On compressible magnetic relaxation in planar symmetry

TL;DR

This paper studies the behavior of compressible magnetic relaxation equations under planar symmetry, establishing well-posedness, relaxation, and stability results without vacuum formation or implosions.

Contribution

It introduces a simplified model derived from compressible MHD with Darcy friction and proves key properties under planar symmetry.

Findings

Local well-posedness for smooth initial data

Magnetic relaxation for perturbations of steady states

No vacuum states or implosions before singularity

Abstract

We consider the compressible Magnetic Relaxation Equations on the three-dimensional torus $\mathbb{T}^{3}$. The system is derived from compressible magnetohydrodynamics (MHD) by replacing the acceleration term with a Darcy-type friction. Under planar symmetry, we establish three main results: (1) local well-posedness for smooth initial data, (2) magnetic relaxation for smooth perturbations of constant steady states, and (3) the absence of vacuum states or implosions prior to and at the time of a potential singularity.