Axisymmetric self-similar solutions to the MHD equations without magnetic diffusion

TL;DR

This paper classifies axisymmetric self-similar solutions to the stationary MHD equations without magnetic diffusion, showing that solutions are trivial or reduce to known solutions under certain conditions.

Contribution

It provides a complete classification of axisymmetric self-similar solutions to the stationary MHD equations without magnetic diffusion, including triviality results in half-space domains.

Findings

In b3^3b7, b1u is a Landau solution and b1B=0.

Solutions are trivial in the half-space with no-slip or Navier slip boundary conditions.

Classifies solutions and proves triviality under specific boundary conditions.

Abstract

We study the axisymmetric self-similar solutions $(\mathbf{u},\mathbf{B})$ to the stationary MHD equations without magnetic diffusion, where $\mathbf{B}$ has only the swirl component. Our first result states that in $\mathbb{R}^3\setminus\{0\}$, $\mathbf{u}$ is a Landau solution and $\mathbf{B}=0$. Our second result proves the triviality of axisymmetric self-similar solutions in the half-space $\mathbb{R}^3_+$ with the no-slip boundary condition or the Navier slip boundary condition.