Local 4/5-law and energy dissipation anomaly in turbulence

A strong local form of the '4/3-law' in turbulent flow has been proved recently by Duchon and Robert for a triple moment of velocity increments averaged over both a bounded spacetime region and separation vector directions, and for energy dissipation averaged over the same spacetime region. Under precisely stated hypotheses, the two are proved to be proportional, by a constant 4/3, and to appear as a non-negative defect measure in the local energy balance of singular (distributional) solutions of the incompressible Euler equations. Here, we prove that the energy defect measure can also be represented by a triple moment of purely longitudinal velocity increments and by a mixed moment with one longitudinal and two tranverse velocity increments. Thus, we prove that the traditional 4/5- and 4/15-laws of Kolmogorov hold in the same local sense as demonstrated for the 4/3-law by Duchon-Robert.