Decay properties of axially symmetric D-solutions to the steady incompressible magnetohydrodynamic equations

Abstract

In this paper, we investigate the decay properties of axially symmetric solutions to the steady incompressible magnetohydrodynamic equations in R3 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 h(r,z)=hθ(r,z)eθ, 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 Brezis–Gallouet inequality and the weighted energy estimate.