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., $\mathbf{u}=\mathbf{B}=0$.