Local 4/5-law and energy dissipation anomaly in turbulence of incompressible MHD Equations

Abstract

In this paper, we establish the longitudinal and transverse local energy balance equation of distributional solutions of the incompressible three-dimensional MHD equations. In particular, we find that the functions \(D_L^\varepsilon (\mathbf {u},\mathbf {B})\) and \(D_T^\varepsilon (\mathbf {u},\mathbf {B})\) appeared in the energy balance, all converging to the defect distribution (in the sense of distributions) \(D(\mathbf {u},\mathbf {B})\) which has been defined in Gao et al. (Acta Math Sci 33:865–871, 2013). Furthermore, we give a simpler form of defect distribution term, which is similar to the relation in turbulence theory, called the “4 / 3-law.” As a corollary, we give the analogous “4 / 5-law” holds in the local sense.