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

2021/02/25 by Shangkun Weng, Yan Zhou, Weng, Shangkun +1
Mathematics · #35Q35(Primary) #76W05 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:76W05

paper · pdf · doi:10.48550/arxiv.2102.12693

arxiv created 2022/03/22 · arxiv updated 2022/03/23

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 \textbf h(r,z)= h_θ(r,z) \mathbf e_θ, 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.

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