On axisymmetric self-similar solutions to the MHD system

TL;DR

This paper classifies axisymmetric self-similar solutions to the stationary MHD equations, showing that under certain conditions, solutions are either trivial or reduce to known Landau solutions with zero magnetic field.

Contribution

The paper proves that all axisymmetric self-similar solutions under specified conditions are either Landau solutions with zero magnetic field or trivial in the half-space.

Findings

Solutions in ^3\setminus\{0\} are Landau solutions with B=0.

Solutions in the half-space with boundary conditions are trivial, i.e., zero velocity and magnetic field.

Abstract

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