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.
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
.
DOI: 10.1007/s10473-025-0408-z
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