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

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 $1 1+d/p$ 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.