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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.13790 |
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| _version_ | 1866909177390039040 |
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| author | Jeong, In-Jee Oh, Sung-Jin |
| author_facet | Jeong, In-Jee Oh, Sung-Jin |
| contents | It has been shown in our previous work that the incompressible and irresistive Hall- and electron-magnetohydrodynamic (MHD) equations are illposed on flat domains $M = \mathbb{R}^k \times \mathbb{T}^{3-k}$ for $0 \le k \le 2$. The data and solutions therein were assumed to be independent of one coordinate, which not only significantly simplifies the systems but also allows for a large class of steady states. In this work, we remove the assumption of independence and conclude strong illposedness for compactly supported data in $\mathbb{R}^3$. This is achieved by constructing degenerating wave packets for linearized systems around time-dependent axisymmetric magnetic fields. A few main additional ingredients are: a more systematic application of the generalized energy estimate, use of the Bogovskiǐ operator, and a priori estimates for axisymmetric solutions to the Hall- and electron-MHD systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13790 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On illposedness of the Hall and electron magnetohydrodynamic equations without resistivity on the whole space Jeong, In-Jee Oh, Sung-Jin Analysis of PDEs It has been shown in our previous work that the incompressible and irresistive Hall- and electron-magnetohydrodynamic (MHD) equations are illposed on flat domains $M = \mathbb{R}^k \times \mathbb{T}^{3-k}$ for $0 \le k \le 2$. The data and solutions therein were assumed to be independent of one coordinate, which not only significantly simplifies the systems but also allows for a large class of steady states. In this work, we remove the assumption of independence and conclude strong illposedness for compactly supported data in $\mathbb{R}^3$. This is achieved by constructing degenerating wave packets for linearized systems around time-dependent axisymmetric magnetic fields. A few main additional ingredients are: a more systematic application of the generalized energy estimate, use of the Bogovskiǐ operator, and a priori estimates for axisymmetric solutions to the Hall- and electron-MHD systems. |
| title | On illposedness of the Hall and electron magnetohydrodynamic equations without resistivity on the whole space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.13790 |