On axisymmetric self-similar solutions to the MHD system

Let $(\mathbf{u},\mathbf{B})$ be an axisymmetric self-similar solution to the stationary MHD equations with magnetic diffusion, of the form $\mathbf{u}=u^r(r,z)\mathbf{e}_{r}+u^{\theta}(r,z)\mathbf{e}_{\theta}+u^z(r,z)\mathbf{e}_{z}$ and $\mathbf{B}=B^{\theta}(r,z)\mathbf{e}_{\theta}$ in cylindrical coordinates $(r,\theta,z)$, where $(\mathbf{e}_r,\mathbf{e}_\theta,\mathbf{e}_z)$ is the orthonormal basis. Under the assumption that $u^r < \frac{1}{3r} + \frac{2r}{3}$ on the unit sphere and on its intersection with the half-space, respectively, we prove two main results. First, for the domain $\mathbb{R}^3\setminus\{0\}$, the velocity field $\mathbf{u}$ must be a Landau solution and the magnetic field $\mathbf{B} \equiv 0$. Second, in the half-space $\mathbb{R}^3_+$ with either the no-slip or Navier slip boundary condition, we establish that all such axisymmetric self-similar solutions are trivial, i.\,e., $\mathbf{u}=\mathbf{B}=0$.