Intermittency and regularity issues in 3D Navier-Stokes turbulence

Two related open problems in the theory of 3D Navier-Stokes turbulence are discussed in this paper. The first is the phenomenon of intermittency in the dissipation field. Dissipation-range intermittency was first discovered experimentally by Batchelor and Townsend over fifty years ago. It is characterized by spatio-temporal binary behaviour in which long, quiescent periods in the velocity signal are interrupted by short, active events' during which there are violent fluctuations away from the average. The second and related problem is whether solutions of the 3D Navier-Stokes equations develop finite time singularities during these events. This paper shows that Leray's weak solutions of the three-dimensional incompressible Navier-Stokes equations can 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. An estimate for the width of the latter is very small and decreases with increasing Reynolds number. It also decreases relative to the lengths of the good intervals as the Reynolds number increases. Within these bad’ intervals, lower bounds on the local energy dissipation rate and other quantities, such as $\mathrm{\parallel }\text{bu}(\cdot ,t){\mathrm{\parallel }}_{\mathrm{\infty }}$ and $\mathrm{\parallel }\mathrm{\nabla }\text{bu}(\cdot ,t){\mathrm{\parallel }}_{\mathrm{\infty }}$, are very large, resulting in strong dynamics at sub-Kolmogorov scales. Intersections of bad intervals for $n\ge 1$ are related to Scheffer’s potentially singular set in time. It is also proved that the Navier-Stokes equations are conditionally regular provided, in a given `bad’ interval, the energy has a lower bound that is decaying exponentially in time.