The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence

Abstract

In this paper, we consider the Littlewood-Paley \begin{document}$ p $\end{document} th-order ( \begin{document}$ 1\le p ) moments of the three-dimensional MHD periodic equations, which are defined by the infinite-time and space average of \begin{document}$ L^p $\end{document} -norm of velocity and magnetic fields involved in the spectral cut-off operator \begin{document}$ \dot\Delta_m $\end{document} . Our results imply that in some cases, \begin{document}$ k^{-\frac{1}{3}} $\end{document} is an upper bound at length scale \begin{document}$ 1/k $\end{document} . This coincides with the scaling law of many observations on astrophysical systems and simulations in terms of 3D MHD turbulence.