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<p>In this paper, we investigate the well-posedness result of the three-dimensional incompressible hyper-dissipative Hall-Magnetohydrodynamic equations with small anisotrop- ic derivative. Making using of anisotropic Littlewood-Paley theory, we conclude that the hyper-dissipative Hall-MHD system has a unique global solution provided that<Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_501_Article_Equ1.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="358" /> </MediaObject> <EquationSource Format="TEX">\(\left(||J_0||_{\mathcal{B_2^1}^{-2\alpha}}+||(\Lambda{_h^-1}\partial_3u_0,B_0^h)||_{\mathcal{B_2^1}-2\alpha}\right)\cdot{F}(u_0,B_0)\)</EquationSource> </Equation>is sufficiently small. Here, <i>F</i>(<i>u</i><sub>0</sub>, <i>B</i><sub>0</sub>) is a bounded function, which depends on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_501_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="119" /> </InlineMediaObject> <EquationSource Format="TEX">\(||(u_0,B_0)||_{\mathcal{B_2^1}-2\alpha}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_501_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(||u{_0^h}||_{H^1}\)</EquationSource> </InlineEquation>.</p>

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Global well-posedness of 3D incompressible hyper-dissipative Hall-MHD equations in anisotropic Besov spaces

  • 期刊论文

  • 发表日期：

    2025年6月20日

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  • Dezai Min,
  • Qingkai Wang,
  • Gang Wu,
  • Zhuoya Yao

摘要

In this paper, we investigate the well-posedness result of the three-dimensional incompressible hyper-dissipative Hall-Magnetohydrodynamic equations with small anisotrop- ic derivative. Making using of anisotropic Littlewood-Paley theory, we conclude that the hyper-dissipative Hall-MHD system has a unique global solution provided that \(\left(||J_0||_{\mathcal{B_2^1}^{-2\alpha}}+||(\Lambda{_h^-1}\partial_3u_0,B_0^h)||_{\mathcal{B_2^1}-2\alpha}\right)\cdot{F}(u_0,B_0)\) is sufficiently small. Here, F(u0, B0) is a bounded function, which depends on \(||(u_0,B_0)||_{\mathcal{B_2^1}-2\alpha}\) and \(||u{_0^h}||_{H^1}\) .