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One of the main tenets in the Kolmogorov theory of 3D turbulence is the direct cascade of energy. This means that the rate of transfer of energy from one length scale to the next smallest is roughly constant over the so-called inertial range of scales. This can be indicated by a large quotient of the averages of enstrophy over energy. Similarly, the Batchelor, Kraichnan, Leith theory of 2D turbulence features an additional direct cascade, that of enstrophy, which in turn is indicated by a large quotient of averaged palinstrophy over enstrophy. In the case of the 2D NSE we have derived bounding curves for these pairwise quantities by combining estimates for their time derivatives. To do so for the 3D NSE, however, is to confront its outstanding global regularity problem.

Beirao da Veiga, following work of Constantin and Fefferman, showed that solutions to the 3D NSE are regular if one assumes that the direction of vortex filaments is Holder continuous with exponent 1/2. Under this assumption we construct in a single bounding curve whose maximal enstrophy is shown to scale as an exponential of the Grashof number. This suggests that even under this smoothness assumption solutions can display extraordinary bursts in enstrophy.