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<p>To uncover the mechanism by which magnetic fields can stabilize electrically conducting turbulent fluids, we investigate the stability of a special three-dimensional anisotropic magnetohydrodynamic (MHD) system. This system features dissipation only in the direction of <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>1</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$x_{1}$</EquationSource> </InlineEquation> and magnetic damping near a background magnetic field. Due to the absence of dissipation in the <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>2</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$x_{2}$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>3</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$x_{3}$</EquationSource> </InlineEquation> directions, establishing the stability and long-time behavior of this MHD system is highly nontrivial. Through a subtle energy estimate and careful analysis of the nonlinearities, we rigorously justify the stability of this MHD system near a background magnetic field and derive explicit decay rates. The primary challenge lies in proving the uniform integrability of <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">∥</mo> <msub> <mi mathvariant="normal">∇</mi> <mi>h</mi> </msub> <mi>u</mi> <mo stretchy="false">∥</mo> </mrow> <msup> <mi>L</mi> <mi mathvariant="normal">∞</mi> </msup> </msub> </math></EquationSource> <EquationSource Format="TEX">$\|\nabla _{h}u\|_{L^{\infty }}$</EquationSource> </InlineEquation> in the time variable.</p>

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Global Existence and Decay Estimates of the Three Dimensional Magneto-Hydrodynamic Equations with Partial Dissipation

  • 期刊论文

  • 发表日期：

    2026年1月19日

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  • Fangfang Jian,
  • Dongxiang Chen,
  • Xiaoli Chen

摘要

To uncover the mechanism by which magnetic fields can stabilize electrically conducting turbulent fluids, we investigate the stability of a special three-dimensional anisotropic magnetohydrodynamic (MHD) system. This system features dissipation only in the direction of x 1 $x_{1}$ and magnetic damping near a background magnetic field. Due to the absence of dissipation in the x 2 $x_{2}$ and x 3 $x_{3}$ directions, establishing the stability and long-time behavior of this MHD system is highly nontrivial. Through a subtle energy estimate and careful analysis of the nonlinearities, we rigorously justify the stability of this MHD system near a background magnetic field and derive explicit decay rates. The primary challenge lies in proving the uniform integrability of ∥ ∇ h u ∥ L ∞ $\|\nabla _{h}u\|_{L^{\infty }}$ in the time variable.