Point Singularities of Solutions to the Stationary Incompressible MHD Equations
Shaoheng Zhang, Kui Wang, Yun Wang
TL;DR
This work analyzes point singularities of stationary incompressible MHD solutions in a punctured ball, showing that under small velocity data and $|x|^{-1}$-type bounds for both fields, the velocity's leading behavior is a Landau solution while the magnetic field's $(-1)$-order term vanishes. The authors develop a framework in Lorentz spaces, leverage Landau solution theory for NS, and use elliptic regularity to obtain sharp $W^{1,q}$ and pointwise bounds near the origin, with the flux defining the Landau datum $b$. They provide axisymmetric corollaries and a Liouville-type rigidity result, demonstrating how symmetry and boundary conditions enforce the vanishing of the magnetic singular part. The results clarify the local structure of MHD near singularities and extend Landau-type descriptions to the coupled MHD system, with potential implications for singularity formation and regularity theory in magnetohydrodynamics.
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 $\mathbf{B} = B^θ(r,z) \mathbf{e}_θ$ or the boundary condition (7) is imposed.