Yaglom's law and conserved quantity dissipation in turbulence

In this paper, we are concerned with the local exact relationship for third-order structure functions in the temperature equation, the inviscid MHD equations and the Euler equations in the sense of Duchon-Robert type and Eyink type. It is shown that the local version of Yaglom's $4/3$ law is valid for the dissipation rates of conserved quantities such as the energy, cross-helicity and helicity in these systems. In the spirit of Duchon-Robert's classical work, we derive the dissipation term resulted from the lack of smoothness of the solutions in corresponding conservation relation. It seems that these results suggest that the Yaglom's law of the hydrodynamic equations holds if an analogue of dissipation term as Duchon-Robert's is obtained. Base on this, the first Yaglom's relation for the Oldroyd-B model and, inspired by the very recent work due to Boutros-Titi, six new 4/3 laws for subgrid scale $α$-models of turbulence are also presented.