover, these functionals deene a projection operator which satis-es 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 Kolmogoroo dissipation small scale in turbulent ows.