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arXiv · 2402.02451

On local well-posedness of 3D ideal Hall-MHD system with an azimuthal magnetic field

Zijin Li
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Abstract

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 $L^2$-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 $H^m$ with $(3\leq m\in\mathbb{N})$ 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.

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Zijin Li. 2025-08-09. On local well-posedness of 3D ideal Hall-MHD system with an azimuthal magnetic field. https://arxiv.org/abs/2402.02451

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