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arXiv · 2504.09607

Point Singularities of Solutions to the Stationary Incompressible MHD Equations

Shaoheng Zhang·Kui Wang·Yun Wang
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Abstract

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 $\mathbf{B} = B^{\theta}(r,z) \mathbf{e}_{\theta}$ or the boundary condition (7) is imposed.

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Shaoheng Zhang, Kui Wang, Yun Wang. 2025-04-13. Point Singularities of Solutions to the Stationary Incompressible MHD Equations. https://arxiv.org/abs/2504.09607

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