A DECREASING PROPERTY OF THE 3D MAGNETO-HYDRODYNAMIC FLOWS ON A TORUS

doi:10.1007/s10473-025-0408-z

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (4): 1384-1390.doi: 10.1007/s10473-025-0408-z

A DECREASING PROPERTY OF THE 3D MAGNETO-HYDRODYNAMIC FLOWS ON A TORUS

Zhaoxia LIU

College of Science, Minzu University of China, Beijing 100081, China

收稿日期:2024-06-25 出版日期:2025-10-10 发布日期:2025-10-10

A DECREASING PROPERTY OF THE 3D MAGNETO-HYDRODYNAMIC FLOWS ON A TORUS

Zhaoxia LIU

College of Science, Minzu University of China, Beijing 100081, China

Received:2024-06-25 Online:2025-10-10 Published:2025-10-10
About author:Zhaoxia LIU, E-mail: zxliu@amt.ac.cn
Supported by:
National Natural Science Foundation of China (12371123).

摘要/Abstract

摘要： Let $(u, B)$ be a strong solution of the magneto-hydrodynamic system on three dimensional torus $\mathbb{T}^3$. In this note, using the properties of the curl operator, we show that $\|(\nabla\times(u-B), \nabla\times(u+B))(\cdot, t)\|_{L^1}+\frac{1}{2\nu}\|(u-B, u+B)(\cdot, t)\|^2_{L^2}$ is decreasing in time $t$ as long as the solution $(u, B)(\cdot,t)$ exists, where $\nabla\times w$ means the curl of the vector function $w$, and $\nu>0$ is the viscosity coefficient.

关键词: magneto-hydrodynamic equations, strong solution, curl operator

Abstract: Let $(u, B)$ be a strong solution of the magneto-hydrodynamic system on three dimensional torus $\mathbb{T}^3$. In this note, using the properties of the curl operator, we show that $\|(\nabla\times(u-B), \nabla\times(u+B))(\cdot, t)\|_{L^1}+\frac{1}{2\nu}\|(u-B, u+B)(\cdot, t)\|^2_{L^2}$ is decreasing in time $t$ as long as the solution $(u, B)(\cdot,t)$ exists, where $\nabla\times w$ means the curl of the vector function $w$, and $\nu>0$ is the viscosity coefficient.

Key words: magneto-hydrodynamic equations, strong solution, curl operator

Zhaoxia LIU. A DECREASING PROPERTY OF THE 3D MAGNETO-HYDRODYNAMIC FLOWS ON A TORUS[J]. 数学物理学报(英文版), 2025, 45(4): 1384-1390.

Zhaoxia LIU. A DECREASING PROPERTY OF THE 3D MAGNETO-HYDRODYNAMIC FLOWS ON A TORUS[J]. Acta mathematica scientia,Series B, 2025, 45(4): 1384-1390.

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A DECREASING PROPERTY OF THE 3D MAGNETO-HYDRODYNAMIC FLOWS ON A TORUS

A DECREASING PROPERTY OF THE 3D MAGNETO-HYDRODYNAMIC FLOWS ON A TORUS

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