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 $B_{p,r}^s\left({\mathbb{R}}^d\right)$, $d\ge 2$, 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 via the so-called Elsässer 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.

本文的目标是双重的。一方面，我们介绍了经典理想MHD系统的准同质版本，并研究了其在临界Besov空间中的适定性${乙}_{p，\mathrm{[R}}^s（{\mathbb{[R}}^d）$， $d\ge 2$，带有 $\mathrm{1个}<p<+\infty$ 并在Lipschitz条件下 $s>\mathrm{1个}+d∕p$ 和 $\mathrm{[R}\in [\mathrm{1个}，+\infty ]$， 要么 $s=\mathrm{1个}+d∕p$ 和 $\mathrm{[R}=\mathrm{1个}$。一个关键因素是通过所谓的Elsässer变量重新构造系统。另一方面，在理想情况和耗散情况下，我们给出准齐整的MHD模型的严格理由：$d=2$，我们将从具有Coriolis力的非均匀不可压缩MHD系统派生出来，该系统具有低Rossby数和恒定状态下较小的密度变化。我们的证明方法依赖于原始系统的相对熵不等式，并根据原始数据的大小，Rossby数的数量级以及粘度和电阻率系数的规律性得出精确的收敛速度。