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

The goal of this paper is twofold. On the one hand, we introduce a quasi-homogeneous version classical ideal MHD system and study its well-posedness in critical Besov spaces Bp,rs(Rd), d≥2, with 11+d∕p r∈[1,+∞], or s=1+d∕p r=1. A key ingredient reformulation via so-called Elsässer variables. other give rigorous justification models, both dissipative cases: when d=2, will derive them from non-homogeneous incompressible Coriolis force, regime low Rossby number for small density variations around constant state. Our method proof relies on relative entropy inequality primitive system, yields precise rates convergence, depending size initial data, order regularity viscosity resistivity coefficients.