A single-component regularity criterion and Inviscid limit of axially symmetric MHD-Boussinesq system
Zhaojun Xing
TL;DR
This work analyzes the 3D axially symmetric MHD-Boussinesq system, proving a sharp Beale-Kato-Majda-type blow-up criterion that depends solely on the horizontal swirl component, and establishing global regularity in $H^m$ for $m\ge3$ under this one-component condition. It introduces a reformulated axisymmetric framework with swirl-related quantities $(\Omega, J, \mathcal{N}, \nabla\mathcal{H})$ to derive robust a priori estimates and a closed, one-component criterion that governs the solution’s continuation. In addition, the paper rigorously justifies the inviscid limit by comparing viscous and inviscid solutions and provides a quantitative convergence rate in $L^2$ norms, revealing how the viscosity parameter $\mu$ influences the dynamics as $\mu\to0$. Overall, it extends Beale-Kato-Majda-type criteria to a single-component setting in MHD-Boussinesq flows and supplies a concrete vanishing-viscosity rate for axisymmetric convection with magnetic effects, with potential implications for stability and blow-up analysis in related fluid models.
Abstract
In this paper, we first give a critical BKM-type blow-up criterion that only involves the horizontal swirl component of the velocity for the inviscid axially symmetric MHD-Boussinesq system. Moreover, we consider the inviscid limit of the viscous MHD-Boussinesq system, and the convergence rate for the viscosity coefficient tending to zero is obtained.