Rigorous derivation and well-posedness of a quasi-homogeneous ideal MHD system. (August 2021)

• Record Type: Journal Article
• Authors: Cobb, Dimitri; Fanelli, Francesco
• Is Part Of: Nonlinear analysis. Volume 60(2021)
• Publication Date: 2021-08
• Subjects: Quasi-homogeneous ideal MHD -- Elsässer variables -- Critical regularity -- Singular perturbation -- Low Rossby number -- Relative entropy inequality; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2020.103284
• Languages: English
• ISSNs: 1468-1218

Abstract

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 ( R d ), d ≥ 2, with 1 < p < + ∞ and under the Lipschitz condition s > 1 + d ∕ p and r ∈ [ 1, + ∞ ], 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.