Point Singularities of Solutions to the Stationary Incompressible MHD Equations
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908317204348928 |
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| author | Zhang, Shaoheng Wang, Kui Wang, Yun |
| author_facet | Zhang, Shaoheng Wang, Kui Wang, Yun |
| contents | We investigate the point singularity of very weak solutions $(\mathbf{u},\mathbf{B})$ to the stationary MHD equations. More precisely, assume that the solution $(\mathbf{u},\mathbf{B})$ in the punctured ball $B_2\setminus \{0\}$ satisfies the vanishing condition (4), and that $|\mathbf{u}(x)|\le \varepsilon |x|^{-1},\ |\mathbf{B}(x)|\le C |x|^{-1}$ with small $\varepsilon>0$ and general $C>0$. Then, the leading order term of $\mathbf{u}$ is a Landau solution, while the $(-1)$ order term of $\mathbf{B}$ is $0$. In particular, for axisymmetric solutions $(\mathbf{u}, \mathbf{B})$, the condition (4) holds provided $\mathbf{B} = B^θ(r,z) \mathbf{e}_θ$ or the boundary condition (7) is imposed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_09607 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Point Singularities of Solutions to the Stationary Incompressible MHD Equations Zhang, Shaoheng Wang, Kui Wang, Yun Analysis of PDEs We investigate the point singularity of very weak solutions $(\mathbf{u},\mathbf{B})$ to the stationary MHD equations. More precisely, assume that the solution $(\mathbf{u},\mathbf{B})$ in the punctured ball $B_2\setminus \{0\}$ satisfies the vanishing condition (4), and that $|\mathbf{u}(x)|\le \varepsilon |x|^{-1},\ |\mathbf{B}(x)|\le C |x|^{-1}$ with small $\varepsilon>0$ and general $C>0$. Then, the leading order term of $\mathbf{u}$ is a Landau solution, while the $(-1)$ order term of $\mathbf{B}$ is $0$. In particular, for axisymmetric solutions $(\mathbf{u}, \mathbf{B})$, the condition (4) holds provided $\mathbf{B} = B^θ(r,z) \mathbf{e}_θ$ or the boundary condition (7) is imposed. |
| title | Point Singularities of Solutions to the Stationary Incompressible MHD Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.09607 |