Rigorous derivation and well-posedness of a quasi-homogeneous ideal MHD system

TL;DR

This paper introduces a quasi-homogeneous ideal MHD system, establishes its well-posedness in critical Besov spaces, and rigorously derives it from non-homogeneous MHD models under specific physical regimes.

Contribution

It provides a new quasi-homogeneous MHD model, proves its well-posedness in critical spaces, and offers a rigorous derivation from physical models with precise convergence rates.

Findings

Well-posedness in critical Besov spaces for the quasi-homogeneous MHD system.

Rigorous derivation of the quasi-homogeneous MHD models from non-homogeneous systems.

Quantitative convergence rates depending on initial data, Rossby number, and viscosity/resistivity.

Abstract

The goal of this paper is twofold. On the one hand, we introduce a quasi-homogeneous version of the classical ideal MHD system and study its well-posedness in critical Besov spaces $B^s_{p,r}(\mathbb{R}^d)$, $d\geq2$, with $1<p<+\infty$ and under the Lipschitz condition $s>1+d/p$ and $r\in[1,+\infty]$, or $s=1+d/p$ and $r=1$. A key ingredient is the reformulation of the system \textsl{via} the so-called Els\"asser variables. On the other hand, we give a rigorous justification of quasi-homogeneous MHD models, both in the ideal and in the dissipative cases: when $d=2$, we will derive them from a non-homogeneous incompressible MHD system with Coriolis force, in the regime of low Rossby number and for small density variations around a constant state. Our method of proof relies on a relative entropy inequality for the primitive system, and yields precise rates of convergence, depending on the size of the initial data, on the order of the Rossby number and on the regularity of the viscosity and resistivity coefficients.