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<p>We consider the incompressible Hall-MHD system on the 3D whole space. Then, it is known that the perturbed system around the constant equilibrium state exhibits a dispersive structure. However, this dispersion is so complicated that the results on the effect of dispersion are known only for the special case <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\nu =\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>=</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation>, where the dispersive relation becomes simpler. Here <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\nu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> are viscosity and resistive coefficients respectively. The purpose of this paper is to improve the previous results and investigate the dispersive effect for the general case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\nu \ne \mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>≠</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> without complicated calculations. Consequently, we may obtain the global well-posedness and time-periodic solvability for <i>large</i> data in critical Besov spaces, provided that the size of the constant magnetic field is sufficiently large.</p>

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Dispersive Phenomena on the Hall-MHD System Around the Constant Equilibrium State in the General Dissipative Coefficients Case

  • 期刊论文

  • 发表日期：

    2026年1月13日

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  • Mikihiro Fujii,
  • Shunhang Zhang

摘要

We consider the incompressible Hall-MHD system on the 3D whole space. Then, it is known that the perturbed system around the constant equilibrium state exhibits a dispersive structure. However, this dispersion is so complicated that the results on the effect of dispersion are known only for the special case \(\nu =\mu \) ν = μ , where the dispersive relation becomes simpler. Here \(\nu \) ν and \(\mu \) μ are viscosity and resistive coefficients respectively. The purpose of this paper is to improve the previous results and investigate the dispersive effect for the general case \(\nu \ne \mu \) ν ≠ μ without complicated calculations. Consequently, we may obtain the global well-posedness and time-periodic solvability for large data in critical Besov spaces, provided that the size of the constant magnetic field is sufficiently large.