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Two related open problems in the theory of 3D Navier-Stokes turbulence arediscussed in this paper. The first is the phenomenon of intermittency in thedissipation field. Dissipation-range intermittency was first discoveredexperimentally by Batchelor and Townsend over fifty years ago. It ischaracterized by spatio-temporal binary behaviour in which long, quiescentperiods in the velocity signal are interrupted by short, active `events' duringwhich there are violent fluctuations away from the average. The second andrelated problem is whether solutions of the 3D Navier-Stokes equations developfinite time singularities during these events. This paper shows that Leray'sweak solutions of the three-dimensional incompressible Navier-Stokes equationscan have a binary character in time. The time-axis is split into `good' and`bad' intervals: on the `good' intervals solutions are bounded and regular,whereas singularities are still possible within the `bad' intervals. Anestimate for the width of the latter is very small and decreases withincreasing Reynolds number. It also decreases relative to the lengths of thegood intervals as the Reynolds number increases. Within these `bad' intervals,lower bounds on the local energy dissipation rate and other quantities, such as$\|\bu(\cdot, t)\|_{\infty}$ and $\|\nabla\bu(\cdot, t)\|_{\infty}$, are verylarge, resulting in strong dynamics at sub-Kolmogorov scales. Intersections ofbad intervals for $n\geq 1$ are related to Scheffer's potentially singular setin time. It is also proved that the Navier-Stokes equations are conditionallyregular provided, in a given `bad' interval, the energy has a lower bound thatis decaying exponentially in time.Comment: 36 pages, 3 figures and 6 Table

Intermittency and Regularity Issues in 3D Navier-Stokes Turbulence