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\nrecently by Duchon and Robert for a triple moment of velocity increments\naveraged over both a bounded spacetime region and separation vector directions,\nand for energy dissipation averaged over the same spacetime region. Under\nprecisely stated hypotheses, the two are proved to be proportional, by a\nconstant 4/3, and to appear as a nonnegative defect measure in the local energy\nbalance of singular (distributional) solutions of the incompressible Euler\nequations. Here we prove that the energy defect measure can be represented also\nby a triple moment of purely longitudinal velocity increments and by a mixed\nmoment with one longitudinal and two tranverse velocity increments. Thus, we\nprove that the traditional 4/5- and 4/15-laws of Kolmogorov hold in the same\nlocal sense as demonstrated for the 4/3-law by Duchon-Robert.\n