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<p>In this paper, we study the local well-posedness of classical solutions to the ideal Hall–MHD equations whose magnetic field is supposed to be azimuthal in the <i>L</i><sup>2</sup>-based Sobolev spaces. By introducing a good unknown coupling with the original unknowns, we overcome difficulties arising from the lack of magnetic resistance, and establish a self-closed <i>H</i><sup><i>m</i></sup> with (3 ≤ <i>m</i> ∈ ℕ) local energy estimate of the system. Here, a key cancellation related to <i>θ</i> derivatives is discovered. In order to apply this cancellation, part of the high-order energy estimates is performed in the cylindrical coordinate system, even though our solution is not assumed to be axially symmetric. During the proof, high-order derivative tensors of unknowns in the cylindrical coordinates system are carefully calculated, which would be useful in further researches on related topics.</p>

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On Local Well-Posedness of 3D Ideal Hall–MHD System with an Azimuthal Magnetic Field

  • 期刊论文

  • 发表日期：

    2025年12月15日

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  • Zijin Li

摘要

In this paper, we study the local well-posedness of classical solutions to the ideal Hall–MHD equations whose magnetic field is supposed to be azimuthal in the L2-based Sobolev spaces. By introducing a good unknown coupling with the original unknowns, we overcome difficulties arising from the lack of magnetic resistance, and establish a self-closed Hm with (3 ≤ m ∈ ℕ) local energy estimate of the system. Here, a key cancellation related to θ derivatives is discovered. In order to apply this cancellation, part of the high-order energy estimates is performed in the cylindrical coordinate system, even though our solution is not assumed to be axially symmetric. During the proof, high-order derivative tensors of unknowns in the cylindrical coordinates system are carefully calculated, which would be useful in further researches on related topics.