Existence of axially symmetric weak solutions to steady MHD with non-homogeneous boundary conditions

TL;DR

This paper proves the existence of axially symmetric weak solutions for steady incompressible magnetohydrodynamics with non-homogeneous boundary conditions, focusing on a special class where the magnetic field has only a swirl component.

Contribution

It establishes the existence of solutions under specific symmetry and boundary conditions, advancing understanding of MHD systems with non-standard boundary data.

Findings

Existence of solutions for a class of steady MHD systems.

Application of Bernoulli's law to the total head pressure.

Analysis of solutions with magnetic fields containing only the swirl component.

Abstract

We establish the existence of axially symmetric weak solutions to steady incompressible magnetohydrodynamics with non-homogeneous boundary conditions. The key issue is the Bernoulli's law for the total head pressure $\Phi=\f 12(|{\bf u}|^2+|{\bf h}|^2)+p$ to a special class of solutions to the inviscid, non-resistive MHD system, where the magnetic field only contains the swirl component.