We study space-time integrals which appear in Caffarelli-Kohn-Nirenberg (CKN) theory for the Navier-Stokes equations analytically and numerically. The key quantity is written in standard notations δ(r) = 1/(νr) Qr |∇u| 2 dx dt, which can be regarded as a local Reynolds number over a parabolic cylinder Q r .First, by re-examining the CKN integral we identify a cross-over scale, at which the CKN Reynolds number δ(r) changes its scaling behavior. This reproduces a result on the minimum scale r min in turbulence: r 2 min ∇u ∞ ∝ ν, consistent with a result of Henshaw et al.(1989). For the energy spectrum E(k) ∝ k −q (1 < q < 3), we show that r * ∝ ν a with a = 4 3(3−q) −1. Parametric representations are then obtained as ∇u ∞ ∝ ν −(1+3a)/2 and r min ∝ ν 3(a+1)/4 . By the assumptions of the regularity and finite energy dissipation rate in the inviscid limit, we derive lim p→∞ ζp p = 1 − ζ 2 for any phenomenological models on intermittency, where ζ p is the exponent of p-th order (longitudinal) velocity structure function. It follows that ζ p ≤ (1 − ζ 2 )(p − 3) + 1 for any p ≥ 3 without invoking fractal energy cascade.Second, we determine the scaling behavior of δ(r) in direct numerical simulations of the NavierStokes equations. In isotropic turbulence around R λ ≈ 100 starting from random initial conditions, we have found that δ(r) ∝ r 4 throughout the inertial range. This can be explained by the smallness of a ≈ 0.26, with a result that r * is in the energy-containing range. If the β-model is perfectly correct, the intermittency parameter a must be related to the dissipation correlation exponent µ as µ = 4a 1+a ≈ 0.8 which is larger than the observed µ ≈ 0.20. Furthermore, corresponding integrals are studied using the Burgers vortex and the Burgers equation. In those single-scale phenomena, the cross-over scale lies in the dissipative range. The scale r * offers a practical method of quantifying intermittency. This paper also sorts out a number of existing mathematical bounds and phenomenological models on the basis of the CKN Reynolds number.

A connection between dissipation anomaly in fluid dynamics and Colombeau's theory of products of distributions is exemplified by considering Burgers equation with a passive scalar. Besides the well-known viscosity-independent dissipation of energy in the steadily propagating shock wave solution, the lesser known case of passive scalar subject to the shock wave is studied. An exact dependence of the dissipation rate ǫ θ of the passive scalar on the Prandtl number P r is given by a simple analysis: we show in particular ǫ θ ∝ 1/ √ P r for large P r . The passive scalar profile is shown to have a form of a sum of tanh n x with suitably scaled x, thereby implying the necessity to distinguish H from H n when P r is large, where H is the Heaviside function. An incorrect result of ǫ θ ∝ 1/P r would otherwise be obtained. This is a typical example where Colombeau calculus for products of weak solutions is required for a correct interpretation. A Cole-Hopf-like transform is also given for the case of unit Prandtl number.

We compare freely decaying evolution of the Navier-Stokes equations with that of the 3D Burgers equations with the same kinematic viscosity and the same incompressible initial data by using direct numerical simulations. The Burgers equations are well-known to be regular by a maximum principle [Kiselev and Ladyzenskaya (1957)] unlike the Navier-Stokes equations.It is found in the Burgers equations that the potential part of velocity becomes large in comparison with the solenoidal part which decays more quickly. The probability distribution of the nonlocal term −u · ∇p, which spoils the maximum principle, in the local energy budget is studied in detail. It is basically symmetric, i.e. it can be either positive or negative with fluctuations.Its joint probability density functions with 1 2 |u| 2 and with 1 2 |ω| 2 are also found to be symmetric, fluctuating at the same times as the probability density function of −u · ∇p.A power-law relationship is found in the mathematical bound for enstrophy dQ dt + 2νP ∝ Q a P b α , where Q and P denote the enstrophy and the palinstrophy, respectively and the exponents a and b are determined by calculus inequalities. We propose to quantify nonlinearity depletion by the exponent α on this basis.