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

Yao Nie; Jia Yuan

doi:10.3934/dcds.2020397

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2021, Volume 41, Issue 7: 3045-3062. Doi: 10.3934/dcds.2020397

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

Yao Nie and
Jia Yuan^,

School of Mathematics and Systems Science, Beihang University, Beijing 100191, China

^* Corresponding author: Jia Yuan
^* Corresponding author: Jia Yuan

Received: June 30, 2020

Revised: September 30, 2020

Early access: December 2020

Published: July 2021

The work is supported by NSF grant No.11871087 and No.11771423. The first author is supported by the Academic Excellence Foundation of BUAA for PhD students and China Scholarship Council No.201906020100

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Abstract

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

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 42B37, 76D03, 76W05.

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