The Kolmogorov theory of turbulence, based on a scaling argument, predicts a smallest significant length scale, beyond which a flow is very smooth. Stated spectrally, there is a largest significant wave number, beyond which the energy decays very rapidly with respect to further increases in the wave number. However, this scaling argument has never made contact with the rigorous mathematical theory of the Navier-Stokes equations. If it were understood in the context of rigorous theory it would have many important consequences, a particular corollary being the global in time regularity of solutions of the Navier-Stokes equations.
Here, we consider a certain infinite system of ordinary differential equations, regarded as a highly simplified model of how energy might be passed up the spectrum in the Navier-Stokes equations, into the smaller scales of motion. Numerical experiments with this system of equations reveal a very striking ``inertial range'' and ``smallest scale'' phenomenon. One observes the apparent determination of a ``largest significant mode number'' by an abrupt change over just a very few mode numbers in the character of the energy decay with respect to mode number. We formulate corresponding mathematical definitions and prove much of what is observed in these experiments, especially regarding the energy decay of steady solutions with respect to mode number. Our results for nonstationary solutions are not as complete as for steady solutions, but their proofs are probably more relevant to Navier-Stokes theory. We conclude by describing the results of further experiments with related systems of equations. In each of them the ``inertial range/smallest scale phenomenon'' is found to persist. The results are quite dramatic, and will be demonstrated during the talk with real time visualizations of the computations. Finding that this phenomenon is generic to a wide class of equations, we speculate on whether the system of spectral Navier-Stokes equations belongs to it.\ Our objective is to begin a rigorous investigation of smallest scale phenomena in simple model problems, hoping for insights that might generalize to the Navier-Stokes equations.