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 $B_{p,r}^s({\mathbb{R}}^d)$, $d\ge 2$, 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 \textsl{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.

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References in corpus (4)

• Relative entropies, suitable weak solutions, and weak strong uniqueness for the compressible Navier-Stokes system
• Asymptotics of fast rotating density-dependent incompressible fluids in two space dimensions
• On the fast rotation asymptotics of a non-homogeneous incompressible MHD system
• Commutator estimates in Besov-Morrey spaces with applications to the well-posedness of the Euler equations and ideal MHD system

Cited by in corpus (1)

• Weak Solutions for a non-Newtonian Stokes-Transport System