Let (u, B) be a strong solution of the magneto-hydrodynamic system on three dimensional torus <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_408_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{T}^3$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. In this note, using the properties of the curl operator, we show that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_408_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="470" /> </InlineMediaObject> <EquationSource Format="TEX">$\|(\nabla\times(u-B), \nabla\times(u+B))(\cdot, t)\|_{L^1}+{{1} \over {2\nu}}\|(u-B, u+B)(\cdot, t)\|^2_{L^2}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo fence="false" stretchy="false">∥</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>−</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mo>∥</mo> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> </mrow> </msub> <mo>+</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>ν</mi> </mrow> </mfrac> </mrow> <mo>∥</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>−</mo> <mi>B</mi> <mo>,</mo> <mi>u</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msubsup> <mo fence="false" stretchy="false">∥</mo> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> </mrow> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation> is decreasing in time t as long as the solution (u, B)(·, t) exists, where ∇ × w means the curl of the vector function w, and v > 0 is the viscosity coefficient.

A decreasing property of the 3D magneto-hydrodynamic flows on a torus

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发表日期：

2025年4月8日

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Zhaoxia Liu

摘要

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