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<p>For the three-dimensional (3D) Navier-Stokes equations with unidirectional dissipation, the stability remains an open problem. Inspired by the intriguing experiments and numerical simulations demonstrating the stabilizing effects of background magnetic fields and temperature, this paper aims to investigate the stability and large-time behavior of perturbations near a background magnetic field and hydrostatic balance. Our analysis is done in the spatial domains <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2107_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega =\mathbb {T}\times \mathbb {R}^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mi mathvariant="double-struck">T</mi> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2107_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {T}=[-{1/2},{1/2}]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo>-</mo> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> <mo>,</mo> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> denotes a one-dimensional periodic torus. In this 3D magnetohydrodynamic Bénard system, the velocity and temperature diffuse only along the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2107_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(x_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-direction, while the magnetic field exhibits full dissipation.</p>

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The large time behavior of the three-dimensional incompressible magnetic Bénard fluid

  • 期刊论文

  • 发表日期：

    2025年8月16日

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  • Xiaokui Zhao,
  • Hang Song

摘要

For the three-dimensional (3D) Navier-Stokes equations with unidirectional dissipation, the stability remains an open problem. Inspired by the intriguing experiments and numerical simulations demonstrating the stabilizing effects of background magnetic fields and temperature, this paper aims to investigate the stability and large-time behavior of perturbations near a background magnetic field and hydrostatic balance. Our analysis is done in the spatial domains \(\Omega =\mathbb {T}\times \mathbb {R}^{2}\) Ω = T × R 2 , where \(\mathbb {T}=[-{1/2},{1/2}]\) T = [ - 1 / 2 , 1 / 2 ] denotes a one-dimensional periodic torus. In this 3D magnetohydrodynamic Bénard system, the velocity and temperature diffuse only along the \(x_1\) x 1 -direction, while the magnetic field exhibits full dissipation.