In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary incompressible Magnetohydrodynamic equations in the whole space, the half-space, or a periodic slab, and presenting the solution must vanish under the condition that for some 0 ≤ δ ≤ 1 < L and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$q = {{6({3 - \delta})} \over {6 - \delta}},\mathop {\lim \,\inf}\limits_{R \to \infty} {1 \over R}\Vert{({{\bf{u}},{\bf{h}}})}\Vert_{R < \vert x \vert < LR}^{3 - \delta} = 0$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>q</mi> <mo>=</mo> <mrow> <mfrac> <mrow> <mn>6</mn> <mo stretchy="false">(</mo> <mrow> <mn>3</mn> <mo>−</mo> <mi>δ</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>6</mn> <mo>−</mo> <mi>δ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <munder> <mrow> <mo form="prefix" movablelimits="true">lim</mo> <mspace width="thinmathspace" /> <mo form="prefix" movablelimits="true">inf</mo> </mrow> <mrow> <mi>R</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mspace width="thinmathspace" /> <mrow> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mrow> <mo>∥</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <mrow> <mrow> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mo>,</mo> <mrow> <mrow> <mi mathvariant="bold">h</mi> </mrow> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <msubsup> <mo>∥</mo> <mrow> <mi>R</mi> <mo><</mo> <mo fence="false" stretchy="false">|</mo> <mi>x</mi> <mo fence="false" stretchy="false">|</mo> <mo><</mo> <mi>L</mi> <mi>R</mi> </mrow> <mrow> <mn>3</mn> <mo>−</mo> <mi>δ</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </math></EquationSource> </InlineEquation>. We also deduce sufficient conditions by allowing shrinking ratio L = 1 + R−α. When in slab with zero boundary condition, stronger decay rate is needed. We do not assume the global bound of the velocity field u and the magnetic field h and investigate the Liouville type theorems by the conditions lim inf rather than lim.

A Note on the Liouville Type Theorem for the Steady-state Magnetohydrodynamic Equations

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发表日期：

2026年3月1日

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Hui-ying Fan,
Meng Wang

摘要

In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary incompressible Magnetohydrodynamic equations in the whole space, the half-space, or a periodic slab, and presenting the solution must vanish under the condition that for some 0 ≤ δ ≤ 1 < L and $q = {{6({3 - \delta})} \over {6 - \delta}},\mathop {\lim \,\inf}\limits_{R \to \infty} {1 \over R}\Vert{({{\bf{u}},{\bf{h}}})}\Vert_{R < \vert x \vert < LR}^{3 - \delta} = 0$ $q = \frac{6 (3 - δ)}{6 - δ}, \underset{R \to \infty}{lim inf} \frac{1}{R} ∥ (u, h) ∥_{R < | x | < L R}^{3 - δ} = 0$ . We also deduce sufficient conditions by allowing shrinking ratio L = 1 + R^−α. When in slab with zero boundary condition, stronger decay rate is needed. We do not assume the global bound of the velocity field u and the magnetic field h and investigate the Liouville type theorems by the conditions lim inf rather than lim.