On Moffatt's magnetic relaxation equations

TL;DR

This paper studies the stability and long-term behavior of Moffatt's magnetic relaxation equations, highlighting their energy structure and differences in solution behavior over time.

Contribution

It provides new results on the well-posedness and stability of Moffatt's models, emphasizing their topological preservation and energy properties.

Findings

Established local and global well-posedness.

Identified differences in asymptotic behavior in weak and strong norms.

Analyzed the stability properties of the equations.

Abstract

We investigate the stability properties for a family of equations introduced by Moffatt to model magnetic relaxation. These models preserve the topology of magnetic streamlines, contain a cubic nonlinearity, and yet have a favorable $L^2$ energy structure. We consider the local and global in time well-posedness of these models and establish a difference between the behavior as $t\to \infty$ with respect to weak and strong norms.