Point Singularities of Solutions to the Stationary Incompressible MHD Equations

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 $\mathrm{\mid }\mathbf{u}(x)\mathrm{\mid }\le \epsilon \mathrm{\mid }x{\mathrm{\mid }}^{-1},\mathrm{\mid }\mathbf{B}(x)\mathrm{\mid }\le C\mathrm{\mid }x{\mathrm{\mid }}^{-1}$ with small $\epsilon >0$ 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.