Decay properties of Axially Symmetric D-solutions to the Steady Incompressible Magnetohydrodynamic Equations

TL;DR

This paper studies how axially symmetric solutions to the steady incompressible magnetohydrodynamic equations decay at infinity, providing decay rates under various conditions and employing advanced mathematical tools.

Contribution

It derives decay rates for D-solutions to axisymmetric MHD equations, including special cases with swirl magnetic fields, using a combination of scaling, inequalities, and energy estimates.

Findings

Decay rates for general D-solutions established.

Improved decay rates obtained for solutions with swirl magnetic fields.

Decay behavior along the axis Oz analyzed for specific D-solutions.

Abstract

In this paper, we investigate the decay properties of axially symmetric solutions to the steady incompressible magnetohydrodynamic equations in $\mathbb{R}^3$ with finite Dirichlet integral. We first derive the decay rates of general D-solutions to the axisymmetric MHD equations. In the special case where the magnetic field only has the swirl component ${\textbf {h}}(r,z)= h_{\theta}(r,z) {\mathbf e_{\theta}}$, we obtain better decay rates. The last result examines the decay rates along the axis $Oz$ also within the special class of D-solutions with only swirl magnetic field. The main tool in this paper is the combination of the scaling argument, the \textit{Brezis-Gallouet inequality} and the weighted energy estimate.