Axisymmetric self-similar solutions to the MHD equations without magnetic diffusion
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916810713989120 |
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| author | Zhang, Shaoheng |
| author_facet | Zhang, Shaoheng |
| contents | We study the axisymmetric self-similar solutions $(\mathbf{u},\mathbf{B})$ to the stationary MHD equations without magnetic diffusion, where $\mathbf{B}$ has only the swirl component. Our first result states that in $\mathbb{R}^3\setminus\{0\}$, $\mathbf{u}$ is a Landau solution and $\mathbf{B}=0$. Our second result proves the triviality of axisymmetric self-similar solutions in the half-space $\mathbb{R}^3_+$ with the no-slip boundary condition or the Navier slip boundary condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_20131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Axisymmetric self-similar solutions to the MHD equations without magnetic diffusion Zhang, Shaoheng Analysis of PDEs We study the axisymmetric self-similar solutions $(\mathbf{u},\mathbf{B})$ to the stationary MHD equations without magnetic diffusion, where $\mathbf{B}$ has only the swirl component. Our first result states that in $\mathbb{R}^3\setminus\{0\}$, $\mathbf{u}$ is a Landau solution and $\mathbf{B}=0$. Our second result proves the triviality of axisymmetric self-similar solutions in the half-space $\mathbb{R}^3_+$ with the no-slip boundary condition or the Navier slip boundary condition. |
| title | Axisymmetric self-similar solutions to the MHD equations without magnetic diffusion |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.20131 |