1D Model for the 3D Magnetohydrodynamics

TL;DR

This paper introduces a 1D model for 3D incompressible ideal magnetohydrodynamics, establishing a regularity criterion and demonstrating global solutions without singularities through analysis and numerical simulations.

Contribution

The paper develops a novel 1D model for 3D MHD and proves a Beale-Kato-Majda type regularity criterion, showing global solutions under certain conditions.

Findings

The 1D model admits global strong solutions without singularities.

Numerical simulations support the absence of finite-time blow-up.

A regularity criterion analogous to Beale-Kato-Majda is established.

Abstract

We propose a one-dimensional (1D) model for the three-dimensional(3D) incompressible ideal magnetohydrodynamics. We establish a regularity criterion of the Beale-Kato-Majda type for this 1D model. Without the stretching effect, the model with only transport effect equipped with a proper sign is shown to have global in time strong solution. Some numerical simulations suggest that solutions of this model with smooth periodic initial data do not tend to develop singularities at finite time.