On local well-posedness of 3D ideal Hall-MHD system with an azimuthal magnetic field
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909729734787072 |
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| author | Li, Zijin |
| author_facet | Li, Zijin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02451 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On local well-posedness of 3D ideal Hall-MHD system with an azimuthal magnetic field Li, Zijin Analysis of PDEs 35Q35, 76D05 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. |
| title | On local well-posedness of 3D ideal Hall-MHD system with an azimuthal magnetic field |
| topic | Analysis of PDEs 35Q35, 76D05 |
| url | https://arxiv.org/abs/2402.02451 |